Projective and Cayley-Klein Geometries (Springer Monographs in Mathematics)

A

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P r o o f . T h e transitivity immediately follows from the transitivity of the action of GL{n + l,K) on the set of all bases. By Proposition 1 we have {al)(f = (p \i and only if there exists a yU, G Z{K*) with the property (6). This obviously holds for aU ^5 G ^ , if it is true for any single one. • D e f i n i t i o n 1. Let i f be a skew field, and denote by Z{n + 1) the subgroup Z{n+l):={{5i)ii\iieZ{K*),i,j=Q,...,n}.

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T h e quotient group PGL{n is called the projective following exercise.

+ 1, K) := GL{n

+ 1, K)/Z{n

+ 1),

linear group of order n + 1 over K. Note also the

Exercise 2. By Lemma 5 the center Z{n + 1) is the kernel of the action under consideration (cf. Definition 1.1.4.2) and hence a normal subgroup. The action of PGL(n + 1) on $ defined via representatives is thus simply transitive, cf. § 1.1.4. Prove that Z(n + 1) is the center of the group GL(n + 1, K) (cf. Example 1.1.4.7). This notion is not to be confused with that of the order of a finite group.

1.2 Homogeneous Coordinates

21

1.2.3 Inhomogeneous Projective Coordinates In this section we will consider the inhomogeneous coordinates associated with a homogeneous coordinate system (p{x) =

{x\x))K*

and calculate their transformation rule. The i-th coordinate hyperplane Bi is defined as the coordinate hyperplane complementary to the base point Oj. Let Ui = P " \ Bi be its complement in P " , cf. Example 1: Ui =

{xeP'''\x'{x)^0}.

Obviously, the i-th chart cpi on Ui, is correctly defined: xeUi^


y^i,j=0,...,n)eK".

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For brevity we set ef (a;) := x^{x){x\x))-^

for x e Ui

and prove the following. Lemma 6. a) Each homogeneous coordinate system (p on the projective space P " determines n + 1 charts, cpi : Ui ^ K", defined by (8), that are bijective maps. b) The domains of definition of these charts cover the projective space, i.e. c) Let, as above, 'ip denote the homogeneous coordinate system associated with the basis (oj), let (a^) be the matrix of the coordinate transformation, and let ''Pi = (Ci) be the charts 'ipi : Vi ^ K" determined by 'ip. Then the coordinate transformations

V-b o ^ - 1 : if^iUa n Vb) —> MUa n 14) defined on the intersections of coordinate neighborhoods, UaC\Vi,, are fractional linear transformations; moreover, it{^)

= a';il{x){a\Ca{x))-'.

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P r o o f . Note that in this formula, according to the sum convention, the sum in the numerator runs over j and that in the denominator over i. Moreover, Q = 1, ^. For arbitrary (y^,...,y") G if" the point x with homogeneous coordinates x°{x) = y\ x\x) = y ^ . . . , x'-\x) = y\ x\x) = 1, x'+\x) = y'+\ . .., x"(a;) = y" is the only point in Ui mapped to (y^,..., y"). This proves a). Since for every point X G P " at least one among its homogeneous coordinates does not vanish, b) holds. The remaining statement c) immediately follows from the transformation rule (5) and the definition of inhomogeneous coordinates. •

22

1 Projective Geometry

E x a m p l e 2. T h e P r o j e c t i v e L i n e . For n = 1 there are two inhomogeneous coordinates essentially coinciding with the corresponding projective scales. For example, the value ^{ai) is not yet defined by ^ := Co- Setting, as seems obvious, ^{ai) := oo yields the projective scale defined in Example 1.3. In this case, formula (9) describes the transformation of projective scales on P ^ , 6A'- K ^ K, entailed by a change of bases in V"^: OA--i^i--={ai

+ h){ci + d)-^ with A = ( ^ J J eGL{2,K).

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To complete t h e picture, we extend 9A in the following way:

^^(_c-id) = oo, ^A(OO) = ('*'^"'' IJ'^^J'' [ oo,

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it c = U. D

Exercise 3 . In the notation of Lemma 6, let (p, ip be homogeneous coordinate systems on P" satisfying the additional condition Uo = VQ. Prove that the coordinate transformation i/>o o ipg-^ is an affine transformation in K", cf. I, (5.7.36). Exercise 4 . Prove that the map of K to itself defined by (10), (11) is a bijection and satisfies the equality (OA)^^ = ^ A - I - (Note that, in general, A" is a skew field and hence determinants cannot be used.) Denote by S{K) the group of all bijective maps on K. Prove that the map F : Ae

GL{2, K) ^ 9A e S{K)

is a homomorphism whose image is isomorphic to PGL(2,K). Show that for K = R, C , H every 9A is a homeomorphism with respect to the topology on K described in Example 1.3. Exercise 5. A projective plane P is said to have Pappos' Property if Pappos' Theorem holds in it, cf. Exercise 1.10, i.e., if the points ci, C2, cs in a configuration as depicted in Figure 1.6 are always collinear. Prove that P^ has the Pappos' Property if and only if the skew field on which it is based is commutative, i.e., it is a field. — Hint. Choose homogeneous coordinates in a way that (z, as, 63; C2) are the base points and the unit point of the coordinate system, respectively, and express then the condition in these coordinates. 1.2.4 T h e P r o j e c t i v e Linear G r o u p over a F i e l d E x a m p l e 3. Let K he a field. T h e n we have Z{K*) = K* = Z{n + 1). Consider t h e canonical homomorphism resulting from Definition 1, p : ( 4 ) G GL{n

+ 1, if) ^ ^ (a})if * G PGL{n

+ 1, K).

The matrices from GL{n + 1, K) contained in the same coset of the factor group PGL{n+l, K) = GL{n+l, K)/Z{n+l) only differ in a factor fie K*.

1.2 Homogeneous Coordinates

23

Now choose a "fixed" projective frame R in the projective space P " over K and, in addition, a variable "moving" projective frame R; let (x*) and (x*) be the homogeneous coordinates associated with R and R, respectively, defined by the bases (Oj) and (Oj) of V. According to Proposition 1, these bases are determined by the homogeneous coordinates only up to a common factor /i G K*. Hence the cosets of GL{n + 1, K) by Z{n + 1), i.e. the elements of PGL{n + l,K), correspond bijectively to the projective frames. Thus, the set of these frames is a geometric reahzation of the group PGL{n + l,K), just hke the set of linear frames, (Oj), is a geometric realization of the group

GL{n + l,K). Now we will try to find a distinguished representative in each coset of GL{n + Ijif) by Z{n + 1) in a natural way. To this aim, we consider the determinant as a function of yU. Looking at the well-known formula det(A/i) = det(A)/i"+^, the obvious condition to impose on a distinguished representative should be Aet{An) = 1 for A = (a|) G GL{n + 1, K) and n e K*

(12)

Hence the intended choice amounts to deciding the solvability of M"+I = (det(A))-i, and, apart from n, this again essentially depends on the algebraic properties of the field K. Let now if = R. If n is even, for given A the equation (12) has exactly one real solution /x. Hence, each coset contains a unique representative with determinant one, and we arrive at the isomorphism PGL{n + 1, R) = SL{n + 1, R) (n even).

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Hence, in this case, the projective linear group is isomorphic to the special linear group, SL{n + 1, R) := {A G GL{n + 1, R) | det(A) = 1}.

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The situation is different in the odd-dimensional case: Now the sign of det(A) is constant on every coset, and this allows to introduce an orientation: Two projective frames, or the homogeneous coordinate systems associated with them, respectively, are called equally oriented if the coordinate transformation A transferring one into the other has positive determinant, i.e. sign(det(A)) = 1. Apparently, there are two classes of equaUy oriented homogeneous coordinate systems (cf. the foUowing exercise); a real projective space of odd dimension is caUed oriented if one of these classes is distinguished. This orientation, introduced within the framework of linear algebra, is in complete accordance with the topological notion of orientation, cf. Appendix 4.2. It can be shown that a real projective space, considered as a differentiable manifold, is orientable if and only if its dimension is odd [103], Exercise 1.4.2. D

24

1 Projective Geometry

Exercise 6. a) Prove that the relation "equally oriented", introduced in the preceding example, is an equivalence relation, and that there are precisely two equivalence classes for odd n. - b) Use the definition \SL\{N,IC):={AeGL{N,IC)\

|det(A)| = l }

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to prove the isomorphy PGL{n

+ l,IC}-^\SL\{n+l,IC}/{±1}

(n odd),

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where / denotes the unit matrix. Exercise 7. In this exercise we consider the action of the group GLiV"^^) of linear automorphisms of the right vector space T^"+^ over the skew field K. Prove: a) Setting (ff,x) e G L ( F " + i ) xP"^gx

:= [ff(y)] e P" for x = [y]

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defines a transitive action; its kernel is isomorphic to the center Z{K*) of K*. - b) If -R" is a field, then the restriction of this action to the special linear group SL{V"+')

:= {g e GL{V"+')\N{g)

= 1}

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is transitive; here N{g) denotes the norm of the linear automorphism g, which is equal to the determinant of the matrix for g with respect to an arbitrary basis of T^"+^. - c) Let (C) be the inhomogeneous coordinates of a: e P", and let (??^), i,j = 1 , . . . , n, be the inhomogeneous coordinates of the image point y = gx under a linear automorphism g e GLiV"^^). Prove that these coordinates are related by a fractional linear transformation of the following form: r?^' = {a' + S^;alC){b + S^;bie)-\

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Here the coefficients a-',al,b,bi are the entries of the matrix for g with respect to the basis for T^"+^ determining both the homogeneous and the corresponding inhomogeneous coordinates, which again depends on the charts chosen so that X e Uk,y e Ui.

1.3 Collineations Collineations are the isomorphisms of projective geometry; sometimes they are even introduced t h a t way, cf. Exercise 5 below. Here we will start with a much weaker definition: a m a p from one projective point space to another is coUinear if it commutes with the operation of joining points. If, in addition, it is bijective, then we will call it a collineation: A bijective m a p transferring lines into lines. This property even implies t h a t a collineation maps projective A;-planes onto projective A;-planes. Since the projective A;-planes are embedded into the subsets lattice of the projective point spaces P " via the canonical m a p TT, each collineation generates an isomorphism of the lattices called projective geometries, and vice versa. The essential result here is the Main Theorem of

1.3 CoUineations

25

Projective Geometry describing the collineations algebraically in dimensions n > 2. One consequence of this is that geometric isomorphy implies algebraic isomorphy: Just the existence of a colhneation / : P{V) i-^ Q{W), where V, W are finite-dimensional vector spaces of dimensions larger than two, suffices to conclude that the scalar domains of the vector spaces are isomorphic, and, moreover, that the spaces have equal dimensions. Note that the isomorphy of the scalar domains for the vector spaces is not a precondition, but a conclusion of the Main Theorem. In Subsection 3 this will lead to complete descriptions of the group of auto-coUineations for the projective point space P " and of the group of automorphisms of the projective geometry ^{V), 2 < d i m F < oo. 1.3.1 CoUinear Maps In this section we consider maps / : P " -^ Q™ preserving the geometric operation of join, i.e. satisfying f{A y B) = f{A) V f{B) for all projective subspaces A,B a P". It suffices to require this property for the join of points (cf. Corollary 5). We define: Definition 1. Let P", Q™ be projective spaces over the respective (possibly different) skew fields Ki and K2 • A map / : P " -^ Q™ is called coUinear if it satisfies the following two conditions: f(o) = o, fix V y) = fix) V / ( y ) for all x,y e P^. A bijective coUinear map is called a coUineation.

(1) (2) •

Any bijective map / : P " -^ Q™ transferring lines into hues can always be considered as a colhneation by setting /(o) := o. Conversely, because of (2), each colhneation maps lines into lines. If the dimension of the image is less than two, D i m / ( P " ) < 2, then, obviously, the definition has not much substance; e.g., every bijective map between projective lines satisfying (1) is a colhneation. For this reason, the one-dimensional case will be treated separately in the following section. The colhnear maps with D i m / ( P " ) > 2 can be represented as the superposition of a colhneation with a (general) central projection to be introduced in the following example. Example 1. General central projections. Let Q™ C P", 0 < TO < n, be a projective subspace, and let B C P " a complementary subspace to Q™. The central projection p from P " onto Q™ with center B is defined to be the map pix):=ixVB)AQ:^, xeP:. (3) In case x ^ B Proposition 1.1 immediately implies

26

1 Projective Geometry Dim(a; \/ B) = Dim x + Dim B + 1 = Dim B + 1 = n - m

as well as Dim((a; V B) A Q™) = 0, so that (3) indeed uniquely determines a point of <5™. Obviously, p-\o) = B. (4) Let W := 7r^^(<5™) and t/ := TV^^{B) be the corresponding vector subspaces. Since Q™ and B are complementary, we have V"+^ = W®U,dimW

= m+l,dimU

= n-m.

(5)

Denoting by pr : V ^ W the projection of vectors determined by the decomposition (5) it can easily be checked that the foUowing equation holds for all ? € V: p{[f]) = br(p)]. (6) This or Definition (3) itself can be used for a straightforward proof that p is a coUinear map satisfying p op = p. If the center B is a point, we obtain the central projection discussed in Example 1.2. For a coordinate m-plane Q™, e.g. <5™ = ao V . . . V a„t, and its complementary coordinate [n — m— l)-plane B = Om+iV.. .Va„ the projectionpr occurring in (6) is just the corresponding vector coordinate projection: pr:i

= aix'e

F"+^ ^ pr{i) = o^x" G VF™+^

Here W™^^ denotes the vector subspace spanned by the m + 1 basis vectors Oa, a = 0 , . . . , m,« = 0 , . . . , n, and the sum convention is in action. Restricting the coordinate functions (x"(x)), a = 0 , . . . , m, to the coordinate m-plane Q™ yields a homogeneous coordinate system for this projective subspace with the base points OQ, whose unit point CQ is the image under the projection of the unit point e for the coordinate system in P " we started from: eg

: p{e) = (e V a™+i V . . . V a„) A (OQ V . . . V a^) = [oo + • • • + a™]. (7)

Definition 2. Consider again the situation from Definition 1. Denote by V and W the vector spaces associated with the projective spaces P " and Q™, respectively. Let, moreover, a : V ^ W and a : Kf ^ K2 be maps such that a(o) = 0 and a(pa) = a(p)o-(a) (p G F , a G if*). (8) Setting x=[^]eP:^

fix)

:= [a(y)] G Q ^

(9)

correctly defines a map / : P " ^ Q™: the map induced by a. If there exists a map
1.3 CoUineations

27

E x a m p l e 2. S e m i - L i n e a r l y I n d u c e d M a p s . The m a p a : V ^^ W considered in Definition 2 is called semi-linear if there is an isomorphism of skew fields, a : Ki ^ K2, such t h a t a(pa + t)/3) = a(p)o-(a) + a(t))o-(/3)

(p, t) G F , a, /3 G i f i ) .

For brevity, semi-linear maps with the isomorphism a will often be called
G K.

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(p G W, a G Ki).

This construction is called replacement of scalars; in Example II.7.1.4 it is defined in greater generality. Obviously, the subspaces of W are nothing but those of W, and the dimension of a subspace is not changed by replacing the scalars. Moreover, the relation a(pa) = a(p)o-(a) = a(p)a

(p G V , a G i f i ) ,

implies t h a t , now, i.e. considered as a m a p from V to W, a is even linear. Hence images as well as pre-images of subspaces under a are again subspaces. In particular, the kernel of a, K e r a := a^^(o) C V, and the image of a, I m a := a{V) C W are well-defined subspaces. As for hnear maps we define the rank of a by r k a := d i m ^ j I m a . For a finite-dimensional vector space V we then have the same relation as in the case of linear maps: d i m ^ i K e r a + r k a = d i m ^ i V^ = « + 1 < 00.

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28

1 Projective Geometry

Now we will prove t h a t the m a p / between the associated projective spaces defined, according to (9), by a semi-linear m a p a is coUinear; relation (6) then shows t h a t , in particular, this holds for a central projection p . P r o p o s i t i o n 1. Let a : V -^ vt^™+ ftg a-linear. f : P " -^ <5™ defined by (9) is coUinear, and, moreover,

Then

Dim/-i(o)+Dim/(P^) = n - l .

the

map

(12)

P r o o f . Since a is semi-linear, we have a(o) = o implying (1). Now take any 33 = [?], y = [t)] € -P"• Then a; V y = 7r([{p, t)}]) is determined by the linear hull, and this implies (2): fix

V y) = ^(a([{y, t)}])) = ^([{a(y), a(t))}]) = fix)

V /(y),

where a([{p, t)}]) = [{«(?), a(t))}] follows from the semi-finearity of a. Hence / is coUinear. T h e dimension formula (12) is a direct consequence of (11). • As a straightforward consequence of Definition 1 we note the following. P r o p o s i t i o n 2. The inverse f^^ of a coUineation f again is a collineation. If f and g are coUinear, then their composition f o g is also coUinear. The class of projective spaces over arbitrary skew fields with the coUinear maps as morphisms form a category. • E x a m p l e 3 . T h e constant maps on P " , i.e. the maps / : P " -^ Q^ with / ( o ) := o and fix) := yg = const for x ^ o, are always colfinear. If yg ^ o, then D i m / ^ ^ ( o ) = —1. In this case, (12) is satisfied if and only if n = 0; this is the only situation where / is induced by a semi-linear m a p (Proposition 1). In case yg = o the m a p / is obviously induced by the zero m a p . Moreover, an arbitrary bijection of projective fines, / : P\ -^ QI, such t h a t / ( o ) = o, is a collineation which, in general, is also not induced by a semi-linear m a p . • Exercise 1. Find an example for a collineation / : Pi -^ Ql of projective lines over non-isomorphic fields. Next we want to deduce some properties of colfinear maps. Definition 1 and Exercise 1.4 immediately imply the following. L e m m a 3 . Let f : P " -^ Q™ be coUinear, and let A C P " , B C Q™ be projective subspaces. Then / ( A ) and f^^iB) are also projective subspaces; so, in particular, / ^ ^ ( o ) and fiP^) are projective subspaces. • L e m m a 4. Let f : P " -^ Q™ be coUinear. Then for all k G Ng and any XieP: fixo \/ ...\/Xk) = fixo) V . . . V fixk). (13)

1.3 CoUineations

29

P r o o f . T h e prove will proceed by induction. For A; = 0 , 1 the statement is trivial or holds by assumption, respectively. Assume t h a t (13) is already proved for k. If Xk+i G H^ := a;o V . . . V Xk, then (13) obviously also holds for {xo,..., Xk+i}. Suppose t h a t Xk+i ^ H^, set M^^^ := XQV .. .V Xk+i = H^ V Xk+i, and take z G M^^^. Then we wiU have to prove the following relation: fiz) G fixo) V . . . V fixk+i) = f{H') V f{xk+i). We may assume z ^ Xk+i as weU as ^ ^ H^, since otherwise the claim holds by the induction hypothesis. As H^ C M^'^ is a hyperplane, the line Xk+iVz intersects it in precisely one point, y = {xk+i V 2) A H^. Since / is coUinear, z e y\/ Xk+i imphes fiz)

G / ( y V Xk+i) = f{y) V f{xk+i)

C f{H') V /(a^^+i),

hence f{xo V . . . V Xk+i) C f{xo) V . . . V / ( a ; fc4 - 1

•

On the other hand, M^'^ is a projective subspace of P " , and, by Lemma 3, so is / ( M ' ' + i ) C g ™ . Since f{xi) e / ( M ' ' + i ) , i = 0 , . . . , A; + 1, t h e statement follows from t h e definition of the join as the smallest of all subspaces containing the one considered. • C o r o l l a r y 5. Let f : P " -^ Q™ be coUinear. Then for any two projective subspaces A, B C P " the restriction, f\A : A -^ Q™, is also coUinear, and D i m / ( A ) < Dim A . Moreover, f{A\/B) = f{A)\/f{B).

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P r o o f . Both claims follow from (13) by considering A , B as generated by projectively independent points. • T h e following proposition characterizes injective coUinear maps: P r o p o s i t i o n 6. Let f : P " -^ Q™ be coUinear. Then the foUowing ments are equivalent: a) / is injective; b) D i m / ( P : ) = n ; c) D i m / ( A ) = Dim A for every projective subspace A.

state-

P r o o f , a) imphes b): Suppose t h a t B := f{P") ^ o and D i m B = k < n. Then there are k + 1 points in general position spanning B: B = 60 V . . . V bk- Choose Oj G f^^{bi) ^ o, i = 0,... ,k. Setting A : = OQ V . . . V a^ we conclude / ( A ) = B = f{P^) and Dim A < n from (13). Therefore, there is an X G P " \ A for which, nevertheless, f{x) G B has to hold. Hence / cannot be injective. li B = o, then, because of n > 0 we obtain f{x) = o for all x G P " , i.e., / is not injective. b) implies a): Consider a m a p / t h a t is not injective. Then there are x,y e P " , X ^ y, such t h a t f{x) = f{y). If, e.g., x = o, then / ( y ) = o and

30

1 Projective Geometry

y y^ o. Set now ao := y and choose additional points to obtain a projectively independent sequence ( a o , . . . , a „ ) in P " . > F r o m / ( O Q ) = O we conclude by Lemma 4 / ( P : ) = / ( a i ) V . . . V / ( a „ ) , hence D i m / ( P : ) < n.

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If a; and y are different from o, denote t h e m by ao := x, ai := y, and once more form a sequence of projectively independent points (OQ, . . . , a „ ) in P " . Since / ( O Q ) = / ( a i ) , this again yields (15). As for each subspace A C P o t h e restriction / | A of the m a p / is also injective, c) immediately follows from the equivalence of a) and b). Since b) is a special case of c), t h e latter is equivalent to b). • R e m a r k . Obviously, for each injective collinear m a p we have / ^ ^ ( o ) = o. As t h e example of a constant m a p (Example 3) / with y^ ^ o and n > 0 shows, this property does not imply t h e injectivity of / . Nevertheless, we have t h e following. P r o p o s i t i o n 7. Let f : P " -^ Q™ be collinear and suppose, moreover, D i m / ( P " ) > 1. The map f is injective if and only if f^^{o) = o.

that

P r o o f . W i t h o u t loss of generality we may assume t h a t / is surjective. Let bj, j = 0,... ,m, be points spanning Q™: Q™ = 6o V . . . V 6 ^ . Choose points Uj G f^^{bj), ttj ^ o, and form A := ao V . . . V a ^ . Relation (13) implies f{A) = <5™, and by CoroUary 5 we have Dim A = m. According to Proposition 6, g := f\A is bijective. Now t h e proof proceeds by contradiction. Suppose t h a t f^^{o) = o and t h a t / is not injective. T h e n there has to be a point c G P " \ A for which there exists a point CQ G A satisfying / ( C Q ) = / ( c ) . Since c ^ o and f^^{o) = o, we have / ( c ) 7^ o as well as CQ ^ o. Consider t h e projective subspace H™^^ := cV A and define the m a p q := g-^ o f\H"'+^

: H"'+^

-^

A,

which is collinear by Proposition 2. Now q\A = i d ^ immediately implies q"^ = q. For each x G H'™+^ there is precisely one XQ G A, namely XQ = q{x), satisfying f{x) = f{xo). By definition, q{x) G A, f{q{x)) = {g o g-'^ ° I){x), and g = f\A is injective. Let x G H\A. T h e n q{x\/XQ) = q{x)\/q{xo) = XQ, and hence a; V a;o C q^^{xo). We are now going to show q^^{xo) = x\/ XQ. To this end, suppose t h a t xi ^ x \/ XQ and q{xi) = XQ- Since x ^ o and q^^{o) = o, this implies XQ y^ o and Dima; V a;o V a;i = 2. The dimension formula then yields Dim(a; V a;o V xi) A A = 2 + m - ( m + l ) = l. Because oi q\A = i d ^ , this contradicts t h e relation Xo = q{{x V a;o V Xi) A A ) = (a; V a;o V Xi) A A .

1.3 CoUineations

31

But by assumption we have m + 1 > 2. Therefore, there has to exist a 2GH'™+^\(AU(a;Va;o)). In fact, take any projective frame ( a o , . . . , am+i', e) for H such that x V xo = aoV ttm+i and A = ao V . . . V a^n- Then z = e has the above property. Let zo := q{z). As shown before, q^^{zo) = z\/ ZQ, and hence ZQ ^ XQ. Moreover, {z V ZQ) A (a; V a;o) = o. Then, yL{z V ^o) A (a; V a;o) has to satisfy q{y)LXo as well as q{y)LZo. Since 030 7^ ^0, this is only possible for y = o. Thus the hnes ZVZQ, XVXQ are skew, and hence L := (zV ZQ) V (a; V a;o) is a three-dimensional subspace of H™^^. Since f f ^ + i = I, V A, this implies Dim L AA = 2. Because of q\A = i d ^ we arrive at the contradiction AAL

C q{L) = q{x) V q{xo) V q{z) V q{zo) =

XQV ZQ.

The converse is trivial. Now we can easily deduce the foUowing.

•

Corollary 8. Let f : P'^^ -^ Q"^ he collinear with D i m / ( P ^ ) > 1. Then f can be represented as a composition, f = g o p, of a central projection p and an injective collinear map g in the following way: For B = /^^(o) and any subspace A complementary to B in P " , let p be the central projection p : P " -^ A with center B. Then g := f\A is injective, and f = g op. P r o o f . First, by construction {gop){x) = f{{xVB)AA)tf{x)Vf{B) = f{x). Since AAB = O, the relation {g op)(x) = o can only occur for p{x) = o, i.e. X e B, and hence {gop){x) = f{x). Moreover,p(a;) G A, thus f{P") = f{A). Hence we have Dim gr( A) > 1, and because of A A B = owe obtain gr^^(o) = o. By Proposition 7 the map g is injective. • Proposition 6 implies Dim A = D i m / ( P " ) , and hence we immediately arrive at the following property corresponding to (12): Corollary 9. Let f : P " -^ Q™ be a collinear map such thatDmif(P") Then, for every projective subspace H C P",

Dim/(H') = D i m H ' - D i m H ' A / " i ( o ) - 1.

> 1.

(16)

Thus for any collineation f the dimensions of a projective subspace and its image always coincide, Dim f(H) = DimH". P r o o f . By CoroUary 8, f = g op. Hence f{H) = {g op)(H), and since g is injective, Proposition 6, c) implies Dim/(H') = T>iinp{H). On the other hand, p{H) = {H V B) A A. Applying the dimension formula twice, the complementarity of A and B = f^^{o) yields:

32

1 Projective Geometry Biinp{H)

= Dim H + Dim B - Dim H AB = BiinH

-BiinHAB

+ Dim

A-n,

- 1.

For a collineation / we have by d e f i n i t i o n / ^ ^ ( o ) = o, hence Dim i J A B = —1; the case D i m / ( P " ) < 1 is triviaUy true for coUineations (and in general false for collinear maps, cf. Example 2). D 1.3.2 T h e M a i n T h e o r e m of P r o j e c t i v e G e o m e t r y In this section we will prove the following Main Theorem of Projective Geometry, which also implies the characterization of collinear maps mentioned in the introduction to this chapter (cf. E. Artin [4], J. Dieudonne [36]): P r o p o s i t i o n 10. Let V, W he {n+1)-dimensional vector spaces, n > 2, over the skew fields Ki and K^, respectively, with associated projective spaces P " andQ". Let, moreover,

f:xeP:^x'

= fix) e Q:

be a collineation. Then there are an isomorphism of skew fields a : Ki -^ K^ and a a-linear bijection a : V ^ W inducing f, i.e. satisfying /([p]) = [«(?)] for all f (^ V. Lf ai : V ^ W is another ai-linear bijection inducing f, then there exists JJ, G Kf such that ai(?) = a(?M),

? e V,

^i(e) = ^(M"'eM),

eeifi-

(17)

(is)

P r o o f . Proposition 6, c) implies t h a t the images of the points in a projective frame ( a o , . . . , a „ : e) of P " , a^ = / ( a i ) , e' = / ( e ) , * = 0 , . . . , n ,

(19)

form a projective frame of Q " , since any n + 1 among t h e m are projectively independent. Let (Oj) and ( o ^ be bases of V and W, respectively, determining the projective frames ( a o , . . . , a „ : e) and ( o g , . . . , a^ : e'). By Proposition 2.2 each of t h e m is uniquely defined up to a factor from if* and if^ •• respectively. We will prove t h a t there is an isomorphism a : Ki -^ K2 for which, in the homogeneous coordinates corresponding to the bases, / is represented by f{[aix%

= [a[a{x%

(20)

(Recall the sum convention!), the a-linear m a p we are looking for is thus a : OjX* eV

^

a'iCrix^) G W.

To see this, consider a coordinate axis OQ V a^, 0 ^ j . Because of (19) we obtain a bijection aj : Ki -^ K^ defined by

1.3 CoUineations

/([oo + oj-e]) = [o^ + o;.a,-(e)], e e K,,

33

(21)

(cf. Example 1.3). We prove: The map GJ is independent of j . Consider, say, <Ti and
c' = ix[iOVx',iO)Aia[Va',) = [(0^ + o'iai(e))5 + (0^ + o^a2(e))^] = [a'i7 + a^^]. Comparing coefficients yields a = - / 3 , 7 = cri(C)a, S = cr2(C)/3 = -c^2(C)«,

and we conclude c ' = [0^^2(6-o'iai(e)]. Since c' does not depend on ^, we may set ^ = 1 and finally obtain from the resulting equation, (TJ{1) = 1, that c' = [02 — a'l]. This together with the last formula imphes the assertion ai{S,) = (^2(0 foi" ^H ^ & Ki. Now we prove (20). First let x ^ HQ, where HQ = a i V.. .Va„ denotes the coordinate hyperplane complementary to OQ. Since f{Ho) is the coordinate hyperplane H'Q complementary to Og, we also have x' = f{x) ^ H'^. Hence the coordinates x'-', x"-' of a; and x' are different from zero. Normalizing these to have x" = 1, x ' " = 1 we obtain the normalized representations

34

1 Projective Geometry a; = [oo + aix^ + . . . o„x"], x' = [OQ + a[x''^ + ... a^x'"].

Let pi,p'i be the respective projections onto the coordinate axes ao V a i and Og V a[. Moreover, denoting by HQI, H'Q-^ the coordinate planes complement a r y to these axes, we have p-i{x) = {xV Hoi) A (ao V a-i) = [oo + aix^] p[{x') The equations from Example other, the fact Definition (21)

= {x' V H'o,) A {a'o V a[) = ^

+

a[x%

on the right-hand side of these formulas immediately result 1, (6). Now, since / maps the coordinate planes into one ant h a t the colhneation / commutes with V and A together with of a implies:

f{pi{x)) i.e. x'^ = a{x^). we obtain

= [a'o +a[a{x')]

=p[{f{x))

= [a'o +

a[x'%

As corresponding relations hold for all the coordinate axes, n

n

/([oo + Yla^^']) = K + E4^(^')]' ^' ^ ^1'

(22)

and this is just assertion (20) for the case x ^ HQ- If a; G HQ, then also x' = f{x) G HQ, and we have x° = x ' ° = 0. In this case, x = [p] with p = Yl^=i l j ^ ^ lies on the line ao V [oo + ?]. Hence, by (19) and (21), f{x) lies on the line a'g V [OQ + Yl"=i a'jCr{x^)]. Thus n

n

This implies JJ, = —v and, since a common factor does not matter, y^ = a(x^). This again is the asserted equation (20). Now it is not difficult to show t h a t a is an isomorphism: T h e point X = [oo + ai(C + v) + I2] lies on the line [oo + aiC] V [air/ + 02]. Consequently, the image point x' = [OQ + Cl[a{^ + r]) + O2] is on the image line [a'Q + a[cr{^)]\/[a[cr{r/) + a'2]. This implies o-{^ + r/) = o-{^)+a{r/). Analogously, the point a; = [oo + Cli^r] + 02??] hes on the fine ao V [oi^ + O2], so the image point x' = [a'o + a[cr{^r/) + a'20-{r/)] belongs to the image line ag V [0iO-(C) + 02]. Comparing coefficients in a'o + a'lCriCl]) + 02Cr(??) = aoJ^ + (a'iCr(C) + 02)^ yields i/ = 1, 1^ = (T{r]), ^(^ry) = (T{S,)a{r]). This completes the proof of the existence part of the proposition. To prove uniqueness consider the m a p b := a^^ o a i : V ^ V . It is a semilinear bijection with the isomorphism T := a^^ o ai of Ki. Since it induces the identity on P " , there has to be a function

1.3 CoUineations

35

A : p G F \ {0} H^ A(p) G K* with 6(p) = pA(p). In particular, we have 6(0i) = Ojaj, a j G Kf.

From

6(0i + Oj) = (Oi + aj)A(oi + Oj) = 6(0i) + 6(0j) = Ojaj + Oj-aj we conclude a j = a^ = : JJ, and hence 6(aj) = ajjj, for a certain /x G Kf and all j . Since Oi is an arbitrary vector t h a t is linearly independent of Oo, and d i m F > 3, we obtain A(p) = /x for ah p G V \ {0}. Thus 6(p) = p/x, i.e. ai(p) = a(p/i) for some /x G i f J. Furthermore, we obtain 6(p^) = TI^T{0 = ?CM) i.e. T ( ^ ) = /X^^CM and cri(C) = cr(/i"^C/i). D For coUinear maps the Main Theorem implies the following. C o r o l l a r y 1 1 . Every coUinear map f : P"^ ^ Q™ such that D i m / ( P " ) > 2 is generated by a semi-linear map between the associated vector spaces b:V ->W. P r o o f . In the notations of Corollary 8 the m a p / can be represented in the form f = g o p; according to (6) the central projection pi is induced linearly. Since g{A) = f{P"), the m a p g : A ^ f{P") is a coUineation satisfying the hypotheses of Proposition 10. Hence it is generated by a semi-Unear m a p a. Consequently, the composition f = g o p \s induced by a semi-linear m a p b = a o pr. D 1.3.3 T h e G r o u p of A u t o - C o U i n e a t i o n s An important and somewhat surprising partial result of the Main Theorem is the isomorphy between the skew fields of two projective spaces t h a t are m a p p e d onto one another by a coUineation. Because of this invariance of the scalar domains, considering coUinear maps we will always suppose Ki = K2, i.e., we will be working within the category of projective geometries (or spaces, respectively) over a fixed skew field K. Since, as already mentioned, the notion of a coUineation is less rich in substance for n < 2, for the description of the automorphism group of projective spaces we will require n > 2 until the end of this section. The projective Unes, for which the cross ratio plays a decisive part, will be studied in the subsequent section. Let us start with the following definition. D e f i n i t i o n 3. For n > 2, let A u t P " denote the group of all coUineations f .pn ^ pn f^^^ pn ^^ -^g^jj^ rpj^^ elements / G Aut P " are called the auto-collineations of P " . Here and in the sequel we will often omit the index o, when there is no risk of confusion; if necessary, a m a p / : P " -^ Q™ will be understood to be extended by / ( o ) := o to a m a p / : P " -^ Q™ . • T h e Main Theorem allows us to describe Aut P " algebraically, since each auto-coUineation is generated by a semi-linear bijection. First we will prove the following.

36

1 Projective Geometry

L e m m a 12. The group of all the f G A u t P " leaving each point of a given projective frame {ai] e), i = 0 ..., n, of P " fixed is isomorphic to the group of automorphisms Aut K of the skew field K for P " . P r o o f . Let a G Aut K, and take a basis (Oj) of the vector space V associated with P " determining the frame (OJ; e ) . Then the m a p ip : a i-^ fa- with faiiaix'])

:= [aM^l]

(23)

defines a homomorphism from Aut i f to A u t P " , for which every f^ belongs to the stationary subgroup of ( a j ; e ) , i.e., it satisfies the conditions f{ai)

= Ui, f{e)

= e, i = 0,...,n,

(24)

since
/<,([oo + aiC]) = [ao + aicr(C)] = [ao + aiC], and this imphes a{S,) = ^ for all ^ & K. Hence f is injective. So we stiU have to show t h a t each auto-colhneation / satisfying (24) has the form (23). First, the equations (24) yield for any semi-linear m a p a inducing / with corresponding automorphism a G Aut K:

a(Oi) = Oi/ij, a ( 2 j 1j) = 22 '^Jf^J ^ ( Z J '^J')'"' i.e. iJ,i = jj, =^ 0 for « = 0 , . . . , n. Hence we have a(OiX*) = Oi/iO-(x*),

(25)

f{[aix'])

(26)

= [aii^a{x')] = [aii^a{x')i^-'].

Setting thus T := cr^i O a, where
(27) this way, and the associated is linear. Any automorphism

1.3 CoUineations

37


Consequently, the matrix (a^) is invertible, and we obtain a m a p

: a G G ( F " + ^ ) i—> <^{a) := ((af), a) G GL{n

+ l,K)x

Aut K.

Furthermore, 'P{a-')

= {{a-Ha\))-\a-').

(28)

A straightforward calculation using ^(6) = ((/3f), T ) leads to
(29)

Defining in the product set G „ + i ( i f ) := GL{n

+ l,K)x

Aut i f

a multiplication by ((/3f),r).((a^),a):=((A'^)(r(a^)),roa),

(30)

results in a group, the semi-linear group Gn^i{K) of order n + 1 over K, as the semi-direct product of the normal subgroup GL{n + 1) x { i d x } with the subgroup {(<5j)} X Aut if, cf. Exercise 1.3.2.3. Moreover, ^ t u r n s out to be an isomorphism: G ( F " + i ) = G„+i(if), n G N o . (31) In Example 2 we introduced the dilation d^ by dn{i) = p/x, p G V , yU, G K*; d^ is semi-linear with automorphism a^i-i. As the following proposition implies, the set of all dilations Z>:=K|MGif*}cG(F) is a normal subgroup oi G{V). neK*

By means of

^ d^-i

G -D,

K* = D,

it is isomorphic to K*. In the sequel we consider K* as canonically embedded into the semi-linear group via ^eK*^


and correspondingly write K* C G „ + i ( i f ) .

a^) G G „ + i ( i f ) ,

(32)

38

1 Projective Geometry

Proposition 13. The map F : a e G(F"+^) ^ fa e A u t P " , n>2, by /a([?]) := [«(?)]; is a surjective homomorphism with kernel Kei F = D,

defined (33)

so that A u t P " = G„+i(if)/if*,

n>2.

(34)

P r o o f . The Main Theorem together with the obvious formula fboa = fb° fa immediately show that F is a surjective homomorphism. Hence, (34) can be deduced from (33) using (31) and (32). To describe the kernel K e r F , choose a basis for V " ^ and consider a map fa e G(F"+^) determined by fa = (P^^{{al),a), for which /„ = idpn. Since /„ in particular satisfies (24), we derive relation (26) as in the proof of Lemma 12; we only have to replace ( " D = (<^iM"^),

/ a = / r w i t h T = cr^-l OCT,

cf. (23). Since the map a i-^ fa- is injective by Lemma 12, T has to be the identity of K, i.e. a = a^. On the other hand, each d^ e D obviously belongs toKerF. D Now we ask whether the sequences (ao, . . . , a „ ; e) we caUed projective frames actually are frames in the sense of the group-theoretic view onto geometry. According to E. CARTAN [28] this means that the group, on which the geometry is based, acts simply transitively on this set of frames. As will be shown below this does not hold, in general for the group of auto-coUineations; in the important case of the field if = R of real numbers, however, it is true. Proposition 14. Let P^, Q" be n-dimensional projective spaces, n>l, over the skew field K, and let a G Ant K be any automorphism of K. Then, for every pair of projective frames, {aj]e), ( a ' ; e ' ) of P^ and Q^, respectively, there exists a collineation f : P " -^ Q " generated by a a-linear map such that f{aj) = a;.,

/ ( e ) = e'.

(35)

/ / K is commutative, f is uniquely determined by (35) and a. Conversely, if such a map is uniquely determined by a couple {aj]e), {a'j]e') together with (7 G Aut K, then K is a field. P r o o f . By Proposition 2.2, we find bases (Oj) of F " + \ (o^) of W"+^ such that (oj-; e) and (a^; e') are the associated projective frames. Considering (20) as a definition for / , the equations
1.3 CoUineations

39

L e m m a 15. The scalar domain of P^, n G N , is a field if and only if there is only one collineation f generated by a linear map a G GL{V ) that satisfies (24), namely f = i d p r i . P r o o f . As in the proof of Lemma 12, relation (24) implies the condition a(Oj) = ajjj,,

jj, G K*,

(36)

for the linear m a p a we want to find. If i f is a field, then each of these maps generates the identity collineation / = i d p r i . Consequently, the latter is uniquely determined by requiring (24) as well as linearity. If K is not commutative, there exist i^, a e K* with i^a/^^^ ^ a. Since n > 1, the associated vector space has at least dimension two. Consider the vector p := Ooa + ai and the linear m a p a determined by (36) t h a t generates / . Then /([?]) = [«(?)] = [loM" + IIM] = [aoM"M"^ + i i ] 7^ [loa + ai] = [p], since the vectors Ooa + fli, Oo/UQ^M^^ + l i ^ire obviously linearly independent. T h u s / is a linearly generated auto-coUineation t h a t is different from the identity and satisfies (24). D C o r o l l a r y 16. Let K = H be the field of real (or rational) numbers. If f : P " -^ <5™ is a collinear map with D i m / ( P " ) > 2, then f is generated by a linear map between the associated vector spaces. The group A u t P " , n>2, acts simply transitively on the set of projective frames. P r o o f . It can easily be shown, cf. Exercise 1.2.1.3, t h a t the the fields of rational and real numbers each have only a single automorphism, the identity m a p . Hence, in these cases there is only one coUineation satisfying (35) by Proposition 14. Hint. To prove the statement concerning the automorphisms a of R, first show that for each of these automorphisms the restriction (7|Q to the field Q of rational numbers is the identity idq. Then, starting from (7(5^) = ((7(5))^ prove that a is monotonous and derive from this its continuity. This, finally, implies a = idR. D Exercise 2. Let / : P" -^ Q™ be a map with the following properties: a) / ( o ) = o; b) f{x V J/) C f{x) V f{y) for all x,y E P"; c) f{P") is a projective subspace of dimension n. Prove that / is an injective collinear map. Exercise 3. Let P", Q™ be projective spaces over the field K, let {aj;e) be a projective frame of P", and let {bj; b), j = 0,... ,n, he a sequence of points in Q™. Find examples to show that there does not always have to exist a collinear map / : P" -^ Q™ , for which / ( o j ) = bj, j = 0,. .. ,n, and / ( e ) = b. Prove, moreover, that, in general, these conditions together with the postulate a = idx for the automorphism of K associated with / do not determine / uniquely. (Compare this, however, with Proposition 1.5.3.4 for afline maps!)

40

1 Projective Geometry

Exercise 4 . Let P", Q"^^', K,{aj;e) be as in Exercise 3, and let, in addition, {bo,... ,br;b), n > r > 2, be a projectively dependent sequence of points from Q™, each (r + l)-element subsequence of which is projectively independent. Prove that for every automorphism a G Aut K there is precisely one collinear map f : P" ^ Q™ with the properties: a) / ( a ^ ) = bp, p = 0,. .. ,r; h) / ( e ) = b; c) / ( c r + i V . . . V a„) = o; d) / is generated by a a-linear map c : F"+^ -^ VF™+^ of the associated vector spaces. Exercise 5. Let \^"+^, W be right vector spaces over the skew fields Ki and K2, respectively, n > 2, and let ^" = ^(V), Q = ^(W) be the associated projective geometries (cf. Definition 1.1). A map F : ^ " ^ Q is called monotonous if ior A,B e ^ , A C B always implies the relation F{A) C F{B). Prove: If F is bijective, and F as well as F^^ are monotonous, then / := F\P" is a coUineation. Conversely, every coUineation / generates an isomorphism of the lattices F : [^", C] ^ [Q, C]. (In R. BAER [5], § III.l, the projective maps are actually defined as such isomorphisms of lattices.) Exercise 6. In analogy to Corollary 8, describe all collinear maps / : P" -^ Q™ with D i m / ( P ^ ) = 0. Exercise 7. Let / G A u t P " , n > 2, and let F be the automorphism of the lattice [^", C] determined by / according to Exercise 5. Prove that / is the identity of P " , if for some k, 0 < k < n, the restriction of F to Pn,k is the identity. Exercise 8. Coarse Classification of Collinear Maps. Let PO,QTJ projective spaces over the skew field K. Define: T:={f:P:^

n,m

> 2, be

Q ^ l / collinear, D i m / ( P : ) > 2},

G:= AutQ" X AutP". Via {{g,h),f) ^GxTi-^gofo hr^ e T the group G acts on JF; denote by ~ the corresponding equivalence relation. Prove that / i ~ /2 if and only if Dim/i(P^) = Dim/2(P^). Exercise 9. Prove that the group A u t P " , n > 2, acts transitively on the Grafimann manifolds P„,k of fc-planes in n-dimensional projective space. (To this end, consider the realization of fc-planes as subsets of point space defined by the canonical map (1.2) as well as the action defined on them via the action of A u t P " on P".) Determine the isotropy group of a fc-plane. (The isotropy group or stationary subgroup of an element x in the transformed space is the set of all those elements g in the transforming group G leaving x fixed, i.e. gx = x, cf. I, § 1.4.)

1.4 Cross Ratio and Projective Maps In afRne geometry, the length ratio of two parallel segments or vectors is an important invariant closely related to the afRne parameter on a line, cf. Exercise 1.4.3.1. In projective geometry, an analogous procedure starting from

1.4 Cross Ratio and Projective Maps

41

the projective scales on a line leads to the notion of the cross ratio of four coUinear points, t h a t can be interpreted as the quotient of two length ratios, which explains its name^. In this section we will first deal with the geometry on the projective line over a skew field K. The cross ratio of four collinear points will be defined by means of a projective scale on the line; nevertheless, it does not depend on the choice of the scale. In general, however, this cross ratio is not invariant under collineations; it is transformed by the automorphism of K associated with the coUineation. Requiring invariance of the cross ratio will lead to a special class of colhnear maps, the projective maps. These form a category whose objects are the projective geometries (or spaces) over the skew field K, t h a t , in general, is smaller t h a n the category based on all collinear maps. For the field i f = R of real numbers all collinear maps / with D i m / ( P " ) > 2 are projective.

1.4.1 T h e G r o u p A u t P ^ In Example 3.3 we already pointed out t h a t arbitrary bijections between lines are collinear. The Main Theorem of Projective Geometry does not hold for n = 1, and the group A u t P is not defined, yet. In order to characterize the projective geometry on a line by a group we will proceed as follows: Consider the Grafimann manifold Pn,i of lines in an n-dimensional projective geometry, n > 1. The group of auto-colhneations of P " acts transitively over Pn,i (Exercise 3.9); denote by H the isotropy group of a fixed line B C P " . T h e elements h e H leave the line B as a whole fixed, i.e., they transform the points of B among themselves. This way, an action of H over B is defined, t h a t allows to develop a rich projective geometry on the line. To describe this action we choose a projective frame (OJ; e) of P " for which B = ao V a i is a coordinate axis. A semi-linear m a p a inducing the coUineation h £ H G Aut P " then has to satisfy the conditions a(oo) = ao^o + a i 6 j ,

a(oi) = 006? + ai6j

(1)

with respect to a basis (Oj), i = 0,... ,n, determining this frame. Since h is a coUineation, the 2 x 2-matrix (60) occurring here has to have rank two. Conversely, each semi-hnear transformation of the vector space V " + ^ associated with P " having property (1) determines a coUineation belonging to H. Since the remaining coefficients 6^, « > 1 or j > 1, do not affect the action of h on B, we arrive at the same initial situation for the projective line as in the case of higher dimensional projective spaces: Formulas (3.27) to (3.33) retain their validity even for n = 1. They describe the action of the restrictions h\B to the line B, where, of course, plenty of different elements h e H yield the same restriction. We obtain a homomorphism F from G(V ) to the group of all bijections on P = B, which is defined as in Proposition 3.13. Setting ^ Specialize Formula (15) below to the case K = R.

42

1 Projective Geometry A u t p i := {U\a e GiV')}

= FiGiV')),

(2)

Formulas (3.33), (3.34) also hold true for n = 1. We call A u t P ^ the automorphism group of the projective line. It is not difficult to see that, formally, the case n = 0 is included here; we have G{V^) = £), and A u t P ° just consists of the unit element. The reader may easily check that Lemma 3.12 and Proposition 3.13 remain valid for n = I, including their proofs. 1.4.2 The Cross Ratio Now we consider the projective line P^ under the action of the group A u t P ^ and look for the invariants of this transformation group. By Proposition 3.14, applied to the case Q^ = -P^, the group Aut P^ acts transitively on the triples (a, b, c) formed by pairwise different points on the hue. Of course, these triples may always be viewed as the points in a projective frame. Thus, an invariant is first to be expected looking at quadruples w = (a, 6, c, d) of points on the line P . The point of departure for the determination of such an invariant is a projective scale on the line, cf. Example 1.3. Let, for the time being, the points b, c, d of the quadruple w be pairwise different. Then, by Proposition 2.2 we find a basis (oo, ai) of the vector space V associated with P that is uniquely determined up to a common factor /x G K* and satisfies [oo] = c, [oi] = d, [oo + ai] = b.

(3)

Let ^ denote the value assigned to a by the projective scale for this basis. If the scalar field K is not commutative, then the factor /x G K*, occurring due to the arbitrariness in the choice of the basis, indeed plays a part. By CoroUary 2.4, the different scale values, obtained by varying the basis satisfying (3), are related by inner automorphisms of K. Hence, it is not the scale value itself but its conjugacy class that is assigned to the quadruple in a way independent of the chosen basis. We denote this conjugacy class by < e >:= {MCM" V e K*}, if e G if, < oo >:= {oo}.

(4)

The last postulate only serves to include the case C = oo that actually occurs, for a = d. If if is a field, we can do without these considerations, since each conjugacy class only consists of the element itself. In general, we agree to set < ^ > = ^, if < ^ > has just a single element. In particular, for a skew field < 0 > = 0, < 1 > = 1, < oo > = oo.

(5)

Now we summarize the observations above in the following definition, that now refers to the n-dimensional space, since, typically, we will have to deal with hues as embedded objects. Definition 1. Let P " be a projective space over the skew field K. A quadruple w = (a, 6, c, d) of points in P " is caUed a throw if the points a, 6, c, d are

1.4 Cross Ratio and Projective Maps

43

coUinear, and at least three of them are pairwise different. A throw is caUed special if its last three entries, b, c, d, are pairwise different. If w is a special throw, then its cross ratio, abbreviated as CR, is defined by CR(w;) = (a, 6; c, d) :=< ^ >, where ^ = ^(a) is the scale value of a with respect to a basis for the vector space of the hne c V d satisfying (3). • According to the above considerations CR(w;) does not depend on the choice of the basis satisfying (3). In particular, because of (3) and (5) we have (6, b; c, d) = 1, (c, b; c, d) = 0, (d, b; c, d) = oo. (6) The CR does, however, depend on the order of the points in the throw. In order to remove the asymmetry in Definition 1, and to define the CR for an arbitrary throw we suppose that the four points are mutually different and investigate the way the CR changes under a permutation of the points in the throw. The case that two points of the throw are equal will be considered afterwards. Note that, by (6) this is the case if and only if the CR attains one of the values occurring in (5). This immediately follows from the properties of a projective scale. Since each permutation in the symmetry group 6*4 can be represented as a product of the transpositions (12), (23), (3, 4) (cf. Exercise 1.1.2.7), it suffices to consider the action of these transpositions. We wiU prove: (6, a; c, d) = (a, 6; d, c) = (a, 6; c, d)^^.

(7)

In this as well as in similar formulas, the algebraic operations are to be applied to aU elements in the conjugacy class. Obviously, it suffices to prove the claim for any representative ^ G (a, b; c, d). Using Definition 1 we conclude from (3) the following, where the second line of the formula refers to the computation of (6, a; c, d):

Here, the vectors each have to represent the points: a=[a],b=[b],c=[c],d=m, 0 = oa, b = b/3, c = C7, 6 = t)(5,

(9) {a, f3,'-/,S e K*).

By (9), equations (8) and (10) hold, if we set 0 = 0, b = b, c = c, 6 = t)^.

The value of ^ has to be computed from

tj = c + (fe)e = c+1),

(10)

44

1 Projective Geometry

leading to ^^ = 1, i.e. claim (7) for the permutation of a, b. The pair c, d is treated similarly. The next formula is proved along the same lines (of. the proof of Proposition 1 below): (a, c,b,d) = 1 — (a, b; c, d).

(11)

So far, we considered throws consisting of four different points, hence, e.g. ^ ^ 0 and ^ = ^^^ make sense. If now C = 0, then a = c, and, interchanging c, d, relation (6) imphes 0 = (c, b; c, d) 1-^ (c, b; d, c) = oo. FormaUy setting 0^^ := oo the equality ^ = ^^^ remains vahd also in this case. Considering in the same way the case a = d we analogously see that oo^^ := 0 again yields the correct result. If, finally, a = d, the exchange of b and c together with (6) lead to oo = (d, b; c, d) i-^ (d, c; b, d) = oo. Setting 1 — oo := oo relation (11) also holds in this case. Summarizing we have the following. Proposition 1. Let (a, b; c, d) be a throw containing four different points of the n-dimensional projective space P " over the skew field K. Then, their cross ratio satisfies the relations (7) and (11). Moreover, (a, 6; c, d) = (c, d; a, b) = (6, a; d, c) = (d, c; b, a).

(12)

Relation (12) provides a unique way to define the CR also for arbitrary throws. Setting 0"^ := oo, oo"^ := 0, 1 - oo := oo, (13) rules (7), (11), (12) remain true for arbitrary throws. If S, €z {a,b;c,d) arbitrary CR, then

e, r\ 1 - e, 1 - r\ (i - 0'\ e(e -1)"'

is an

(w)

represent all the possible cross ratios for permutations of the four points in the throw. P r o o f . We first prove formula (11). Since b and c are interchanged, the second line in formula (8) now reads as

0 = b + 6C,

c = b + 6.

Inserting (10) into these equations, using the first line of (8), and comparing coefficients in the representations with respect to the basis (c, ti) of the vector space associated with the hue yields

1.4 Cross Ratio and Projective Maps a = /3 = 7 = -(5,

45

C = 1 - C,

and this is just claim (11). For a special throw with only three mutually different points, in all the three cases for a = b,c, d, it is straightforward to check (11) taking into account (13). Decomposing the permutation, 1234 34 12

(2 3)(3 4 ) ( 1 2 ) ( 2 3),

the first equation of formula (12) follows from (7) and (11). The two remaining equations immediately follow by applying (7) to both pairs of the CR in the resulting equation. If a throw contains only three mutually different points, then, by applying one of the permutations occurring in (12), we can always achieve a situation where the last three points are different from one another. The resulting CR is then correctly defined by extending equation (12) to this case, so t h a t (7), (11), and (12) hold for the CR of an arbitrary throw. Since, according to (12), for each of the CRs corresponding to the 24 throws obtained by permuting four points, there are three others with the same value, altogether at most six different values are possible. These can be obtained as the expressions mentioned in (14) by applying (7) and (11) once or twice. Of course, in particular cases even more of these values may coincide; in the case K = Z2, e.g., there are only three values at all: 0 , 1 , 00. D Exercise 1. Let P be a projective line over the skew field K, let ^ : P^ ^ K be an arbitrary projective scale on P^, and let bj e P ^ , j = 1 , . . . , 4, be four different points with scale values /3j := ^{bj) / 00. Prove the following formula for the CR: (61, 62; b3, 64) = < (/3i -/33)(/3i -/34)"'[(/32 -/33)(/32 -/34)

(15)

(Hint. Apply transformation formula (2.10) from Example 2.2.) Exercise 2. Let K he a field, let P^ he a projective line over K, and let (61, 62, ^3, ^4) be a throw in P^. Denote by {xj,yj) the homogeneous coordinates of the point bj with respect to an arbitrary projective frame ( a o , a i ; e ) of P , J = 1 , . . . ,4 (cf. Corollary 2.3). Prove the following expression for the CR in terms of determinants containing the homogeneous coordinates: Xl

(61,62:^3,64)

Xs

X2

X3

yi ya

2/2 J/3

Xl

X4

X2

yi

yi

2/2 J/4

=

(16)

X4

Formulas (15) and (16) are the classical expressions for the CR; they explain the name "cross ratio". T h e behavior of the CR under coUinear maps is described in the following. P r o p o s i t i o n 2. Let f : P" -^ <5™ be a coUinear map induced by the a-linear map g : F " + i ^ W""^ with a G Aut K, and let (a, b, c, d) a throw in P". Then either

46

1 Projective Geometry

Dim(/(a)V/(6)V/(c)V/(d))
f{d)) = a{{a, b; c, d)).

(17)

In particular, the CR is invariant under any collineation induced by a semilinear map with inner automorphism a = a^. P r o o f . Consider, say, the case c ^ d and H" = c V d. If /(c) = / ( d ) , then Dim/(H') < 1, and we are in the first situation. For /(c) ^ f{d) we know that H := f{H) = f{c) V / ( d ) is a hne, and, by Proposition 3.6, f\H is a colhneation. Hence {f (a), f (b), f (c), f (d)) again is a throw. Let U C V, tj C W be the two-dimensional subspaces associated with H and H, respectively. Then g{U) = U. Because of (12) we can suppose that b, c, d are mutually different, which then has to hold for their images as well. Choosing the basis (oo,ai) for U as in Definition 1, we have, using the notation introduced there,

0 = oo + aiC und gr(o) = ^(oo) + giai)a{^), since <; is a a-linear bijection. As the images of the base points correspond to the image of the throw, a{S,) corresponds to the CR of this image, proving (17). The last assertion follows from the fact that every inner automorphism leaves each conjugacy class invariant. • 1.4.3 Projective Maps The observations preceding the definition of the CR show that the CR is a fundamental notion for projective geometry, comparable to the ratio of segment lengths in afRne geometry and to the distance in Euclidean geometry. Indeed, in the following section on afRne geometry and in Sections 2.5, 2.6 on metric geometries we will see that both these notions can be based on the CR. Hence it should only be reasonable to expect a projective map to leave the CR invariant. Thus excluding the a-linear maps with an automorphism a that is not an inner one: Definition 2. A map / : P " -^ Q™ is called projective if it is colhnear and leaves the CR invariant. More precisely, for every throw {a,b,c,d) in P " , such that / ( a ) V f{b) V /(c) V / ( d ) is again a line, (/(a),/(b);/(c),/(d)) = (a,6;c,d).

(18) D

Corollary 3. The class of spaces over the skew field K together with the projective maps as morphisms forms a category. •

1.4 Cross Ratio and Projective Maps

47

P r o p o s i t i o n 4. Let K be afield, and let P^, Q " be n-dimensional projective spaces over K, n > 0. A map f : P " -^ Q " is a projective bijection induced by a linear isomorphism of the associated vector spaces if and only if it has the following properties:

1. / ( o ) = o; 2. / maps each throw again into a throw; 3. relation (18) holds for every throw in P " , i.e., f leaves the C R

invariant.

P r o o f . 1. and 2. immediately imply t h a t / is injective. To see this, consider a throw (6, b, c, d) in P " . Since the image of this throw also is a throw, the three image points have to be mutually different. For each point 6 G c V d, c 7^ d, we have f{b) G f{c)\/f{d) for the same reason. Hence f{c\/d) C f{c)\/ f{d). Let now ^ G -ft' be arbitrary and take d to be the point in / ( c ) V f{d) for which e=(a,/(6);/(c),/(d)). Since i f is a field, and as the projective scale is bijective, d is uniquely determined by ^ G if, and vice versa. T h e same is true for ^ = (a, 6; c, d) with a G c V d. Because of Properties 2 and 3, we thus have / ( a ) = d, and hence / ( c V d) = / ( c ) V / ( d ) . So / is colhnear, and by Proposition 3.6 it is a coUineation. For n = 1 the fact just proved means /([oo + aiC]) = [ao + aiC], where the bases (oo, a i ) , (oo, ai) of the vector spaces corresponding to P ^ , Q^ are chosen to determine the CR as in Definition 1. Thus it is clear t h a t / is induced by the linear m a p transforming (oo, fli) into (oo, &i). For n > 2, the Main Theorem implies t h a t / is induced by a a-linear bijection of the vector spaces. In the case of a field, the values of the CR belong to K. As there are no proper inner isomorphisms, because of Proposition 2, (17), condition 3 implies a = i d ^ , and hence / has to be a linear isomorphism. Conversely, it is clear t h a t each of the maps obtained this way has Properties 1-3. • Propositions 4 and 3.14 immediately imply the following: C o r o l l a r y 5. Let K be a field, and let P " , Q " be projective spaces over K. Then for every pair of projective frames [aj] e) of P " , [dj] e) of Q " , j = 1 , . . . , n, there exists a unique projective bijection f : P " -^ Q " such that f{aj) = dj, / ( e ) = e. D Exercise 3 . Let K he a field. Prove that assigning to each linear a : T^"+^ -^ |y™+i the projective map it induces defines a covariant functor the category of linear maps between finite-dimensional vector spaces over K the category of projective maps between finite-dimensional projective spaces K.

map from onto over

Exercise 4 . Let K he a skew field, and let P", Q™ be projective spaces over K. Consider throws (6j), (bj), j = 1,...,4, in P" and Q™, respectively. Prove that

48

1 Projective Geometry

there is a projective map / : P " -^ Q™ with f{bj) = 6j, j = 1,... , 4 if and only if (61,62:^3, 64) = (61, 62:^3, 64).

Proposition 6. Let K be a skew field satisfying the following condition: I. Each automorphism a G Ant K such that ) = < ^ > for all conjugacy classes < S, >C K is an inner automorphism. Then for every projective space P " , n >2, over K the group G of projective auto-collineations is isomorphic to the projective linear group PGL(n+ 1, K) (cf. Definition 2.1). This statement always holds, if K is a field or the skew field H of quaternions. P r o o f . According to the Main Theorem, every auto-coUineation / G A u t P " is induced by a a-linear bijection a of the associated vector space V . Since / is projective, a leaves each conjugacy class in K invariant, and so, by condition I, it is an inner automorphism of K. Choose a basis (Oj) of V"^^, « = 0 , . . . , n. With respect to this basis the map a inducing / has the representation where yU, G K* determines the associated inner automorphism a = a^. But then / is also induced by the linear automorphism of V"+ corresponding to the matrix (af/u): f{[aix']) = [aMiix% (19) Hence we already obtain all projective auto-collineations by restricting the homomorphism F from Proposition 3.13 to the subgroup GL{V^^ ) x {idx} of Gn+i{K); denote this restriction by F. Then by (3.33) we have KeiF = (KerF) n (GI,(F"+^) x {idx}) = {(id-^.+i M,idK)|M e Z{K*)} ^ Z{K*),

(20)

where Z{K*) denotes the center of K*. Since F is surjective, we deduce from the Homomorphism Theorem for groups (see Proposition 1.3.1.6) the claim, cf. Definition 2.1: G = GL{n + 1, K)/Z{n + 1) = PGL{n + 1, K).

(21)

In the case of a field. Property I is satisfied for trivial reasons; the skew field H of quaternions also has it, cf. Example II.8.8.5. • Nevertheless, there are skew fields not having Property I, cf. G. K O T H E [68], R. B A E R [5], § HI.4. For these skew fields the following definition is geometrically not compelling; it provides, however, a unified foundation for the further elaboration of projective geometry: Definition 3. Let P " be the projective space associated with the vector space V"+ over the skew field K. Then a projectivity or projective transformation of P " (or from P " to a projective space Q " over K) is always

1.4 Cross Ratio and Projective Maps

49

understood to be a collineation that can be induced by a linear isomorphism. The group of projectivities from P " to itself will be denoted by PZ((P") or, for short P i „ ; it is called the projective group of P " . • With this definition we now have the general relation PLn = GL{V"+^)/Z{GL{V"+^)) = PGL{n + 1, K).

(22)

This immediately implies: The projective group PLn acts transitively on the Grafimann manifolds Pn,k of projective k-planes in P " , hence in particular on the projective space P " , since the linear group acts transitively on the Gra£mann manifolds Gn^i^k+i of (k + l)-dimensional subspaces in the (n + l)-dimensional vector space V"^^. To see the latter, consider two subspaces U,W e Gn+i,k+i and choose bases ao, • • • ,Clk for U, b o , . . . , bfc for W, and complete both to obtain bases (Oj), (bj), i = 0,.. .n, for V. Then the linear transformation g G G L ( V " ^ ) defined by g{ai) = bj, « = 0,.. .n, satisfies gU = W. Propositions 4 and 6 can be considered as geometric characterizations of the projective group. In accord with the Erlanger Programm [65] by F. KLEIN we want to consider projective geometry as the geometry of the transformation group [PLn, P " ] . Since the group PGL{n+ 1, K) is the group of coordinate transformations of P " over K (cf. Definition 2.1), we again obtain the correspondence between geometric and coordinate transformations already described in § 1.5.7: The homogeneous coordinate systems of P " form an atlas adapted to the transformation group [PLn,P"] (cf. Definition 1.5.7.5 and Exercise 1.5.7.6). In the case of a field, the group PLIP"") is identical to the group of projective bijections of P " onto itself. For a skew field with Property I this holds by Proposition 6 for n > 2; in the case n = \ the group of projective bijections is larger, in general. E x a m p l e 1. The group PLi is isomorphic to the group GK of fractional linear transformations (2.10), cf. Exercise 2.4. This fact leads to models of projective lines P {K) as transformation groups [GK, K] that are simple and, in the cases if = R, C, H, easy to work with. D E x a m p l e 2. By Corollary 3.16, each collineation / : P " -^ Q " , n > 2, over the field if = R of real numbers is a projectivity. In this case, we thus are in the very convenient situation that each bijective map transforming lines again into lines also leaves the CR, and hence all projective properties, invariant. • Exercise 5. Prove that the only continuous automorphisms of the field C of complex numbers are the identity and the conjugation a : z t-^ z. Hint. Prove first that every continuous automorphism r of C has to satisfy T(R) = R, cf. Exercise 1.2.3.3. E x a m p l e 3. If if = C is the field of complex numbers, then there are infinitely many automorphisms. This follows from Proposition V.6.1 in the algebra book by N. BOURBAKI [19]. From the geometric point of view, it is natural

50

1 Projective Geometry

to confine oneself to the continuous collinear maps / : P " -^ Q™, n > 2; here, continuity refers to the topology introduced via inhomogeneous coordinates. If / is continuous, then its restriction to an arbitrary projective line also has to be continuous. Interpreting the C R occurring in (17) as projective scales, the arguments in the proof of Proposition 4 show t h a t ^{f{x)) = a{^{x)), or a = ^o f o^^^. Hence a is continuous. According to Exercise 5, a is either the identity — then the collineation / is a projectivity — or a is the conjugation. In this case, we caU the colhneation / an anti-projectivity, cf. also W . BuRAU [24], § III.20. In analogy to Proposition 4, anti-projectivities are characterized by the property (/(a),/(b);/(c),/(d)) = (a,6;c,d). In the sequel, we will only consider continuous collinear maps for complex scalars K = C. This correspondingly also applies to the automorphism group A u t P of the complex projective line. Using a projective scale z = z{x{^)) (cf. Example 2.2) to describe the group, we see t h a t the projectivities are represented by fractional linear transformations, f(z)

=

with ad — be =^ 0, cz + a

and the anti-projectivities correspond to fractional conjugate-linear transformations, f{z)

= -^ with ad — be =^ 0. cz -\- d D

E x a m p l e 4. The sole non-commutative skew field t h a t is important for us is the skew field H of quaternions'^ , cf. § 1.2.3. Since by Example II.8.8.5 or Exercise II.8.9.5 every automorphism of H is an inner one. Propositions 2 and 6 together with the Main Theorem imply: Each collineation f : P " -^ Q " , n > 2, between quaternionic projective spaces can be induced linearly; in addition, it preserves the CR. More generally, this holds for every collinear m a p / with D i m / ( P ^ ) > 2, cf. Corollary 3.11. D E x a m p l e 5. Exercise II.7.4.7 immediately implies the rather obvious isomorphy of centers Z ( G I , ( F " + ^ ) ) = Z{K*), by assigning to JJ, e Z{K*) the corresponding dilation d^i G GL{V^^ ) . Denoting as usual the special linear group by SL{m, K), i.e. the group of square matrices of order m with determinant one over the field K, then (22) and the computation of the kernel for the homomorphism p:ge

GL{n

+ 1, R ) i—> {det{g))

— ^ SL{n —

+ 1, R ) ,

n = 2ke

No,

An elementary introduction into the quaternions (and the octonions) can be found in I. L. KANTOR, A . S . SOLODOWNIKOW [58] and in many textbooks on algebra.

1.4 Cross Ratio and Projective Maps

51

show the relation PLn = SLin+l,^.),

n = 2ke'No.

Denote, moreover, as usual the unit matrix of order rn hy Im- In addition, let \SL\('m, R ) be the group of real square matrices of order m whose determinant has absolute value one. Then, for odd n, we analogously obtain PLn

= |^L|(n+ l,R)/{±/„+i},

n = 2k +

leN.

For K = C denote by Km the group of m-th roots of unity, of. Proposition 1.2.3.7. Dividing g e G I , ( F " + i ) by an (n + l ) - t h root of the norm N{g) and multiplying the result by i f „ + i we obtain PLn = SL{n + l,C)/Kn+i

{K=C).

Here, i f „ + i is embedded into SL{n + 1, C ) as the set of dilatations d^, e G Kn+l. • 1.4.4 H a r m o n i c P o s i t i o n In general, the six CRs corresponding to all permutations of the points in a throw are mutually different according t o (14). As already noted, this does not hold in the case (13) t h a t the throw contains two identical points. If the CR of a throw is (a, b;c,d) = —1, then the throw is called harmonic; this is also expressed by saying t h a t the pairs (a, b), (c, d) are in harmonic position. In this case, permuting the points of the throw we only obtain the values — 1, 2 , 1 / 2 by (14). T h e following exercise describes a geometric construction for the fourth point being in harmonic position with three given ones: Exercise 6. Let P be a projective plane over the skew field K. A complete quadrangle is understood to be any configuration consisting of four points, the vertices Ui e P^, i = 0 , . . . , 3, no three of which are collinear, together with their six connecting hues, also called the sides of the complete quadrangle, cf. Figure 1.8. The sides are arranged into three pairs of opposite sides, {ao V a i , 02 V 03}, {ao V 02, oi V 03}, {ao V as, ai V 02},

(23)

intersecting in the diagonal points of the complete quadrangle do

(ao V 03) A (ai V 02),

di

(ai Va3) A(ao Va2),

d = d2

(02 V 03) A (ao V a i ) .

(24)

Prove: a) do, di, d2 are not collinear if and only if char K ^ 2. - h) The connecting hues di \/ dj, i / j , of diagonal points do not pass through the vertices of the quadrangle. (Hint. Compute the homogeneous coordinates of di with respect to a basis for the projective frame ( a o , a i , a 2 ; e ) with e := as..) - c) Each side of the

52

1 Projective Geometry

Fig. 1.8. Harmonic Position.

complete quadrangle contains precisely one diagonal point, e.g. d := d2 G oo V ai. Denote by s the point of intersection of the side with the connecting line of the diagonal points not lying on it; e.g., for the side ao V a i this point of intersection is s = (ao V a i ) A (do V di). Prove that (s, d; ao, oi) = —1.

(25)

d) Consider a complete quadrangle in the real affine plane, in which the sides ao Vai, a2 V as are parallel and prove that s then cuts the segment ao, a i into halves. R e m a r k . T h e requirement t h a t there exists a complete quadrangle in P whose diagonal points are not coUinear, is known in synthetic projective geometry as the Fano Postulate, cf. R. B A E R [5], Ch. II. It imphes t h a t the characteristic of the synthetically constructed scalar domain is different from two. If the scalar domain has characteristic two, then the relation 1 = — 1 together with (25) imply s = d; harmonic position thus is of no interest in this case. Exercise 7. Let char A" / 2. Using the notations of Exercise 6 prove that the diagonal points (do, di) together with the intersection points p, q of the lines do Vdi with the sides of the complete quadrangle not passing through do, di, p := (do V di) A (ao V a i ) ,

q := (do V di) A (a2 V as),

form a harmonic throw: (p, q; do, di) = —1. Exercise 8. Permuting the points in a throw results in at most six different CRs by Proposition 1. Prove that the number of values ^ of the CR under the permutations of the points in a throw is less than six precisely in one of the following cases:

1.4 Cross Ratio and Projective Maps

53

a) 5 = 1,0,00;

(26)

6)C = - 1 , 2, i ;

(27)

c) 5 is a root of the equation x — a; + 1 = 0.

(28)

In the cases a) and b) the specified values are attained. For char K = 2, a) and b) coincide. If A" = R, then only the cases a) and b) are possible. For A" = C, in the situation of case c), the throw is called equi-anharmonic; in this case, only the two values C=(l±iV^)/2 (29) can occur. Prove by means of Exercise II.8.9.5 that for the quaternions, K = H, all solutions of (28) are conjugate; there are infinitely many of them. So the value < C > of the CR just consists of a single conjugacy class. Exercise 9. As in Example 1.3 we identify the complex projective line with the Riemann sphere: P^{C) = C. Prove: If zo,zi,Z2 are the vertices of an equilateral triangle in the complex number plane C, and m is its center, then (zo, zi, Z2,m) and (zo, zi,Z2, 00) are equi-anharmonic throws. 1.4.5 S t a u d t ' s M a i n T h e o r e m For the projective geometry over a field K we are now going to prove the following result due to K. G. C H R . V . S T A U D T (1798-1867) t h a t completes the Main Theorem of Projective Geometry and is therefore sometimes called Staudt's Main Theorem: P r o p o s i t i o n 7. Let P he the projective line over a field K with characteristic char i f ^2. A surjective map f '• P ^ P is a projective automorphism if and only if it preserves harmonic position. Thus, for the real projective line f is a projectivity. P r o o f . Obviously, the condition is necessary. To prove the converse, we choose a projective scale ^ in P and identify the points x e P with the correspondingly labeUed scalars x = ^{x) G K. Since any pair of different points on the line can always be completed to a harmonic throw by Exercise 6, and because of char K y^ 2, the points of a harmonic throw are mutually different, the m a p / is bijective, and /(O), / ( I ) , / ( o o ) are three different points. Let a denote the projectivity satisfying a ( / ( 0 ) ) = 0, a ( / ( l ) ) = l, a ( / ( o o ) ) = oo. By Corollary 5 it is uniquely determined. Then, <; := a o / is a m a p from P to itself preserving harmonic position with fixed points 0 = g(0), 1 = g{l), 00 = g{oo). Now the proof is finished once we have shown t h a t g\K is an automorphism of the field K, since then / = a^^ o gr is a projective automorphism on P^. In the case i f = R this has to be a projectivity according to Exercise 1.2.1.3 (note also the hint foUowing Corollary 3.16), because oi g = idR.

54

1 Projective Geometry Relation (16) implies t h a t for all x,y ^ K, x ^ y: (x, y- (x + y ) / 2 , oo) = {g{x), g{y)- g{{x + y)/2),

oo) = - 1 ,

(30)

hence giix + y)/2) = igix) + giy))/2,

x^y.

(31)

Setting here y = 0 we conclude from g{Q) = 0 g{x/2)

= g{x)/2,

x e K,

(32)

and from x = 2y t h a t g{2y) = 2g{y),

y e K.

(33)

(31) and (33) imply g{x + y) = g{x) + g{y),

x,y e K.

(34)

Once again from (16) we obtain {-z,z;l,z^)

= {g{-z),g{zy,l,g{z^))

= -l,

(z e K).

(35)

Since (34) immediately implies g{—x) = —g(x), we deduce from (35) t h a t giz^)=gizf,

zeK.

(36)

Inserting z = x + y into this last equation and applying (33) and (34) yields

g{x-y) = g{x)- g{y),

x,y e K.

(37)

Since / and a are bijective, t h e same holds true for g; hence, by (34) and (37) the m a p g is an automorphism of the field K. D For i f = R t h e assertion can be proved without assuming the surjectivity of / . This is a consequence of two facts: The continuity of each endomorphism of R , and the density of the rational numbers in R . R e m a r k . T h e proof of Proposition 7 presented here can also be found in E . S P E R N E R [98], Bd. II, or W . K L I N G E N B E R G [67]. A more general result for a skew field K with char i f =^ 2 contains E . A R T I N [4], Theorem 2.25, cf. also R. B A E R [5], § III.4. Exercise 10. Let P^ be a projective line over the field K, char if / 2, and let, in addition, f : P -^ P be a surjective map preserving harmonic position and leaving the point oo e if = P^ fixed. Here we identify if with P^ as in the proof of Proposition 7. Prove that there exist a G Aut if, a e if* and /3 G if such that fix) = aa{x) +l3ior xeK

= P^\ {oo}.

Exercise 1 1 . Let I n t i f be the group of inner automorphisms of the skew field if. Then Int if is a normal subgroup of the group Aut if of automorphisms of if. Prove that for every projective space P", n > 2, over if the group PLn is a normal subgroup in A u t P " , and that the following quotients are isomorphic: Aut P"/PL„

-^ Aut if/ Int if.

1.4 Cross Ratio and Projective Maps

55

1.4.6 P r o j e c t i v e E q u i v a l e n c e of C o U i n e a r M a p s . I n v o l u t i o n s We conclude this section by some considerations concerning the of coUinear maps. They wiU be based on the fohowing.

classification

D e f i n i t i o n 4. Let P " , Q" be projective spaces over the skew field K. Two maps / i : P " -^ P " , /2 : <5" -^ Q " ^^^ called projectively equivalent, if there exists a projectivity g : P " -^ Q" such t h a t f2 = 9 ° fi° 9^^^ It is obvious t h a t projectively equivalent coUinear maps have to satisfy the condition Dim/i(P:) = Dim/2(g:); if the fj are induced by the o-j-linear maps a^, then the 0 projective subspaces Ai C P", where a) Ai A \J Aj = o, r

b) y^(DimAi + 1) < n + 1. i=l

In the case of a projective line, any projectivity / / id p i can thus have at most r < 2 fixed points; accordingly, the map / is called elliptic, parabolic or hyperbolic if r = 0,1, or 2, respectively. Exercise 1 3 . Let P^ be a projective line over the field K, char A" / 2. Prove: a) If there are two points a,b & P^, a ^ b, that are transformed into each other by the projectivity / , / ( a ) = b and f{b) = a, then / is involutive, i.e. /^ = i d p i . - b) If / e PLi, f / i d p i , is involutive, then / either has two difi'erent or no fixed point at all. If a, b are two different fixed points of the involution f, then we have (a, b; x,f{x))

= -1

(a: / a, b).

(38)

c) Let a,a',b,b' e P^ be four different points. Prove that there is a unique involutive projectivity with / ( a ) = a' and f{b) = b'. Exercise 14. Again consider the situation of Exercise 13. Prove that every projectivity / e PLi can be represented as the product of two involutions.

56

1 Projective Geometry

Now we want to describe the involutive

coUineations f G A u t P " , / ^ =

idpn.

Exercise 1 5 . Let P" ,n > 2, be a (right) projective space over the skew field K. Consider an involutive and fixed-point fi-ee / G A u t P " : /

= idpn,

/ ( * ) / X for all x e P",

Prove that n is odd, and / can be induced by a cr-linear map a of the associated vector space V^''\ 2k = n + 1, which has the following properties: a) There is A e K* such that a^ = d\; here the following relations hold (7(A) = A and a = a^ ,

(39)

where cxiS,) = A^A^ denotes the inner automorphism of K corresponding to A. b) There is a basis (o^, Otk+K)K=i,...,k of V'^^ such that, in this basis, a has the following representation:

a{ar.) = ak+r., a{ak+r.) = ar.\.

(40)

c) There is no /it G K* with X = IJ,a{iJ,).

(41)

Conversely, each collineation induced this way is involutive and fixed-point free. Classify all these coUineations in the cases K = H and K = C, a the identity or the conjugation of C. (Hint. Note that p / 0 and a(p) are always finearly independent, since f{x) / a: = [p]. Moreover, by /^ = i d p n the subspace W C V they span is invariant under a.) T h e classification result, whose proof we outhned in Exercise 15, may be summarized as follows: In the case K = H, the fixed-point free involutions f G Aut P " are projectively equivalent to the normal form (40) with A = —1, a = idR. Thus, in homogeneous coordinates, / can be represented in the form k

/ ( [ ^ ( o « x « + ak+.x^+^)])

k

= [ ^ ( - o « x ' ^ + « + Ofc+«x«)],

(42)

i.e. / is the simultaneous "rotation through 7r/2" in all the two-dimensional subspaces W^ := £ ( 0 K , afc+fj), "- = 2A;. In the case K = C, the automorphism a has to be conjugation, and up to projective equivalence all fixed-point free involutions are of the form k

k

/([^(o«x« + ak+.x'^+n]) = [J2(-a.x''^^ n=l

+ ak+.x^)].

(43)

n= l

For K = C, a = i d c , and in the case of the quaternions K = ii there are no fixed-point free involutions at all. This is a consequence of the following. Exercise 16. Prove: a) If K is an algebraically closed field, then each projectivity / G PLn has at least one fixed point. - b) Over the skew field of quaternions H, each collineation / G A u t P " , n > 2, has at least one fixed point.

1.4 Cross Ratio and Projective Maps

57

Exercise 17. Let K he a skew field, char A" / 2, let P" be a (right) projective space over K, n > 2, and let / G Aut P " be an involutive collineation having at least one fixed point. Prove that under these conditions / even has at least n + 1 fixed points in general position. More precisely, prove that there is a basis (Oi), i = 0,.. . ,n, of the associated vector space V such that a cr-linear map a inducing / has diagonal form in (cu), and a{ai) = OiPi, Pi(T{Pi) = l,a^=

idy, a^ = idx .

(44)

(Hint, a^ = d\ and a{b) = b/x together imply IJ,IJ{IJ) = A. Consider, in addition, the map a' := d^ o a also generating / , and prove that the maximal subspace of V spanned by the eigenvectors of a' is all of V.) Starting from (44) and specializing K as well as a, it is often possible to obtain a complete classification of t h e involutions with fixed points. Below we will quote some of the related results, b u t leave the proofs to the reader. E x a m p l e 6. Under the assumptions of Exercise 17 let, in particular, K be a field a n d a = i d ^ - Then, with a suitable numbering of t h e basis elements, (44) imphes t h a t for each of these involutions / ^ i d p n there exists a number A ; G N o , 0 < A ; < n / 2 , such t h a t , for an appropriate choice of the basis, a{o.a) =-0.a, 0(0^) = OK,

a = Q,...,k, K = A; + 1 , .. . , n .

,

,

Denoting by aj := [Oj] t h e points corresponding to the basis elements, the A;-plane A := ao V . . . V a^ as well as its complementary [n — k — l)-plane B := Uk+i V . . . V a „ are point-wise fixed; AU B is the set of fixed points of / . T h e m a p / is called a reflection of P " with respect to the pair of complementary subspaces A, B. Obviously, for every such pair of subspaces there is precisely one reflection, and two reflections of this kind are projectively equivalent if and only if the dimensions k,n — k — I are the same. Choosing one of t h e spaces, say B, as a subspace of the hyperplane at infinity H of an afRne space A " = P " \ H results in an affine reflection in a A;-plane A in the direction of t h e complementary space B; compare this also with the results of t h e subsequent section. D E x a m p l e 7. In t h e case K = C, apart from t h e cases already treated in Example 6, we now consider the involutive anti-projectivities J with t h e conjugation a = T : z 1-^ z as inducing automorphism having, in addition, a fixed point. According to Exercise 17 then (44) holds, i.e. /3j/?j = |/3j|^ = 1, and hence f3j = e' '-^^. Transforming t h e basis according to b,- = 0jei'^^'/2,

^- = 0,

,n.

we arrive at bj = a(bj); hence all of these involutions are projectively equivalent. Moreover, in an adapted basis (bj) as just described, they aU have the normal form

58

1 Projective Geometry n

n

J{[J2b,x^]) = [J2b,x^]. 3=0

(46)

3= 0

So the homogeneous coordinates {TJ) of the fixed points satisfy TJ = ^^z, where ^^ G R and z G C*. Obviously, they form a real projective space P " ( R ) C P " ( C ) , for which the points bj = [bj],j = 0 , . . . , n, and e = [J2j bj] form a projective frame. This sequence of points is a projective frame for the complex space P " ( C ) as weU. For this reason, a n involution J with normal form (46) is caUed a real structure on P " ( C ) , cf. Definition II.8.9.5; the set P " ( R ) C P " ( C ) of fixed points of J is caUed a Staudt chain. Consequently, all real structures, and hence all Staudt chains of a given dimension are projectively equivalent. • E x a m p l e 8. Let now K = ii denote the skew field of quaternions. In order to classify the involutions of P " ( H ) we could again start from (44). Instead, we prefer to make use of a fact already mentioned in Example 4: Each collineation can be induced by a linear m a p . According t o Exercises 16 and 17 the involution / has a t least n + 1 fixed points in general position. Thus we find a basis (o;) of the associated vector space V as weU as a linear m a p a inducing / t h a t is in diagonal form with respect t o this basis: a{ai) = aiqi,

qi G H .

(47)

According t o (22), involutivity means t h a t a? = d\ has t o b e a dilation by A G .^(H) = R . This again implies: ?;2 = A G R * ,

/ = 0, . . . , n .

(48)

Turning to the equivalent representation of / by a!:=d^oa,

/x : = |A|^2

(49)

we can assume without loss of generality t h a t A = ± 1 . Recalhng how the multiphcation is defined in H , cf. 11.(8.9.4), it is easy t o see t h a t in H the equation q^ = \ only has the solutions
(50)

Looking a t the basis bi := aipi, / = 0 , . . . , n, we immediately see t h a t a{bi) = aiqipi = bii.

(51)

1.5 AfEne Geometry from the Projective Viewpoint

59

Hence any involution of P " ( H ) , n > 2, is either a reflection or a projectivity J , which by (51) has the normal form

Ji[J2b,x']) = [J2b,Ax'].

(52)

Note t h a t , according to Exercises 15-17 and the subsequent examples for the cases i f = R , H , the involutive colhneations, and for i f = C the continuous involutive coUineations are completely classified. • Exercise 18. Find an example showing that the assertion in Exercise 17 is wrong without the assumption char if / 2. Exercise 19. Let A, B be complementary projective subspaces in P " , both different from o, and consider xi, y^ e P"\{AUB), {xi\/y-^^)AA = ai, (xi\/y{}/\B = bi. a) Prove that there is a unique projectivity / G PLn with the properties f\A = idj^, f\B = i d j j , and f{xi) = y^. - b) For n > 2, find a geometric way to construct fix) for any point x e P " \ (A U B). (Hint. Recall Exercise 2.1.) For n = 2, such maps are also called central coUineations, cf. J. BOHM et al [18], Bd. II. - c) Prove that / is involutive if and only if (a:i, j/j^, a i , 6i) is a harmonic throw.

1.5 AfRne Geometry from t h e Projective Viewpoint In Section 1 we started from the afhne plane and obtained the projective plane by adjoining a "line at infinity". This idea motivated the definition of the projective geometry ^ " as the lattice of subspaces of an (n + l)-dimensional vector space V . This way we avoided to pursue the rather awkward process of extending the n-dimensional afhne geometry towards the projective one. Now we want to proceed the other way round: AfRne geometry is shown to arise from the projective one by distinguishing a projective hyperplane H in J-*". All affine notions are derived from the projective ones by relating the latter to this distinguished hyperplane. For simplicity, we will use the symbols introduced in the course of the axiomatic approach to affine geometry in Chapters 1.4, 1.5 to denote the corresponding projective notions as well. This should not lead to any confusion. Note also t h a t this time we wiU be more general t h a n there, since now the scalar domain may be a skew field. D e f i n i t i o n 1. Let P " denote the projective space associated with the (n + l)-dimensional right vector space V " + ^ over the skew field K, and let H C P " be a fixed hyperplane in P " . Then for n > 1 its complement A " := P " \ H" is called an n-dimensional affine space over K. For n = 0, — 1 we set A ° := P ° , A^^ := 0. In the case n > 1, H" is called the hyperplane at infinity of the affine space A " ; it is also called the absolute hyperplane, the ideal hyperplane, or the improper hyperplane. •

60

1 Projective Geometry

Proposition 3.14 immediately implies that the projective linear group acts transitively on the set of hyperplanes: Just assign a frame to each hyperplane in a suitable way and consider the case a = id^- Hence it does not matter which particular hyperplane is distinguished. To say it lies "at infinity" is suggested by the geometric interpretation described in Section 1. The fundamental property special to afRne and not belonging to projective geometry is parallelism: Definition 2. In the notation of Definition 1, let B be a projective A;-plane, A; > 0. If B'^ C H, then B^ is called improper, and otherwise proper. For every proper A;-plane we have B'' r\ A^ ^ 0; so M'' := B'' n A " is called the affi,ne k-plane corresponding to B . Let B , C he proper projective subspaces of P " , whose dimensions satisfy 1 < / < A;. Denote, moreover, the affine /-plane corresponding to C ' . £)' is said to be parallel to M'' (or C ' parallel to B'', respectively) if

Hn&cHnB''.

(1)

Starting, conversely, from an affine A;-plane M , the proper projective A;-plane B to be defined is uniquely determined; it is called the projective closure of tv

— k

M and will be denoted by M . I t is thus the projective subspace for which M'^ = M ' ' n A " . D The dimension formula (Proposition 1.1) yields the following. Corollary 1. For every proper projective plane B BimH

AB'' = k - l .

(2)

In particular, a proper k-plane B , k > I, is parallel to a proper hyperplane C " - i if and only if B'' C C " - ^ or B'' A C " - ^ CH. D Definition 2 is in accordance with the idea that two different lines in the plane (or, more generally, hyperplanes in P " ) are parallel if they intersect "at infinity". It can be shown (cf. Exercise 4 c ) ) , that the A;-dimensional vector space associated with an affine A;-plane M'' = B'^HA" (cf. Proposition 1.4.5.1) is precisely the vector space UnW C V"^^; here U and W denote the vector subspaces determining B'' and H, respectively. Hence Definition 2 exactly corresponds to the Definition 1.4.5.3 of parallelism in affine geometry. Since parallelism is to be an afRnely invariant property, we can only permit those projectivities as affine transformations that map H (and hence also A") into itself: The affine group 2l(n) = 2l(A") is thus defined to be the stationary subgroup of H: 2l(n) := {/ G PLn\f{H) = H}. (3) In order to obtain the coordinate representation of affine transformations in its usual form (cf. 1.(5.7.46)) we first have to grasp the notion of an affine

1.5 AfEne Geometry from the Projective Viewpoint

61

coordinate system from the projective point of view. A basis (Oj) of the vector space V or a projective frame (OJ; e) wih be called affinely adapted if H = aiV ...Van

(ai = [Oi]),

(4)

i.e., if the points Oj, « = 1 , . . . ,n, are improper. The only proper base point ao = [oo] is called the origin of the affine coordinate system. The afhne space A " is then just the domain of definition UQ for the zero chart cpo that is defined by x° 7^ 0 and determined by the inhomogeneous coordinate system belonging to the homogeneous coordinate system for the affinely adapted basis (cf. Lemma 2.6). Hence we will consider the inhomogeneous coordinates defined by (2.8) for « = 0 as the affine coordinates of a point x G A " with respect to the affinely adapted basis Oo, • • •, On: For each x G A " there is precisely one n-tuple (x^) G if", j = 1,... ,n, such that

Mx) = (x^) e K",

(5)

n

hence a; = [oo + Y J

ttjxJ].

(6)

j=i

In particular, the unit points [oo + flj] on the axes ao + [Oj] coincide with those of the affine geometry, cf. Fig. 1.7 (for n = 2). Now let (Oj), « = 0 , . . . , n, be a fixed affinely adapted frame in P " , and let / G PLn; take a linear map bf G GL{V"+^) inducing / . Obviously,

f em{n) ^=>bf{aj)eQiai,...,a„),

j = l,...,n.

(7)

Denote by LA{n + 1, K) C GL{n + 1, K) the set of block matrices of the form '/3oo o'' ^ ^ ,, /3oo € if *, b G if", B G GL{n, K). (8) Then (cf. (4.22)) ^{n)^LA{n+l,K)/Z{n+l).

(9)

Here Z{n + 1) denotes the center of GL{n + 1, if), i.e. the group of diagonal matrices /„+iyU,, yU, from the center Z{K*) of the multiplicative group of if. If if is a field, then we normalize the matrices via multiplication by /Jgg to have the form {b,B):=[l°^\

beK^\Be

GL{n, if).

(10)

This immediately implies that the recent definition is compatible with the earlier Definition 1.5.1.2 of the affine group; since (10) is the coordinate matrix of the affine transformation / , using the normalization x" = 1, y" = 1 we obtain for the coordinates of the image point y = f{x):

J i

\ RO RJ I \ -r.J I '

^^^•^

62

1 Projective Geometry

thus n

y^ =/3^+ J2/3fx\

j = l,...,n.

(12)

Up to notation, this is the coordinate representation 1.(5.7.42) for the affine map / . Exercise 1. Let A" be a field. Taking into account definition (10), prove the niultiphcation rule in 2l(n,-R'): ( b , B ) - ( c , C ) = (b + B c , B - C ) .

(13)

Calculate the expression (p, X) = {b, B)^^, and show that 2l(n, K) is the semi-direct product of the normal subgroup {{b,I)\beK"}^[K",+],

I = {Si),

(14)

with the subgroup {{o,B)\B

e GL{n, K)} ^ GL{n, K)

(15)

(cf. § 1.5.3). Exercise 2. Consider the affine space A " over the field K. a) Let g^ be a proper line, Coo := g^ A H, and denote by t the affine parameter on the affine line g^ Pi A " with zero point co and unit point ci / CQ. Prove the relation X = co+ coci t •<=^ t = (x, a; Co, Coo). Thus the affine scales on g^ are just the projective scales assigning the value oo to the improper point g^ A H. Moreover, the CR t is equal to the ratio of the parallel vectors CQX and coci. - b) Let char A" / 2. Prove for three coUinear points co,ci,a: e
1.5 AfEne Geometry from the Projective Viewpoint

63

form an n-dimensional vector space over K.li f ^ id^ri is a translation, then all lines x V f{x) pass through the same point of H, and we have {x\/f{x))AH=[b],

(16)

if y = a; + b is the affine representation of / (cf. I. (4.5.3)). According to (16), translations by proportional vectors b, bA, A G if*, determine the same improper point. Hence the points of the improper hyperplane can be interpreted as directions in the affine geometry, i.e. equivalence classes of parallel lines. Viewing translations as vectors it is easy to verify the axioms (I), (II), (III) of affine geometry (§ 1.4.3). This completes the projective incorporation of affine geometry. Exercise 4 . Let (cu), i = 0, . . . , n , be an affinely adapted frame; then W" := -C(Oi,... , On) C F"+^ defines the absolute of the affine space A", a) Prove that [A", W,_ftr] satisfies axioms (I), (II), (III) of Definition 1.4.3.1 for an affine geometry, if the action of W" on A " is as follows:

tf, :« = [?] e A"^^tf,(a:):=[j;+b] e A",

beW".

- b) Each it| with b / 0 is a translation in the sense of Exercise 3, i.e., it, = / | A " is the restriction of a projectivity / on P " with f\H = idjj that has no fixed points in A". - c) If M'' = A"n B^ is an affine fc-plane, and C/'=+^ C F " + i denotes the vector subspace associated with B^, then the set of vectors corresponding to the affine fc-plane M (cf. Proposition 1.4.5.1) is V ( M ' = ) = {b e W"|if,(M'=) = M ' ^ } = W " n C7'=+\ Exercise 5. Find an example of two affinely skew planes M'^ , B^ G A^ C P* in the four-dimensional affine space over a field K whose defining projective planes intersect. (Remark: "affinely skew" means that the affine planes are disjoint and not parallel, cf. § 1.4.6.) Exercise 6. Prove (in the notations of Definition 2): If 7) is parallel to M _D' n M'^ / 0, then 7)' C M'=.

and

To conclude this section we want to consider ajfine coUineations, i.e. bijecfive maps / : A " -^ A between affine spaces over skew fields, not necessarily isomorphic, mapping hues into hues satisfying f{x\/y) = f{x)\/f{y). Here the operation V denotes the restriction of the corresponding projective operation to the affine subspaces; similarly for A: £>' A M'' := {D^ A M'') n A " ,

D'

V M ' ^ := {D^ V M'')

D

A".

(17)

We will prove t h a t each affine collineation is the restriction of a uniquely determined colhneation between the associated projective spaces. Using the Main Theorem of Projective Geometry we can draw some interesting conclusions from this fact.

64

1 Projective Geometry

Lemma 2. Let f : A ^ A be an affine coUineation between affine spaces A " C P " , A C P over the skew fields K, K, char if ^ 2, char if ^ 2. Then there is a unique coUineation f : P " -^ P such that f = / | P " . P r o o f . For n = 0,1 the statement is triviaL Let now n > 2. If / is a coUineation extending / , then we obviously have f{H) = H for the improper hyperplanes, and for every affine subspace D C A " the relation fiDAH)=fiD)AH

(18)

has to hold. This allows the following construction for the image f{x) of any improper point x G H: Choose an affine line h in the direction of x, i.e. satisfying x = h A H, and set f{x) := f{h) A H. To justify this definition one has to show that the image f{x) does not depend on the choice of the affine line h with improper point x. In other words, the following lemma has to hold: Lemma 3. Under the assumptions of Lemma 2, every affine coUineation f : A ^ A maps pairs of parallel lines into pairs of parallel lines. P r o o f . Since the closures hi, 1x2 of paraUel hues, hi 7^/12, intersect in an improper point x G H, they span a plane, i.e., they are not skew. Consider a parallelogram a, b, c, d with a V d = /ii, 6 V c = /i2, whose sides a\/ b,c\/ d are paraUel, too. The diagonals a\/ c,b\/ d then intersect in a proper point s = (a V d) A (6 V c) G A". Of course, this intersection point lies in the plane spanned by the parallels. This plane intersects the improper hyperplane H in the line spanned by the diagonal points of the complete quadrangle a, 6, c, d corresponding to the pairs of paraUel sides of the parallelogram. Because of char if ^ 2, the diagonal points are not coUinear, cf. Exercise 4.6, and this implies s G A". Since the affine coUineation / is bijective, the image hues intersect in the proper point / ( s ) = / ( a V d) A f{b V c). So the image points / ( a ) , /(6), / ( c ) , / ( d ) , and hence also the associated image lines f{hi) = f{a V 6), /(/i2) = / ( c V d) lie in a plane. As they are disjoint due to the bijectivity of / , they have to be paraUel. • Now we complete the proof of Lemma 2. Since the map / is uniquely determined according to Lemma 3, all we have to show is that it also maps improper lines into improper lines. So let x,y, z £ H he three different coUinear points. Consider a projective plane M intersecting H in the line a; V y. In this plane we choose three lines /ii, /i2, /i3 such that x = H Ahi, y = H Ah2, z = H A hs. These lines determine a proper triangle in the plane M . The image points / ( a ) , /(6), / ( c ) of the vertices a, b, c in this triangle determine ^—2

a plane M , and for the images of the improper points we have fix) = f{hi) n H, f{y) = f{h2) n H, f{z) = f{hs) n H.

1.6 Duality Because of f{hi)

C M

65

, « = 1, 2, 3, these points lie on the line of intersection

M A H, hence they are collinear. The bijectivity of / follows directly from the bijectivity of / . Hence / is a coUineation. D T h e Main Theorem of Projective Geometry immediately leads to a series of coroUaries: C o r o l l a r y 4. Let m > 2. Under the assumptions of Lemma 2, the dimensions of the affine spaces coincide, n = m, and there is an isomorphism a : K ^ K such that the affine coUineation f has a basis representation of the form n

fix)

n

= f{a + J2 l i ^ ' ) = / ( a ) + E

''i^(^')-

If K is a field, and f preserves the affine ratio of parallel segments, an affine bijection with K = K and a = i d ^ .

(19) then f is

P r o o f . By Corollary 3.9 and Lemma 2 we have n = m. Hence n > 2, and we may apply the Main Theorem of Projective Geometry. For affinely adapted frames we obtain representation (18) starting from the basis representation of the (T-linear m a p inducing / . By Exercise 2, the segment ratio is a special case of the cross ratio, and this implies K = K as well as a = i d x • C o r o l l a r y 5. For n > 2 and a field K with char i f ^ 2, the affine group 2l(n) is the group of affine coUineations for A " leaving the ratio of parallel segments invariant. • For a later application, we point out the real case i f = R . There is only the trivial isomorphism a = idpt, and this leads to a particularly simple description of the affine group: P r o p o s i t i o n 6. The affine group 2l(n) of the real affine space A " , n > 2, consists of all bijective maps from A " to itself mapping lines into lines. • Exercise 7. Using the Main Theorem of Projective Geometry prove that the isometry group^ of the Euchdean point space E", n > 2, is the Euchdean group. (A different proof of this statement can be found in Exercise 1.6.2.13.)

1.6 D u a l i t y In projective geometry, there is a fundamental conceptual symmetry under which the notion pairs join and meet, points and hyperplanes, or, more generally, A;-planes and (n—A;—l)-planes, correspond to one another. This symmetry. ^ This is understood to be the set of all bijective maps from a metric space to itself preserving distance. Obviously, this is a group with respect to composition, see also Section 2.5.6 below.

66

1 Projective Geometry

called duality, allows to produce new geometric results without new proofs, just by formally replacing each notion by its dual counterpart. It reflects the duality of vectors and linear forms, the covectors, cf. § 1.5.6; we will apply this relation to study the duality for projective geometries of arbitrary dimension, see 1.6.2 below. Let us start with the simplest interesting case: 1.6.1 D u a l i t y in P l a n e I n c i d e n c e G e o m e t r y In plane incidence geometry (cf. Example 1.4) there are two basic sets, the set of points and the set of lines, together with a relation, incidence, between points and lines. Performing the dual replacements point I—> line, line I—> point, incidence i—> incidence, in the axiom system P. 1-3, Axioms P . l and P. 2 of plane incidence geometry are transformed one into the other: P . l I—> P.2,

P.2 I—> P . l ,

whereas P.3 changes into a new statement P.3': P.3'. There are four lines no three of which pass through the same point. Statement P . 3 ' can, however, be proved using Axioms P. 1-3, cf. Exercise 1.7. Adding P . 3 ' as a (superfluous) axiom to the axioms P.1-3 leads to an axiom system of plane incidence geometry t h a t is transformed into itself by the dual replacements. This conceptual symmetry is called the duality of the projective plane. Points and lines form a mutually dual pair, and the incidence is a notion t h a t is dual to itself^. Since all deflnitions and propositions of plane incidence geometry are exclusively based on the three fundamental notions mentioned together with the quoted self-dual axiom system, dual replacement assigns a corresponding dual notion to every derived notion. Substituting the dual notion for each notion occurring in a proposition (i.e. a proved statement) again produces a valid statement, a proof of which could just proceed by replacing the notions in the original proof of the flrst one by the dual ones. Hence there is no need to explicate it. In special cases, the dual notion or the dual statement may, of course, coincide with the original one, cf. Exercise 1. T h e formal rule t h a t for any true proposition in plane incidence geometry there is a dual one also holding is called the duality principle for the projective plane. In fact, since in the algebraically deflned plane projective geometry over a skew field K the Axioms P.1-3 can be derived as propositions, the plane duality principle can be applied in this case as well. In the subsequent In fact, incidence is defined as including or to be included, cf. Definition 1.1.1, and, from the point of view of lattice theory, to contain and to be contained form a dual pair, see also Equation (2) below.

1.6 Duality

67

Section 1.6.2 we will discuss the n-dimensional case, t h a t will establish the plane duality principle once again, this time as a special case. T h e book by L. H E F F T E R [48] commences with a self-dual axiom system for the projective geometry of three-dimensional space, from which the duality principle for this geometry follows. Exercise 1. Prove that, in the projective plane P over a skew field K, the figure in DESARGUES' Theorem (Exercise 1.9, Fig. 1.5) is self-dual. Starting from this, conclude the following self-dual proposition: Let (01,02,03), (61,62,^3) be two triangles, i.e. triples of non-collinear points, and let A i = 02 V 0 3 , A2 = 03 V o i , A3 = o i V 02

B i = 62 V 63, B2 = b3 V 61, B3 = bi V 62 be the corresponding trilaterals, i.e. triples of non-concentric lines. Then the following holds: The connecting lines of corresponding points, oi V 61, 02 V 62, 03 V 63, are concentric if and only if the intersection points of corresponding lines, Ai A Bi, A2 A B2, A3 A B3, are collinear. Exercise 2. Dualize PAPPOS' Theorem (Exercise 1.10), and sketch the corresponding situation. Exercise 3 . Let P^ be a projective plane over a skew field K, char A" / 2. Dualize the notion of a complete quadrangle (Exercise 4.6) to define a complete quadrilateral. As the support of the point series the line dually corresponds to the point as the support of the bne pencil, i.e. the set of all lines of the plane passing through that point. According to Example 1 one obtains the notions of cross ratio and harmonic position for lines in the pencil by dualizing the corresponding notions defined for the point series. Find a construction for the fourth line that is harmonic to three concentric hues by duabzing Exercise 4.7. 1.6.2 P r o j e c t i v e a n d A l g e b r a i c D u a l i t y Introducing projective geometry by starting from linear algebra, the geometric duality results from the algebraic duality between vectors and covectors. In this section, we will only consider finite-dimensional vector spaces V over a skew field K. As in the commutative case, the dual V is defined as the set of linear maps u : V ^ K; these are also called linear forms or covectors. Defining addition and multiplication by scalars argumentwise, the dual again is a vector space over K of the same dimension. Note t h a t the dual of a right (left) vector space is a left (right) vector space (Proposition II.7.3.7); in fact, in the non-commutative case linearity of the form can be ensured only if scalar multiplication takes place from the opposite side: /iu(j;a + t)/3) = /iu(j;)a + /iu(t))/3,

/x, a, /3 G if, p, t) G V , u G

V.

As in the commutative case, the dual of the dual space, (V)', again is a vector space t h a t is canonically isomorphic to the original space V. As a rule, both

68

1 Projective Geometry

spaces are identified via the canonical isomorphism: (V')' = V. This can be described most easily using the symmetric notation for the scalar product of a vector and a covector: (u, T)eV'

x V ^

(UIP)

:= u(j;) e K.

The canonical isomorphism V -^ {V)' is obtained by fixing the vector p G V in (UIP) and considering u G V ' as the variable. It is easy to see t h a t , for any basis (Oj) of V, the set of projections onto a single component,

uniquely defines a basis (o^) of V, which is called the dual basis to (Oj); here the sum convention is in action, and Sf denotes the Kronecker symbol.^ If p = OiC,

U = WjO*

are the basis representations of a vector and a covector in mutually dual bases, then their scalar product has the simple expression

It is clear t h a t the propositions of projective geometry hold for right as well as for left vector spaces over the skew field K. In fact, projective geometry is determined by the lattice of subspaces, and, according to Example II.7.2.14, every left vector space over K can be considered as a right vector space over the opposite skew field KQ. SO the lattice of subspaces does not change. This is the way to understand the projective geometry ^(V'): T h e projective geometries of equal dimension over opposite skew fields are isomorphic. Conceptually, projective duality is described by means of the annihilator relation and, in contrast to synthetic projective incidence geometry, accessible to calculations. As is well-known, the annihilator (cf. Definition 1.5.6.1) is defined as a m a p _L : « P ( F ) -^ V(V) in the following way: A G * P ( F ) ^ A^

:= {u G V'\

(UIP)

= 0 for all ? G A } G ViV)-

(1)

Here the subspaces A G ^(V), A G ^(V) are viewed as vector sets; considering t h e m as projective subspaces, the annihilator relation makes sense for b o t h arguments due to the homogeneity of the scalar product; the definition u=

[u] e A ^ :<^=>

(UIP)

= 0 for all x = [f] e A

is independent of the chosen representatives u, p of u,x. > F r o m what we proved in linear algebra, cf. § 1.5.6 and Proposition II.7.3.7, Exercise II.7.3.7 in the non-commutative case, the following is immediate. ^ In Chapter II.7, these results are obtained by specializing the propositions concerning free modules over non-commutative rings. An elementary presentation of vector algebra over skew fields can be found in Chapter III of the book by H. REICHARDT

[91].

1.6 Duality

69

Proposition 1. Let ^ " = ^(V), ^ ' " = ^(V') be the projective geometries associated with the (n + I)-dimensional vector space V over the skew field K and its dual V , respectively. Then the annihilator relation _L : ^ " ^ ^ " is a bijection of lattices reversing inclusion C, and hence preserving incidence. In detail: AcB-^^ B^ CA-^, (A-VB)^ = A^AB^, {AAB)^ -= (A^)^ = A, Dim A = k •<==> Dim A^ == n — k — I, o^=P'",

P"^=o'.

A^\/B^,

(2) (3) (4) (5) (6) D

According to Proposition 1, projective duality can be described as follows: The map _L : ^ " -^ ^ ' " assigns to each basic notion, more precisely, to each element of the projective geometry ^ " , the dual basic notion for the dual projective geometry ^ ' " so that formulas (2) to (6) hold. Since each derived notion as well as every proposition are obtained starting from these basic notions through finitely many steps, each concept leads to a pair of mutually dual notions, and each result leads to a pair of mutually dual statements, that are either both true or both false. In this connection, recall (4). Since all n-dimensional geometries over the same or the opposite skew fields are isomorphic, according to this duality principle, for each proposition there is a proposition dual to it, which, because of (5), as a rule has a different geometric content. As in plane incidence geometry it may, of course, happen that the dual proposition coincides with the original one. Note that, by its very definition, the map _L always depends on the space it is based on, hence on its dimension, which becomes particularly evident in Formula (5). In the examples to follow we will apply the duality principle. Example 1. Let « = [u] G P " be a "point" of the dual projective space. Since by (4) the annihilator, viewed as a map _L : ^ ' " ^ ^ " , is the inverse to (1), u represents the hyperplane «^={a;=[y]GP"|(u|y)=0}.

(7)

As the map _L defines a canonical bijection from the dual point space P ' " onto the Grafimann manifold P „ , „ - i of hyperplanes in the original space P " , it is reasonable and customary to interpret the elements u = [u] G P ' " as hyperplanes in P " , even to call them this way. In (7), the hyperplane u^ is characterized as the set of points whose representatives satisfy the equation (UIP) = 0; fixing, conversely, a point x = [f] ^ o oi P " and looking at all hyperplanes u = [u] satisfying this equation, the dual configuration x^ turns out to be the set of all hyperplanes through the point x. In general, the Grafimann manifold P„,fc of A;-planes in the projective space P " corresponds

70

1 Projective Geometry

to the G r a £ m a n n manifold P'„ n-k-i oi (n — k — l)-planes in P ' " . Since P ' " itself is a projective point space, dualizing the notions and propositions on projective configurations of points we obtain the dual notions and propositions concerning projective configurations of hyperplanes. If, e.g. n = 3, then points are dual to planes, and lines are dual to lines. Each hue B determines the set of points incident with it; we identified this set via the canonical m a p n (cf. (1.2)) with the line itself. W i t h o u t this identification, the object "line" has to be distinguished from this set, which, just for this reason, is sometimes caUed a point series. The associated dual notion is the set of all planes incident with the line, i.e. containing it. The totality of all these planes is called the plane pencil with support B. Let a, 6 G .B be two different points on the line; then we have B = ayb=

(a^ A6^)^,

i.e., the line appears as the join of two points, and, dually, as the intersection of two planes. Transferring the notion from the point series onto the plane pencil, which, in fact, is nothing but a line in P' , the latter is thus equipped with a CR. Hence, e.g., there is the notion of harmonic position for four planes in a plane pencil. For n = 2, the set of all lines through one point dually corresponds to the set of aU points in a line. This is caUed a line pencil., and in this set the C R is defined as well. For arbitrary n > 1, the pencil of all hyperplanes through a fixed (n — 2)-plane corresponds to a line in the dual space P ' " ; the latter is again called the support of the hyperplane pencil. Algebraically, viewing a A;-plane as the join of points corresponds to its parameter representation, and considering it as the intersection of hyperplanes corresponds to its representation as the solution set of a homogeneous system of equations. • Exercise 4 . a) Dualize the notions collinear, throw, CR of a throw to refer to hyperplanes. - b) Prove for a throw of points in P " , (a, b, c, d), that ( a ^ , b^ ,c^, d^) is a throw of hyperplanes in the space P ' " , and that the CR of these throws coincide. - c) Formulate and prove an analogue of Exercise 4.4 for throws of hyperplanes. E x a m p l e 2. In (1.2) we defined a canonical m a p n : V ^ P " t h a t , applied to subspaces, aUowed to reahze the projective subspaces B G ^ " as point sets preserving inclusion C; using the incidence relation this is expressed as (,: Tr{B) = {xeP"\x i B}. (8) T h e analogous canonical m a p TV : V' ^ P ' " can be used to consider each projective subspace U C P ' " as a set of points in P ' " , hence, by duality, as a set of hyperplanes in P " . By (6), the nopoint o' = 7r(o) G P ' " then corresponds to the whole space P " . Duahzing (8) leads to the foUowing geometric interpretation of the annihilator relation _L: T{B):={ueP"'\uLB}

= B^,

(9)

1.6 Duality

71

since T(B) by definition is nothing but the incidence in the set of aU hyperplanes u containing B, for which we thus have {u\B) = 0. But this is just B ; T(B) is called the hyperplane pencil with support B. (Still to be included is the whole space P " corresponding to the nopoint o' t h a t always has to be added to all spaces of hyperplanes. In the sequel we will usually not mention this explicitly.) The canonical m a p n preserves the relation C, whereas it is reversed under T by (2); for this reason, to avoid confusions we will not use T as an identification like TT. D Exercise 5. Let (Oi)i=o,...,n be a basis of \^"+^, and let (o^) the basis of V to it. The hyperplane Hj corresponding to 0^ is defined by

dual

X e Hj •<=^ X = [p] and (a^lp) = 0; Hj is the face opposite to the point aj = [ttj] in the coordinate simplex for this basis. The object dual to the unit point e = [Oo + . . . + fln] is the unit hyperplane n

E:=[a°

+ ... + a^] = {xe

PZ\x = [ajx'] and Y^ x' = 0}.

Choose the unit hyperplane E C P " of a projective frame as the hyperplane at infinity for the affine space A " and prove

= MeA"^^^xV J= 0

moreover, the normalization v

:= X

\Y.xT'

{xeA-)

defines the barycentric coordinates of x with respect to the n-simplex ([flo],... , [Un]), cf. Exercise 1.4.5.5. 1.6.3 P r o j e c t i v e P e n c i l G e o m e t r i e s Let ^ " be an n-dimensional, e.g. right projective geometry over the skew field K, and let A G ^ " be an arbitrary, fixed A;-plane. The pencil of projective subspaces with support A'' is defined as the set of all subspaces containing *P"/A'^ : = { B G * P " | A ' ^ C B } .

(10)

Now we prove P r o p o s i t i o n 2. Under the assumptions of Definition (10), letV"^ D W ^ be the vector spaces over the skew field K corresponding to P" and A , respectively. Denote by V/W the quotient space. Then

72

1 Projective Geometry p:Be

*P"/A'^ I—> p{B) := {b + W\b e B} e ^i{V/W)

(11)

defines a canonical bijection, which is an isomorphism of lattices. This way, a projective geometry of dimension n — k — 1 is defined in ^ " / A . The space dual to the projective point space P{'V/W), i.e. the space of its hyperplanes, is canonically isomorphic to A . P r o o f . For simplicity, let B denote the vector subspace and, at the same time, the projective subspace; depending on the context, it thus has to be interpreted as a set of vectors or points. Considered as a map between vector subspaces, the map pi is obviously generated by the canonical projection p-.feV^f

+W

eV/W,

hence it is surjective. Since its kernel is W, we have for the zero vector 0 G V/W the preimage p^^{0) = W, and this is the nopoint in ^ " / A ' ^ . Thus the map (11) is also injective. Obviously, it preserves the relation C and, in addition, satisfies

p{Bi)Vp{B2)=p{BiVB2). Therefore, it is a coUineation as well, where the relative dimension in the pencil has to be taken into account: Dimrel(B) : = D i m B - A ; - l .

(12)

Furthermore, A is the pencil of hyperplanes in P " containing A . For any of these hyperplanes, u = [u], we have u\W = 0, and vice versa. Hence, u(j; + W) := u(j;) is a well-defined linear form on V/W. The map u e W 1-^ fi G (V/Wy is hnear and has kernel o; thus it is injective and, since the dimensions of both spaces coincide, dim W = dim V/W = n — k, it yields a canonically defined linear isomorphism, W^ = {V/Wy,

W (ZV & subspace, AimV < oo.

This immediately implies the second assertion.

(13) D

E x a m p l e 3. In order to illustrate Proposition 2, which may appear a bit formal at first glance, we recall the heuristic description for the projective plane from Example 1.1. There we defined the projective plane by means of the pencil of all rays through a point, cf. Fig. 1.1. The relative dimension of a line as an object in the pencil is zero; they are the "points" of the pencil. Intersecting the pencil with support A by a plane H not passing through A, the assignment h:xeH<—>h{x):=xV

AeP'^/A

(14)

is a projectivity; in Example 1.1 we had n = 3. In the general case of a pencil in an arbitrary P " with support A one has to choose an (n — A; — l)-plane

1.6 Duality

73

complementary to A as H; H is called a section (or a complete system of representatives) for the pencil P " / A . In fact, the m a p S : X G * P " / A I — > s{X):=X

AH

CH

(15)

is the inverse of the m a p between the lattices of projective subspaces defined by h. T h e m a p (14) transfers the (n—A; —l)-dimensional projective geometry from H to the pencil through A. Transferring a homogeneous coordinate system from H onto the pencil h can be represented as assigning equal coordinates. Hence it is a projectivity. If / : F ^ P"'/A is an analogous projectivity from another [n — k — l)-plane F complementary to A onto the pencil with support A , then f^^ oh : H ^ F defines a projectivity generahzing the construction of the central projection in Example 1.2, Fig. 1.3. For this reason, we call it the central projection from H onto F with center A. Denoting by q the central projection from P" onto the subspace F with center A defined in Example 3.1, we h a v e / " ^ o/i = i^lH". D 1.6.4 D u a l M a p s For a linear m a p a : V ^ W the associated dual map a' : W' -^ V' is defined by the following well-known formula: (a'(w)lj:) := (W|a(j:)),

^eV,roeW',

cf. Definition 1.5.2, where a' was caUed the m a p transposed to a. For the applications in projective geometry a slight generalization of this notion will be needed. RecaU, (cf. Definition II.7.2.7), t h a t a m a p a : K ^ L between skew fields is called an anti-isomorphism if a is an isomorphism for the additive groups of the skew fields and, moreover, a(al3) = a(l3)a(a). A m a p a : V ^ W from the right vector space V over K into the left vector space W over L is called a-linear if we have a{fa + t)/3) = a{a)a{f)

+ o-(/3)a(t)),

^,t)eV,a,l3eK,

where a : i f ^ L is an anti-isomorphism. It satisfies a(j; a/3) = o-(/3)o-(a)a(j;), p, G V , a, /3 G K. (There is an analogous definition for maps a from a left into a right vector space.) D e f i n i t i o n 1. Let V be a right vector space over the skew field K, let W be a right (or left, respectively) vector space over the skew field L,\et a : K ^ L be an isomorphism (or anti-isomorphism, respectively) of the skew fields, and let, finally, a : V ^ W he a-linear. Then the m a p a' : W' -^ V' defined by (a'(w)|j:):=a-i((w|a(j:))), or(a»|y):=a-i((a(y)|w)) is called the dual map to a.

^eV,roeW',

(16) (17) D

74

1 Projective Geometry

In Equation (17) we reversed the order in which the vector t) (^ W and the covector tn G W enter the canonical scalar product (t)|tti), since this way W' becomes a right vector space. The following is straightforward from the definitions. Lemma 3. Equations (16) or (17), respectively, define a a^^-linear

a' -.W -> v.

map

a

This lemma suggests the following definition: Definition 2. In the notations of Definition 1, let P " and Q™ be the projective spaces associated with the vector spaces V"+^ and W™^^ over the skew fields K and L, respectively. Moreover, let / : P " -^ Q™ be a colhnear map induced by the a-linear map a : V ^ W. Then the collinear map / ' : <5o™ -^ P Q " induced by the dual a^^-linear map a' is called the dual collinear map associated with the map f. • The following proposition contains an interesting application of the dual map: Formula (18) proved below, written a bit differently in the form

shows the possibility to express the inverse images of the coUinear map / through the images of / ' . Proposition 4. Under the assumptions and in the notations of Definition 2, the dual collinear map f is independent of the choice of the map a inducing f, and, moreover, f-\H)^=f'{H^),

He^{W^+').

(18)

P r o o f . Consider the case that Vt^ is a left vector space and (17) holds. Let U C W be an arbitrary subspace. We wiU then prove the equation a-\U^) = a!{U)^.

(19)

In fact, p G a^^{U ) if and only if a(p) G U , and this is the case if and only if every tn G L/" satisfies the equation (a(p)|tti) = 0. By Definition 1, this is equivalent to (a'(tti)|p) = a^^{a{f)\vo) = 0 for ah vo e U, hence p G a'{U)^. Since a induces the map / , we obtain for the projective subspaces

f-\U^)=a'{U)^=f'{U)^, so that / ' only depends on / , and is independent of the chosen inducing map a. Setting U = H in the last equation and applying _L we arrive at Equation (18). D Proposition 4 justifies the definition of the dual colhnear map / ' . The following properties are immediate.

1.7 Correlations (/')' = /,

{fogy

75 (20)

= g'of'.

(21)

Exercise 6. Let / ' be the map dual to / : P " ^ Q™ . Prove: a) X L f'{H)^^

f{x) L H

{xeP",

H eQ""),

/'-i(o')=/(P")^=r(/(P")).

(22) (23)

b) / determines a coUineation / from the pencil ^ " / / ^ ^ ( o ) onto the projective geometry of the image / ( P " ) : f:Xe

W r \ o )

^

f{X)

:= f{X)

e *P(/(P"),

and applying the canonical cr^^-bijection alV)' = W'/a(V)^

yields

(/)' = (?)•

(25)

E x a m p l e 4. In the notations of Definition 1 suppose t h a t a : V " + is bijective. The contragredient map to a is defined as a* := (a-y Obviously, a* = {a')-^ we have

(24)

: V -^ W.

-^ Vt^"+

(26)

(cf. Exercise 11.7.2.12). (18) implies: For ah B e * P ( F ) r{B^)

= f{B)^,

(27)

where / * denotes the coUineation induced by a*. Setting M = B (27) can also be written in the form

equation

r ( M ) = (/(M^))^. Thus it is clear t h a t the contragredient m a p does not yield anything essentially new for projective geometry. For all B G ^{V), the set f{B)

=

{f{x)\xeB}

is the image of B as a point set, and, dually, f*{B^)

= {f*{W-^)\W-^

D B}

is the image of B as the support of a pencil of hyperplanes.

•

1.7 Correlations In this chapter we will place special emphasis on the lattice structure of the projective geometry ^ " . According to Exercise 3.5 coUineations can be characterized as monotonously increasing bijections F : ^ " -^ Q " between projective geometries whose inverse F^^ is monotonously increasing as well. T h e dualistic structure of projective geometries suggests to consider monotonously decreasing maps likewise.

76

1 Projective Geometry

1.7.1 Definition. Canonical Correlation Definition 1. Let F : ^ " , Q™ be (right or left) projective geometries over the skew fields K and L, respectively. A bijective map F : ^ " -^ Q™ is called a correlation if it satisfies the following condition: AcB-^=^

F{A) D F{B), A,Be

*P",

i.e., if F and F^^ both are monotonously decreasing.

(1) D

Example 1. Forming the annihilator

_L : *P" ^ *P'", _L : *p'" ^ *P"

(2)

is a correlation from a right to a left (or from a left to a right) projective geometry that we will call the canonical correlation, since it is uniquely determined by the very structure of the projective geometry. For these structural reasons we admit the obvious notational misuse. In fact, since the two maps considered in (2) are mutually inverse, strictly, we would have to distinguish -Lqj from -Lqj' and to write J-qj' = -^qy'-' As the correlations are anti-isomorphisms for the lattice structures, the characterization of /\ as the infimum and that of \/ as the supremum lead to the following rules: Fi/\A,)

= \jFiA,),

Fi\jA,)

= /\FiA,),

A, G *p".

(3)

Now we ask under which conditions there exists a correlation between two projective geometries. Since for n = 1 the order C is rather useless — each bijection F : *P^ -^ Q^ with F{o) = Q^ and F{P^) = o is a correlation — we will suppose n > 2, as a rule. First we note two facts immediately following from the definition. Corollary 1. The composition FoG of two correlations is a collineation. The composition of a correlation and a collineation is a correlation. • In order to discuss the existence question, consider two right projective geometries; if one of the geometries is a left one, then by Example II.7.2.14 we may pass to the opposite skew field and vector space, and this way reduce everything to the case of right geometries. Proposition 2. Let ^ " , Q™ be right projective geometries over the skew fields K and L, respectively, n > 2. There exists a correlation F : ^ " -^ Q™ if and only ifn = m and if, in addition, there is an anti-isomorphism a : K ^ L. If both these conditions are satisfied, then each correlation F : ^ " -^ Q " has the form F = JL o G, where G : ^ " -^ Q ' " is a collineation induced by a a-linear bijection, where a runs through the set of all anti-isomorphisms from K onto L.

1.7 Correlations

77

P r o o f . Let F : ^ " -^ Q™ be any correlation. According to Example 1 and Corollary 1, G := _L o _F is a collineation from the right projective geometry ^ " over K onto the left projective geometry Q'™ over L. Let L denote the skew field opposite to L, and denote by Q ' the right projective geometry t h a t arises by passing to the opposite module structure W over the vector space W associated with Q ' . Since the lattices Q ' and Q ' coincide, G : ^ ^ £i' is a collineation of right projective geometries. Hence, by Proposition 3.6 we have n = m, and from the Main Theorem 3.10 we conclude the existence of an isomorphism a : K ^ L and a a-linear m a p a : V ^ W inducing G due to G([j;]) = [a(j;)]. Returning now to the original structures, because of a{a • f3) = a{a) x
= a ( A ) ^ , A e *P";

(5)

a is called a a-linear m a p inducing F. a : K ^ L is an isomorphism or an anti-isomorphism depending on whether V and W are opposite or like vector spaces. By Proposition 2 the m a p F is a correlation if a is a a-linear bijection (in the case n > 2); in case n = 1, we always suppose t h a t correlations are correlative bijections. • For brevity, we only formulated an algebraic definition for correlative maps; it is clear t h a t , just as colhnear maps (cf. Section 1.3.1), they should also have a geometric definition. Composing them, instead, with the canonical correlation and, this way, reducing the situation to the case of collinear maps turns out to be easier. By Proposition 2, every correlation is a correlative m a p as well. Obviously, each correlative m a p is monotonously decreasing: A c B

implies F{A)

D F{B),

A, Be

*p".

(6)

78

1 Projective Geometry

Exercise 2. Let F : ^ " -^ £3™ be the correlative map induced by the cr-linear map a : V ^ W'. Prove that if ai : \^ ^ W' is a ui-linear map also inducing F, then there is /it G K* such that (4) holds. E x a m p l e 2. Let F : ^ " -^ Q " be a correlation. Proposition 2 with G := J- o F implies for each A;-plane H G ^ " : BiinF{H'')

= Biin{G{H'')^)

= n - k - l .

In particular, the image of a line H is an (n — 2)-plane, and the set of points incident with H is transformed into the pencil of hyperplanes incident with F{H^). By Exercise 6.4, these hyperplanes are m a p p e d into the "points" of a line L, on which, consequently, a C R is defined. For a correlation to be projective, i.e. to preserve the CR, it is necessary t h a t K = L and a = i d ^ ; moreover, if ^ " and Q " b o t h are right projective geometries, then K has to be a field, in addition. D E x a m p l e 3 . Let i f = L = R be the field of real numbers. Then there is just the isomorphism a = idpt, cf. Exercise L2.1.3). Each correlative m a p F : *P" ^ Q™ is thus induced by a hnear m a p a:V^W'. D E x a m p l e 4. In the case of complex numbers, K = L = C, as for collineations, we confine ourselves to the continuous automorphisms of C, cf. Example 4.3. Accordingly, there are two types of correlative maps: Taking a = i d c we obtain the (proper) correlative maps, and choosing the conjugation as a leads to the anti-correlative maps. • E x a m p l e 5. Let K = L = ii he the skew field of quaternions, and let ^ " be a right projective geometry over H . If Q™ is another right projective geometry over H , and the correlative m a p F : ^ " -^ Q™ is induced by a
= m — 1 •^=> x = [p] and p ^ K e r a .

(8)

1.7 Correlations

79

More generally, the kernel of F defined by Ker F := Kera is a projective subspace of dimension D i m K e r F = n — rka. (9) If rka = 0, i.e. KevF = P " , then F is called the correlative null map; it is characterized by F{A) = Q™ for all A G ^ " . For any correlative map F and each projective subspace H G ^ " we have DimF(H') = m - D i m H ' + Dim(H'AKerF).

(10)

Note that, in this relation, F{H) appears as the support of the hyperplane pencil T{F{H))

= {F{x)\x

e

H};

in particular, F is surjective if and only if F{P") = o. Definition 3. Let F : ^ " -^ Q™ be a correlative map induced by the
F'{X)

:= a'{X)^

G *P".

(11) D

Proposition 3. Let F : ^ " -^ Q™ be correlative. Then the dual map F' does not depend on the choice of the a-linear map a inducing F. Moreover, F'(g™) = KerF, F o F'{Y) DY, F'O (F'y = F.

F{X)

D

X, for Y G Q™, X e *p",

(12) (13) (14)

/ / F is a correlation, then F'= F-\

(15)

P r o o f . The first statement follows from Proposition 6.4. Furthermore, by (11) with X = g™ and (6.19) with U = W we have F'iQ"") = a'{W)^

= Kera = KerF.

The first relation in (13) also follows from the definitions using (6.19), FiF'iY))

= (a(a'(Y)^))^ = ( a ( a - i ( Y ^ ) ) ) ^ D Y,

since a{ar^{Y )) CY . The second part of (13) is a consequence of the first combined with equation (14), which immediately follows from Definition 3. If F is a correlation, then this is also true of F', and, for dimensional reasons, equality has to hold in both parts of (13), implying (15). •

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1 Projective Geometry

1.7.3 i^-Correspondences and cr-Biforms Now we want to express correlative maps in terms of the points x G P " , y G <5™. To this end, we start with the fohowing definition. Definition 4. Let F : ^ " -^ Q™ be a correlative map. Then the pair (x, y) G P " X <5™ is said to correspond under the map F, or to belong to the F-correspondence if y iF{x). D Corollary 4. F-correspondences have the following properties: a) The set {y G Q^\y iF{x)} is the hyperplane F(x) C Q™ considered as a point set, if x ^ Ker F, and it is the whole space Q™, if x ^ K e r F . h) In general, yiF{x)^^XiF'{y)(16) hence for fixed y the set {x G P^\y iF{x)} is the hyperplane F'{y) C P " considered as a point set, if y ^ F(P"), and all of P", otherwise. • >From now on, we will again suppose that if = L is an arbitrary skew field and a an anti-automorphism of K. Let *P" = *P(F"+i), Q™ = *P(VF™+^) be projective geometries associated with right vector spaces over K, and let F : ^ " -^ Q™ be a correlative map induced by the a-linear map a : V ^ W'. Then we define 6(y,t)):=(a(y)|t)), fet))GFxVF. (17) L e m m a 5. The map h : V x W ^ properties: a) b is linear in t), i.e.

K defined by (17) has the following

6(p, t)a + 1(3) = 6(p, t))a + 6(p, j)/?, b) b is a-linear in p, i.e. b{^a + u/3, t)) = cr(a)6(y, t)) + cr(/3)6(u, t)).

(18) D

Definition 5. Let V, W be right vector spaces over the skew field K. The map b : V X W ^ K is called a biform or, more precisely, a a-biform if it has properties a) and b) from Lemma 5. • Bilinear or hermitian forms are examples of a-biforms. In the literature, instead of a-biforms the term sesqui-linear form is frequently used. If ai : V ^ W' is a map that also induces the correlative map F, then (4) holds (cf. Exercise 2), and ai determines the ai-biform 6i(y, t)) := (ai(p)|t)) = a(/i)6(p, t)).

(19)

1.7 Correlations

81

Two biforms b, 61 differing only in a factor v G K* will be called proportional. If 6 is a (T-biform, and 61 = lyb is proportional to 6, then 61 is a ai-biforni with (Ti(a) = i/(T(a)i/^ . Obviously, each biforni 6 7^ 0 uniquely determines the corresponding anti-automorphism of K. Biforms are particularly well suited to describe correlative maps as correspondences: Proposition 6. For each correlative map F : ^ " -^ Q™ between right projective geometries over the same skew field K there exists a biform b : V X W ^ K, which is uniquely determined up to proportionality, such that M iFm ^^ 6(y, t)) = 0, (y, t))eVxW. (20) Conversely, according to (20), to each class of mutually proportional biforms there corresponds a correlative map F : ^ " -^ Q™. P r o o f . The first part of the assertion follows from Lemma 5 combined with the argument leading to (19). If, conversely, 6 is a biform, then it defines a
a(y)(t)) := 6(y, t)), t) G W.

(21)

But proportional biforms define the same correlative map. In the above notations, CoroUary (4) and (20) immediately imply y,F(a;)^^6(j:,t)) = 0 ^ ^ a ; . F ' ( y ) ,

• (22)

so that we can also describe the dual correlative map by b. On the other hand, applying (17) to the dual map a' we see that, because of (6.16), the transposed biform corresponds to F', 6'(t),y):=a-i(6(y,t))), {T,t))eVxW.

(23)

In fact, we have (a'(t))|p) = cr^-^((a(p)|t))). Obviously, b' is a^-^-linear in t) and linear in p. Next we want to describe the representation of correlative maps in homogeneous coordinates. Consider a pair of dual bases,

(Oi), (a^'), iai\a^) = Sf, i,j = 0,...,n, {b^)Ab^), {b^\b^) = Si a,/3 = 0,...,m, determining coordinate systems on P " and Q™, respectively. Here we will write the coordinates (x*) of a vector p as a column and those of a dual vector as a row; the sum convention will again be used. The matrix (ojo,) describing the (T-linear map a : V ^ W is defined by a(Oi) = ttjab", « = 0 , . . . , n, a = 0 , . . . ,

TO.

(24)

If the coordinates of p G V and u = a(p) G W are (x*) and (««), respectively, then

82

1 Projective Geometry Ma = o-(x*)ajo,;

(25)

(MQ,) are the homogeneous coordinates for the hyperplane F(x) corresponding to the point x = [p] with homogeneous coordinates (x*). At the same time, (ttjo,) G Mn^i^m+iiK) is the matrix of the a-biform b corresponding to F: 6(p, t)) = (a(p)|t)) = a{x')aiay",

ai^ = b{ai, ba).

(26)

Here, (y") are the coordinates of t). For the m a p F' dual to F a straightforward calculation using (24) and (6.16) yields a'(ba) = aLil* = <7^\a,io,)a\

(27)

i.e., the matrix (a^^) of a' coincides with the transposed of the matrix obtained by transforming t h a t of a by means of a^^:

K.) = (^"'Ka))'-

(28)

Exercise 3 . Let {aia) be the matrix of the correlative map F : ^ " -^ Q™ according to (24). Prove the relation Dim Ker _F = n — r, Dim Ker _F = m — r,

(29)

where r is the rank of {aia) (cf. Exercise II.7.3.7). This implies that r does not depend on the chosen coordinates; the integer r is called the rank of F. Exercise 4 . Let ^ := ^ ( ^ " , Q ™ ) denote the set of all correlative maps from *P" to Q " . Consider the action of the group G := PL{P") x PL{Q'^) of pairs of projectivities on VB defined by

(fli,fl2) e G, F e * 8 H^(72oFo(7-i e * 8 .

(30)

Prove that two correlative maps F, F E ^ are equivalent under this action of G if and only if they have equal rank and determine the same class a in the group Aut{K)/lnt{K), cf Exercise 4.11 and (4). T h e following proposition will allow to define the restriction auto-correlative m a p F : ^ " -^ ^ " to a subspace A C P " :

F\A

of an

P r o p o s i t i o n 7. Consider *P" = * P ( F " + i ) , A G *P" with A = n{W) for a subspace W C V of the vector space V. Let F : ^ " -^ ^ " be an autocorrelative map induced by the a-linear map a : V ^ V'. Then xeA^F\A{x):=F{x)AAcA defines an auto-correlative

map F\A ^eW^a{^)\W

If b is a a-biform

(31)

that is induced by the a-linear eW.

describing F, then the restricted b^ := b\W X W

describes the restriction

F\A.

map (32)

a-biform (33) D

1.7 Correlations

83

T h e proof immediately follows from the above definitions. Looking at examples it is obvious t h a t F\A may well be the correlative zero m a p , even for KevF = o (cf. also Section 1.10 below). A notion dual to itself is t h a t of a mixed throw: D e f i n i t i o n 6. Let ^ " , n > 2, be a projective geometry over the skew field K, let a,b e ^ " be points, and let C,D e ^ " be hyperplanes, a ^ b, C ^ D. Set c := C A ( a V 6), d:= D A{aV b). The quadruple (a, b, C, D) is called a mixed throw if (a, b, c, d) is a throw. T h e cross ratio of the mixed throw is defined as (cf. L. Heffter [48], §18) {a,b;C,D)

•.= {a,b;c,d).

(34) D

Exercise 5. Under the assumptions of Definition 6 show the following: A quadruple (a, b, C,D) is a mixed throw if and only if {A,B,C, D) is a throw of hyperplanes, where A := a V (C A _D), B := 6 V (C A _D) (cf. Exercises 6.3, 6.4). In addition, for the CR we have {a,b;C,D)

= {A,B;C,D).

(35)

Exercise 6. Let ^ " , Q™, n > m > 2, be right projective geometries over the skew field-R"; moreover, let {a,b,C,D) and {ai,bi,Ci,Di) be mixed throws in ^ " and Q™, respectively. Prove that there is a projective map F : ^" -^ Q™ with F{a) = ai, F{b) = bi, F{C) = C i , F{D) = Di, if and only if their CRs coincide: (a,b;C,D)

= (ai,bi;Ci,Di).

(36)

Exercise 7. In the notations of Exercise 6 with the additional assumptions that n = m and K is a field, show the following: There is a projective correlation F:^" ^Q" with F{a) = Ci, F{b) = Di, F{C) = ai, F{D) = bi, if and only if (36) holds.

84

1 Projective Geometry

1.8 Symmetric Auto-Correlative Maps In this section we will study auto-correlative maps F : ^ " -^ ^ " . Such a m a p is called symmetric if it coincides with its dual, i.e., if F = F'.

(1)

This name is suggested by the fact t h a t the associated ^-correspondence is a symmetric relation over P": By (7.16) F is symmetric if and only if X iF{y)^^y

iF{x),x,yeP:;.

(2)

There are two types of symmetric auto-correlative maps: the null systems and the polar maps. Now we are going to obtain a projective classification of these maps; for the null systems this will be completely accomphshed in this paragraph, whereas for the polar maps we will achieve a classification of important special cases in the next paragraph. Null systems form the basis for symplectic geometry; we will also interpret t h e m geometrically as linear line complexes. Remarks concerning the terminology. The obvious name "symmetric" for property (1) appears to have rarely (or even never?) been used until now. The reader may forgive us, that we add still another one to the various names already to be found in the literature. The name "reflexive" chosen by J. Dieudonne [36], § 1.6, does not seem very fortunate to us, since the associated F-correspondence is reflexive only for the null systems (compare Deflnition 1.0.2.7, Property 1, with Deflnition 1 below). W. Burau [24] calls the correlative maps satisfying (1) "involutory", even though by 7.3, (7.15), only the symmetric correlations share the property FoF = idojn characterizing involutions; these symmetric correlations are called "polar maps" by R.Baer ]5], thereby possibly making the notational confusion complete. On the other hand, R. Baer uses the name "dual map" instead of correlation, and "self-dual map" instead of auto-correlation, so that we preferred to avoid the obvious name "auto-dual" for (1). 1.8.1 N u l l S y s t e m s a n d P o l a r M a p s D e f i n i t i o n 1. Let F : ^ " -^ ^ " be a symmetric correlative m a p . T h e points in the kernel, x G Ker F, are called singular and the remaining ones, X ^ K e r F , regular points of F] a point x ^ K e r F is called a pole of the hyperplane F{x), and F{x) the polar of x for the correlative m a p F. An element X e ^ " is caUed auto-polar if X tF{X). Let QF C P " denote the set of auto-polar points of F. If Qp = P " , then F is caUed a null system; otherwise, F is a polar map. Moreover, we will call F non-degenerate if Ker F = o, i.e., if F is a correlation; a polar correlation is a polarity . • Note t h a t , according to Definition 7.2, every auto-correlative m a p F is induced by a semi-linear m a p a : V ^ V'. Obviously, the polar of a regular

1.8 Symmetric Auto-Correlative Maps

85

point is uniquely determined; conversely, for the pole this only holds in the case of a non-degenerate F: li x ^ Ker F, then every point y G (a; V Ker F)\ Ker F is a pole of F^x); in fact, P " 7^ F{y) D F{x) A F(KerF) = F{x). Corollary 1. An auto-correlation F : ^ " -^ ^ " is symmetric if and only if it is involutive: F^ = idqj^ . P r o o f . The assertion immediately follows from Definition (1) and (7.15): F = F' = F-K D The symmetric correlations F thus are the involutions in projective geometry *P": The equality A = F{B) holds if and only ii B = F{A); in this case, A, B are called mutually polar under F. For the dimensions of a polar pair we have (cf. Example 7.2) m\nA

= k-i=^m\nF{A)=n-k-l.

(3)

By means of the following proposition, the symmetric correlative maps will be reduced to the symmetric correlations: P r o p o s i t i o n 2. Let F he a symmetric correlative map of the projective geometry ^ " . Then Kei F = F{P"), (4) and, moreover, Kei F c F{A) for all A eV''-

(5)

F determines the symmetric correlation of the bundle of subspaces through KeiF: F : X G * P " / K e r F i — > F{X) e ^"/Ker F. (6) P r o o f . By Proposition 7.3, (7.12), (4) holds because of F = F'. By (7.6), Equation (5) for A C P " foUows from (4). Now recaU the definition of the projective bundle geometries. Section 1.6.3. Because of (5), the map F is well-defined. Let F be induced by the a-linear map a : V ^ V' with Ker F = Ker a =: W; then F is induced by the canonically associated, a-linear bijection a : V/W -^ (V/W)' = W^ that is defined by iaiT+W)\t)

+ W):=iaiT)\t)).

(7)

We still have to show that F is symmetric, i.e. (F)' = F. By definition of the dual map, 6.1, (6.16), it is induced by the dual semi-linear map, which is (T^^-linear: ((a)'(j: + W)\t) + W)=

a-\iait)

+ W)\T+ W)) = a-i((a(t))|j:)) = (a'(y)|t)).

Since F is symmetric, the dual semi-linear map a' also induces F. Hence, according to the uniqueness statement in the Main Theorem 3.10, (3.17),

86

1 Projective Geometry

there is yU G i f * such t h a t a'(p) = a{iij). Inserting this into the last equation yields ((a)'(y +W)\^

+ W) = (a(yM)|t)) = {h{w +W)\^

i.e., (a)'(p + W) = a((p + W)i^),

+

W),

implying the symmetry of F.

D

1.8.2 E q u i v a l e n c e of A u t o - C o r r e l a t i v e M a p s Now we define the projective equivalence of two auto-correlative maps: D e f i n i t i o n 2. Two auto-correlative maps Fa, a = 1,2, of the projective geometries ^ " are called projectively equivalent, Fi ^ F^, if there is a projectivity gr :
(8) D

Again, we may and will consider only the case ^ " = ^ 2 = ^ ( ^ ) - Let ai : V ^ V be a ai-linear m a p inducing Fi, and let the linear transformation c G G L ( V ) induce the projectivity g; by c* := c'^^ we denote the transformation contragredient to c. Since (c*t)|cy) = (t)|y),

(9)

we obtain c ( X ) ^ = c*{X^),

c{Y^)

= c*{Y)^,

XCV,Y

cV.

(10)

Because of (8) and (10) the likewise ai-linear m a p c* o a i o c^^ induces the m a p F2; in fact, c* o a i o c-\X)^

= c*(ai(c-i(X)))^ = c(ai(c-i(X))^) = 3 0 ^ 0

g-\X),

cf. (7.5). According to the uniqueness statement in the Main Theorem 3.10 we know: If 0,2 is any (T2-linear m a p inducing F2, then Fi '~ F2 if and only if there are c G G L ( V ) and 1^ e K* such t h a t 02 = c o a i o c "^ o d ^ and 0-2 = o"! o o - u ^

(H)

here d^i denotes the dilation by the factor jj, in V, and
(12)

1.8 Symmetric Auto-Correlative Maps

87

hence there has to be a projectivity transforming the associated correspondences into one another. Choosing a basis (Oj) in V and denoting by (7*j), (ttij), and (bij), respectively, the matrices of c^^, 61 and 62 with respect t o this basis, then (12) and (7.26) immediately imply the corresponding properties of these matrices (Note again the sum convention!):

«(ai,-) = (^2(7^))'(&fcO(7S-)-

(13)

Formulas (12) and (13) are generahzations of the transformation rules (1.5.9.45) and (1.5.9.81) for bilinear and Hermitean forms, respectively. These considerations apply to an arbitrary correlative m a p F from ^ " to itself; by Proposition 7.6, (7.23), the symmetric correlative maps are characterized by the condition «6(t),j:)=a-i(6(j:,t))),

(14)

or, in matrix form, by «(«ij) = (c^ H^ij))'-

(15)

Apart from the symmetric bilinear forms, the Hermitean biforms and matrices, respectively, are special cases, cf. (1.5.9.77). T h e correlative m a p F is nondegenerate if there is no p G V , p 7^ 0, such t h a t 6(j;, t)) = 0 for all t) e V; biforms with this property are also called non-degenerate. Exercises 7.3 and 7.4 immediately yield the following necessary condition for projective equivalence: C o r o l l a r y 3. Projectively equivalent correlative maps from ^ " to itself have equal rank. Such a map is a correlation if and only if it is non-degenerate, i.e. its rank is n+ I. D Exercise 1. Let Fa, a = 1,2, be symmetric, auto-correlative maps of the projective geometries ^ ^ . Prove that -Fi ~ F2 if and only if the correlations Fi, F2 associated with them according to Proposition 2 are projectively equivalent. (Hint. Consider subspaces Aa e ^ ^ complementary to KeiFa; identify ^{Aa) with ^^/KeiFa by means of

B e^iAa)^^BVKej:Fa

e*P::/KerF„,

and reahze the Fa as correlations of ^ ( A Q , ) . )

1.8.3 Classification of Null Systems Let now F be a null system. If 6 7^ 0 is a a-biform defining F, then for all ^ e V the relation 6(y,y)=0 (16) has t o hold, from which, by inserting p + t) for p, we immediately obtain 6(p,t))+6(t),y)=0.

(17)

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1 Projective Geometry

Since 6 7^ 0, by the linearity in t) we find a pair (p^, t)g) e V x V with 6(y„,t)J = l.

(18)

As 6 is a cr-biform, (17) and (18) imply K?oC, t)o) = ^ ( 6 = -K^o,

?oC) = e

for aU ^ e K. Hence a = id^, if is a field, and b has to be an alternating bilinear form over V, cf. Proposition 7.6 and Definition 1.5.9.3. Thus we arrive at (cf. also Exercise 1.5.9.13) Lemma 4. An alternating bilinear form on a finite-dimensional vector space V over the field K always has even rank 2r. There is a basis (Oj) of V with respect to which the matrix (bij) of b has quasi-diagonal form with r block matrices

0 r -1 0 along the principal diagonal, and zeroes everywhere else (cf (I.5.9.82)J; /f 0 11\ ^ ^\ o o -1 0 0 1 o o o -1 0

ih

(19)

V

o o

o

0 1 -1 0

o

P r o o f . Because of (16), the assertion is trivial in the case dim V = 1: Since 6 = 0, we have rk6 = 0. For d i m V = 2 the assertion follows from (18): In case 6 = 0 we have r = 0, and for 6 7^ 0 we set Oi = po, O2 = t)o- Suppose that we already found vectors Oi,..., 02fc € V such that

6(o2j-i,a2j) = -b{a2j,a2j-i) = 1 for j =

l,...,k,

6(0i, a;) = 0 for i,l = 1, .. ., 2k, \i - l\ y^ 1. Consider the hnear span W'^'' := £ ( o i , . . . , 02fc) and its orthogonal complement W •.= {^e F|6(t),?) = 0 for aU t) G W}. Since 6| Vt^ x Vt^ is non-degenerate, we have W DW = {0}, and a straightforward consideration of ranks implies dim W = dim V—2A;. Hence V = W(BW. lib\W xW = 0, then we complete Oi,..., 02fc by means of an arbitrary basis of W to obtain a basis for V. This leads to the normal form (19) with k = r.

1.8 Symmetric Auto-Correlative Maps

89

Otherwise, we find vectors a2k+i,0^2k+2 & W with b{a2k+i,0^2k+2) = 1 and proceed hke before. • Lemma 4 and what was said before immediately lead to the projective classification of all nuU systems: P r o p o s i t i o n 5. Let ^ " be an n-dimensional projective geometry over the skew field K, n > 2. There exists a null system F o / ^ " with rkF > 0 if and only if K is a field. The biforms determining a null system F are bilinear, alternating and of even rank. Two null systems are protectively equivalent if and only if they have the same scalar field K and equal rank. Non-degenerate null systems only exist for projective geometries of odd dimension. P r o o f . Apart from what was just said it suffices to note t h a t nuU systems of equal rank also have coinciding normal forms (19). Choosing bases for Fi, F2 in which each has normal form, and then assigning corresponding vectors defines a linear isomorphism mapping the null systems into one another. Hence it provides the projective equivalence g in (8). T h e converse is obvious by Corollary 3. • 1.8.4 Linear Line C o m p l e x e s A geometric interpretation of null systems can be obtained by considering the Graf^mann manifold Pn,i of lines in ^ " . Let F be a null system of rank 2r > 0. Then the set of lines RF:={HePnAH iF{H)}

(20)

is called a linear line complex., if n > 3. Concerning the case n = 2 note Exercise 2. Exercise 2. If _F is a null system of rank 2r > 0 in the projective plane ^ , then r = 1, Kei F = Zo is a point, and ^F = T(KeiF) is the pencil of lines through Zo. No we will show t h a t Rp is the set of all connecting lines a; V y for which X is incident with F{y). L e m m a 6. / / F is a null system in ^ " , « > 3, then SiF = {xVy\x,ye^",Xy^y,x

LF{y)}.

(21)

P r o o f . For H = xV y e Sip (20) implies xV y t F{xV y) C F{x) A F{y); in fact, F is monotonously decreasing. Hence, in particular, x t F{y). Conversely, assume t h a t x t F{y). Since F is a null system, y L F{X), X LF{X), and y t F{y). Let now a : V ^ V be a linear m a p inducing F; then X = [p] and y = [t)] imply t h a t the hnear span is £(j;, t)) = H. Thus F{H) = (a(£(j;, t))))^ = £(a(j;), a(t)))^ is characterized by

90

1 Projective Geometry z=[i\e

F{H)

^ ^

(ay|3) = 0 und (atjli) = 0.

From the incidences above we conclude (ap|p) = (at)|p) = (ap|t)) = (at)|t)) = 0. Hence x t F{H), y t F{H), and, since F{H) is a projective subspace, xy y iF{H),i.e. H e^Fn For a better understanding of the concept of a linear line complex we need an embedding of the G r a £ m a n n manifold of all lines P „ , i = Gn+i,2, cf. Definition 1.1, as an algebraic submanifold of a certain higher-dimensional projective space. To obtain such an embedding we need some simple tools from exterior algebra, see e.g. II.8.3. Let A^V denote the space of aU alternating contravariant tensors of degree 2 associated with V, also called bivectors. It is generated by the splitting bivectors j;At):=j;(g)t)-t)(g)j;;

p, t ) G F .

If (Oj), « = 0 , . . . , n, is a basis for V, then the set of all bivectors Oj A Oj with 0 < i < j < n IS a basis for A^V; hence dim7lV= (n+l)n/2. Thus the associated projective space P'^{A'^V)

has dimension

2 Now let p = OjX*, t) = Ojy* be the basis representations of two vectors p, t) G V . Then we have p A t) = ^

Oi A ttjp^^ with p^^ = x^y^ - xJy\

(22)

Obviously, p A t) 7^ 0 if and only if the vectors are linearly independent. Set H = x\/ X e Pn,i for X = [p], y = [t)]. Then the coordinates p^^ of p A t) are called the Pliicker coordinates of the line H. li H = a\/ b, a = [0], b = [b] is another representation of H, then 0 V b = p V t)K for K G K* . Hence the m a p h:H

= x\/ye

Pn,i

-^ h{H)

:= [p A t)] G P^(A^V)

(23)

is correctly defined. Since each splitting bivector p A t) 7^ 0 uniquely defines the line: z = [^] e H if and only if p V t) V 3 = 0, the m a p h is injective and defines the embedding of P " ' as a manifold of points in P (A^V) we were looking for. In the case n = 2 the m a p h can be shown to be even surjective. For n > 3 this is not true. A general bivector

1.8 Symmetric Auto-Correlative Maps

91

T = i o i A ajp^^, p^^ +p^^ = 0, is splitting, i.e. has the form T = o V b for some 0, b G V if and only if its coordinates satisfy the Pliicker relations piii2pi3i _^_pi2i3 pill _^_ pish pi2i = 0 , 0 < n < ii < ^3 < n , / = 0, . . . , n .

(24)

Formula (24) is a special case of II.8.3.20. The equations (24) are not independent. For n = 3 they reduce to a simple quadratic equation defining the Pliicker quadric Q C P^iA^V^): / V 3 + / V ' + P V

= 0,

(25)

(see the next section). According to Proposition 5, the null systems F of ^ " correspond to the alternating bilinear forms & € / \ V . T h e latter are uniquely determined by F up to a factor K G if*; in other words: The null systems of ^ " are described bijectively by the points of the projective space P{/\ V). Let Uij be the homogeneous coordinates of F (or b); then we have a,ij + a,ji = 0 , «, j = 0, .. ., n,

(26)

and this implies: L e m m a 7. Let F be a null system corresponding to the alternating with coordinates aij. Then M.F consists of all lines H = xV y whose coordinates (22) satisfy the linear equation

J2aijP''=0.

2-form Pliicker

(27)

i<j

a In fact, according to Lemma 6 we have aijxy^ = 0, and (25) follows by a straightforward computation from (22) and (26). For 6 7^ 0 it is obvious from (25) t h a t a linear line complex is the intersection of the Grafimann manifold, according to (23) embedded into P ^ , with a hyperplane of P ^ . More generally, apart from the linear line complexes there are more general line manifolds, whose definition is not based on the single linear equation (25), but on a system of algebraic or even just differentiable equations. In the older literature, the geometry of the line space (for different transformation groups) is also called line geometry, and, accordingly, the terms line coordinates, line, or ray complexes etc. are used. Exercise 3 . Prove that under the canonical embedding (23) each hyperplane A C - P ^ ( / \ V) intersects the Grafimann manifold (23) in a proper subset M.:= An /i(Pn,i) / 0, and that M. = M.F for a suitable null system F of ^". Exercise 4 . Prove that for a symmetric auto-correlative map F oi ^ and any subspace A C P" the restriction F\A (cf. Proposition 7.7) is a symmetric autocorrelative map in ^ ( A ) . In particular, the restriction of a null system is again a null system.

92

1 Projective Geometry

1.9 Polarities and Quadrics In this section we consider polar maps F : ^ " -^ ^ " (Definition 8.1). First we will show that the associated correspondences can be described by (T-Herniitean biforms 6, which are generalizations both of the symmetric bilinear and the Hermitean forms. According to Definition 8.1, the set QF C P " of auto-polar points of F is a proper subset of projective space, QF T^ - P " ; this set QF will be caUed the quadric in P " determined by F. If if is a field and b is bilinear, this definition coincides with the classical one considered in § 1.5.9. In this case, a quadric is the set of points whose coordinates satisfy a quadratic equation of the form (1.5.9.1). Since projective geometry is described by means of homogeneous coordinates, the homogeneous equations occurring are of second degree: If a point with coordinates (x*) is a solution, then for any ^ & K the point with coordinates (x*)^ is a solution as weU. Hence, in a projective geometry over the field K with a = id^ the equation for a quadric in P " corresponding to (1.5.9.1) has the form n

where not all coefficients aij are equal to 0. Note that the upper i,j are coordinate indices and no powers. If char K ^ 2, then one may additionally assume that the coefficient matrix is symmetric; in fact, the coefficients aij may be replaced by a,ij := (aij + aji)/2 without changing the solution set. Of course, this set may well be empty. In the general case, the anti-automorphism a also comes into play, cf. (1) below. The aim of this section is to obtain a classification of polar maps and, consequently, for the corresponding quadrics in projective geometries whose scalar field are the real, the complex, or the quaternionic numbers. Starting from this classification we will introduce the most important Cayley-Klein geometries in the next chapter. 1.9.1 cr-Hermitean Biforms Let 6 be a a-biform determining F, and let (ajj) G Mn+i{K) be its matrix in a homogeneous coordinate system defined by the basis (Oj) of V"+ . Then we obtain the equation for the associated quadric QF C P " from (7.22), (7.16) (Note that the sum convention is in action!): a; = M G QF -^^ &(?,?) = 0 •<^^ a{x')aijx^

= 0 , {x y^ o);

(1)

here x* denote the homogeneous coordinates of the point x with respect to the basis (Oj), i = 0,... ,n. Definition 1. Let V be a vector space over the skew field K, and let a be an anti-automorphism of K. A a-biform b : V xV ^ K is called a-Hermitean if it satisfies

1.9 Polarities and Quadrics 6(j:,t))=a(6(t),j:)), y, t ) G F .

93 (2)

P r o p o s i t i o n 1. Let ^ " = ^^(V) be a projective geometry over the skew field K. Then: a) For every polar map F : ^ " -^ ^ " there are an anti-automorphism a of K, a a-Hermitean biform b over V determining F, and an p^ G V such that K?o,?o) = lConversely,

each a-Hermitean

biform of this kind determines

(3) a polar map F

ofV"b) The anti-automorphism involutive, i.e.

a corresponding a oa =

Because of (2), in

\AK

to a a-Hermitean

biform b ^ Q is

•

(4)

particular, a(6(y,y)) = 6fey), J G F .

(5)

P r o o f . Let bo be any biform defining F. Since F is polar, there is an Xo = [po] G P " with Xo ^ F{xo), hence A := 6O(?OJ ?O) ¥" 0- Setting b := A^^6o we obtain (3). According to Proposition 7.6, the biform b also defines F; let
K = K6(yo,yJ = cr"^(6(Po,pJ) = 1. Thus (2) immediately foUows from (8.14). Conversely, (2) is a special case of (8.14). Hence the correlation determined by b is symmetric, and due to (3) a polar m a p (Definition 8.1). b) Since b is non-zero and linear in t), for every ^ G i f one can find a pair (p, t)) G V X F with 6(p, t)) = ^. From (2) we now conclude (4): e = a(6(t),y))=a(a(6(y,t))))=aoa(e).

Exercise 1. Let F,b,a,^^ be as in Proposition 1. If 6i is a cri-biform also defining F, then 6i = Kb, where K E K is a scalar for which there is an t) e T^ with K = b{\), t)) / 0; 6i is cri-Hermitean, ai = a^^ o a, and crK(C) = KS,K^^. Conversely, if K = 6(t), t)) / 0 holds for some t) E V, then 6i = K6 is a cri-Hermitean biform also defining F. E x a m p l e 1. Let K he a field of characteristic 2 and consider V = K".

Then

94

1 Projective Geometry

is a symmetric bilinear form of rank 2 for which 6(j;, p) = 0 holds for all p G V ; hence it does not determine a polar m a p in the associated projective geometry. T h e following lemma is obtained in a way similar to the argument in the proof of Proposition 1 showing, in the case char i f ^ 2, t h a t every (T-Hermitean biform b =^ 0 defines a polar m a p . • L e m m a 2. Let K be a skew field, char K ^ 2, and letb y^O be a biform over V. Then there is an f (^ V with 6(p,p) y^ 0.

a-Hermitean

P r o o f . Assume t h a t 6(p,p) = 0 for all ^ eV. Because of 6 7^ 0, there are p, t) G V such t h a t 6(p, t)) 7^ 0; due to the linearity of 6 in t) we may suppose 6(p, t)) = 1. Then for every A G i f 0 = 6(p + t)A, p + t)A) = 6(p, t)A) + 6(t)A, p) = A + cr(A). Thus o-(A) = —A for all A G if, hence, in particular, cr(l) = 1 = —1. But this implies char i f = 2. D D e f i n i t i o n 2. Let b : V x V ^ b is the subspace

K he a-Hermitean. T h e defect subspace of

VF6:={pGF|6(p,t)) = 0 f o r a l l t ) G F } .

(6)

T h e defect of b is defined by def(6) := dim Wb.

(7)

If a is the a-linear m a p fromV to V defined by 6, and if, moreover, dim V < 00, then the rank of b is defined by the equation rk(6) : = d i m F - d e f ( 6 ) .

(8)

Obviously, the defect subspace of b is equal to Wb = Ker a = Ker F, if F denotes the polar m a p defined by 6, cf. (7.9). Hence we have rk(6) = rk(a). D 1.9.2 C l a s s i f i c a t i o n of P o l a r M a p s The following proposition can be proved in a way similar to Proposition L5.9.5: P r o p o s i t i o n 3. Let b be a a-Hermitean biform on the (n + I)-dimensional vector space V over the skew field K with char i f ^ 2. Then there is a basis (Oj) for V in which the matrix ofb has diagonal form:

1.9 Polarities and Quadrics //3oO 0 •

0\

13, r-l

b{ai,aj

95

(9)

0

Vo

0/

f^h = <J{M 7^ 0 / o r /i = 0 , . . . , r - 1; r = rk(6).

(10) D

Of course, the numbers /3ft, G i f in (9) are not uniquely determined by b. Replacing o^ by o^ = a^A^, A^ G K*, relations (9), (10) remain valid, and hence /3ft := 6(0ft,0ft) = cr(Aft)/3ftAft. (11) T h e transformation (11) can be used to obtain normal forms for polar maps and (T-biforms, respectively. These normal forms depend upon the properties of both the skew field K and the anti-automorphism a. For the classification one first has to describe all the involutive anti-automorphisms a of K, and then t o determine, for each a, corresponding normal forms for the a-Hermitean biforms b. In the subsections to foUow we wiU discuss the cases i f = R , C, H separately. Exercise 2. Let i^ be a polar map of the projective geometry ^ " , n > 2 and char if / 2. Prove: a) There are n + 1 points at e P",i = 0 , . . . ,n, such that Oi e F{aj) for i / j and ai ^ F{ai) if and only if i^ is a polarity. - b) Formulate the dual of Statement a). - c) The points ai with the properties stated in a) are in general position; choosing a projective frame {ai;e) with at as base points, the matrix of F has diagonal form (9) with r = n + 1. A simplex (a^), i = 0,... ,n, with the properties from a) is called a polar simplex for the polarity F. Exercise 3 . Let i^ be a polarity satisfying the assumptions from Exercise 2, and let ai e P", i = 0 , . . . ,n, be a sequence of points in general position. Prove that the following statements are equivalent: a) (ci) is a polar simplex;

h) Fiai) =

\/aj;

= A^(« j^i

96

1 Projective Geometry

1.9.3 T h e R e a l P o l a r M a p s As is well-known, the only anti-automorphism of the field of real numbers is the identity a = idR. Hence, by Sylvester's Law of Inertia, Proposition 1.5.9.6, polar maps are classified by their rank and index. Its projective formulation is P r o p o s i t i o n 4. Let K = H be the field of real numbers, and let ^ " be an n-dimensionalprojective geometry ower R , n > 2. Choose any basis and denote by Fr^i the polar map of ^ " whose matrix in this basis has normal form (9) (cf (i.5.9.49);, where^ fJx = - 1 , A = 0 , . . . , / - ! ; fJ^ = 1, /i = / , . . . , r - 1 , 0 < / < [r/2], 0 < r < n + 1 . (12) Denoting by x* the homogeneous point coordinates, and by yi the homogeneous hyperplane coordinates dual to them, the polar map y = Fr^i{x) has the following normal form: Vj = -x^ for j = 0,...,l1; Vk = x'' for k = I,. . .,r -1, {0
(13)

to one of the Fr^i, and no

P r o o f . According to Exercise 1.2.1.3, cf. also CoroUary 3.16, a = idR, and the biform 6 determining F has to be bilinear; moreover, because of (2) it is also symmetric. In (9), the /3ft, 7^ 0 are arbitrary. Choosing the transformation (11) with Aft := |/3ft|^^'^ and renumbering the basis elements in such a way t h a t the first / diagonal entries are equal to —1 followed by r — / times the entry 1, we arrive at Sylvester's Law of Inertia 1.5.9.6; the matrix of h has the normal form (9) with 0 < / < r. For / > [r/2] the form (12) is obtained via multiplication of 6 by K = —1 and a suitable numbering of the basis elements. Choosing again the indices in the range i = 0 , . . . , n implies t h a t each polar m a p of ^ " is projectively equivalent to one of the Fr^i. By (8.13) and Proposition 1.5.9.6, no two of the Fr^i are projectively equivalent. •

1.9.4 T h e C o m p l e x P o l a r M a p s As in the case of colhnear maps (Example 4.3), the continuity of correlative maps implies t h a t we only have to consider continuous automorphisms of the field C of complex numbers, cf. also Example 7.4. Hence there are two types of polar maps to be classified: those corresponding to a = i d c and the ones associated with the conjugation T : z ^^ z. For i f = C we agree t h a t the As usual, for 5 e R the greatest integer less than or equal to ^ is denoted by [5].

1.9 Polarities and Quadrics

97

t e r m polar map always refers to a = i d c - On the other hand, the polar maps associated with a = T will be called antipolar. Starting from (9) the proof of the analogue to Proposition 4 is straightforward. P r o p o s i t i o n 5. Two polar maps of a complex projective geometry ^ " , n > 2, with a = i d c are projectively equivalent if and only if they have equal rank r, 1 < r < n + 1. Hence the coordinate representations Fr : yj = x^ for j = 0,...

,r - I, yk = 0 for k = r,...,

provide a complete set of normal forms for these polar maps. C o r o l l a r y 6. All polarities of the complex projective with a = i d c are projectively equivalent.

geometry

n,

(14) •

^ " , n > 2, •

T h e following proposition dealing with r-Hermitean biforms and the corresponding anti-polar maps is proved along the same lines as the Law of Inertia (cf. Proposition 1.9.13). As the proof works without any change also in the case of quaternionic projective geometries, we simply combine b o t h situations into one statement. P r o p o s i t i o n 7. Let T be the conjugation in the field C of complex numbers or in the skew field H of quaternions. Denote by Fr^i the polar map of the that, , in projective geometry ^ " that in a fixed coordinate system, has the normal

form % = --xJ for 3 = 0, yk =- x'' for k =-l, yh = 0 for h =-r,

.1-1] r - 1 , (0 < r < n + l ) ; ,n,

(15)

where the matrix (9) satisfies (12). Every polar map of the projective geometry ^ " determined by a r-Hermitean biform is projectively equivalent to one of the maps Fr^i, and no two of the latter are projectively equivalent. P r o o f . According to Proposition 3 we may start from normal form (9) for the matrix of F. Because of (10), we have f3h G R*. As in the proof of Proposition 4 we obtain the diagonal form for the matrix of F with property (12). The following lemma characterizes the number / in an invariant way: L e m m a 8. Let i f = R, C, H. Take a to be the identity in the case of the real numbers, and take conjugation as a for K = C,ii. Consider a a-Hermitean form b whose matrix has the diagonal form (9) in a suitable basis (Oj). Then the number I of negative values among the diagonal entries pi is equal to the maximal dimension of a subspace W C V " for which the restriction h\W^ X W is negative-definite, i.e., 6(j;, p) < 0 for all p G W^,f ^ o.

98

1 Projective Geometry

P r o o f . Because of (2), even in t h e case i f = H we immediately have /3ft, G R . Take f3k < 0 for A; = 0 , . . . , / — 1, and /3ft > 0 otherwise. Obviously, t h e restriction of b t o t h e subspace U := £(oo, • • •, a ; - i ) spanned by Oo, • • •, a ; - i is negative-definite. Consider t h e complementary subspace B := £ ( o ; , . . . , 0„). For every subspace W C V whose dimension satisfies dimVF > / there is a vector p G BnW such t h a t p 7^ o by t h e dimension formula. Since we obviously have 6(j;, p) > 0 for all ^ e B, b cannot be negative-definite on W. D Lemma 8 implies t h a t t h e index I of 6, defined t o be t h e number of negative diagonal entries in (9), does not depend on t h e chosen basis in which b has this diagonal form. T h e same holds for t h e dimension n — r of t h e defect subspace of 6, hence also for t h e rank r. Summarizing, we arrive at t h e analogue of Sylvester's Law of Inertia L5.9.6 for r-Hermitean biforms in t h e cases i f = C , H , and complete t h e proof just as we did proving Proposition 4. D Exercise 4 . Let T^"+^ be a complex vector space, and let r be conjugation in C. A T-biform b is called skew Hermitean if it satisfies the condition 6(t),p) = -6(p,t)), p , t ) e F .

(16)

Prove: a) Each skew Hermitean r-biform determines an anti-polar map of ^"(V), that can also be described by a r-Hermitean biform. - b) Classify the skew Hermitean biforms with respect to the action (8.12) with K = 1 of the linear group of \^"+^. To this end, prove that b(j, t)) is skew Hermitean (or Hermitean, respectively) if and only if i6(p,t)) (with i^ = — 1) is Hermitean (or skew Hermitean, respectively). 1.9.5 T h e Q u a t e r n i o n i c P o l a r M a p s It is well-known t h a t t h e center of t h e skew field H of quaternions is t h e real subfield R . Denote by R ^ t h e real subspace of H spanned by t h e imaginary units i , j , k G H . First we prove: L e m m a 9. The only involutive anti-automorphisms a of the skew field H of quaternions are the conjugation T : q 1-^ q and the anti-automorphisms defined by r , ( A ) : = - < / A < / , < / G R ^ , | < / | = 1.

(17)

P r o o f . Let a be an arbitrary anti-automorphism of H . Then, according t o Example II.8.8.5, T o a has t o be an inner automorphism of H . Hence there is (/ G H*, \q\ = 1, such t h a t T o a = a^. This implies a = T o a^. Since a is involutive, we obtain for all A G H q^^qX =

Xq^^q.

Consequently, q^^q must belong t o t h e center R of H . Now |
1.9 Polarities and Quadrics

99

and o-(A) = —qXq. A straightforward computation shows that each of these anti-automorphisms is involutive. • The polar maps corresponding to r-Hermitean biforms were already classified in Proposition 7. For later application, we write out the formula for such a biform over an n-dimensional vector space V^ below; according to Sylvester's Law of Inertia this is characterized by its index / and rank r and thus, in a suitable basis (Oj),« = 1 , . . . n, has the form: I

r

6(y, t)) = -J2 ^"y" + E ^V, 0 < / < r. a=l

(18)

c=l + l

Here x^,y^,i = 1 , . . . , n, are the vector coordinates of p, t) in this basis. Now we turn to the Tg-Hermitean biforms and start by noting Lemma 10. Let Tq be defined by (17). Then a biform b : V x V ^ H is Tq-Hermitean if and only if b = qb is skew Hermitean in the sense of (16). • The proof is nothing but a straightforward calculation. Since Tq = Tq = T^q, we immediately obtain Corollary 11. / / b is a Tq-Hermitean biform over V, then b := ±iqb is a T^-Hermitean biform over V. D First we consider the Tj-Hermitean biforms. Later, the general case will be reduced to this situation. A simple calculation shows: The set of fixed elements of Ti is the real subspace o / H spanned by l,j,k.' {/3GH|ri(/3)=/3} = £ R ( l , j , k ) .

(19)

Let now 6 be a Tj-Hermitean biform, and let {aj),j = 1 , . . . ,n, be a basis in which the matrix of b has the form (9). In this case, transformation (11) takes the form /3=-iAi/3A. (20) Choosing first A = |/3|^^/^, which is possible by /3 7^ 0, we obtain |/3| = 1. Thus we may already assume |/3| = 1 and henceforth only consider transformations with|A|^ = AA = 1, i.e. A = A^^. Because of (19), the element i/3 lies on the unit sphere in R^. According to Exercise II.8.9.5, each element of the special orthogonal group SO(R^) can be represented as an inner automorphism ax, |A| = 1. As is well-known, SO(R^) = SO(3) acts transitively on the unit sphere; hence we can find A such that Ai/3A = i, i.e. /3 = 1. Performing these transformations for each of the diagonal entries f3h in (9), Proposition 3 together with Corollary 11 implies Proposition 12. Let V be an n-dimensional vector space over the skew field H of quaternions, and let Tq be the anti-automorphism defined by (17). Lfb is

100

1 Projective Geometry

a Tq-Hermitean biform, then there exists a basis (a,,), v = 1,... ,n, ofY that b has the following normal form: r

such

r

6fet)) = ^ r , ( x " ) f i y " = - < ? ^ x " i y " . a=l

(21)

a=l

Here r is the rank of b, x" and y" are the coordinates of p and t) with respect to the basis (Oj/). P r o o f . By the above, for
S(y,t)) = ^ x " i y " .

(22)

The proof is a consequence of (21) and Lemma 10. Note that (21) and (22) provide sets of normal forms with respect to the action (8.12) with K = 1 of the quaternionic linear group for the Tg-Hermitean and the skew Hermitean biforms, respectively; no two of these normal forms are equivalent. In fact, the rank of b is obviously invariant, and to be Tg-Hermitean or skew Hermitean, respectively, is a property preserved by any linear transformation. Combining all these results immediately leads to the classification of polar maps: Proposition 14. Let F be a polar map in the n-dimensional projective geometry ^ " over the skew field H of quaternions, n > 2. Then F is projectively equivalent to one of the maps Fr^i by Proposition 7, or F is projectively equivalent to one of the maps Jr with normal form Jr '• Va = x"" i for a = 0, . . . , r — 1; yc = 0 for c = r,... ,n] 0 < r < n. (23) In both cases, r is the rank of F, and the corresponding anti-automorphism is the conjugation T in H. NO two of these normal forms are projectively equivalent. P r o o f . According to Proposition 7.6 F can be described by aa-biform (7.20). Proposition 1 implies that a is an involutive anti-automorphism of H. By Lemma 9 the anti-automorphism a has to be either conjugation, a = T, or
1.9 Polarities and Quadrics

101

we can always require this to hold. From Proposition 12 we derive the following set of normal forms for the polar maps with a = T;: ija = Tiix"") = — ix" i for a = 0, . . . , r — 1; yc = 0 for c = r,..., n] 0 < r < n. (24) The transition to 6 = i 6 discussed in Lemma 10 leads to the skew Hermitean normal form (22) and this again to (23). No two of the normal forms (15) (with (12)) and (23) (or (24) instead) are projectively equivalent: According to Proposition 7, and because of the invariance of the rank, it suffices to show that no Tj-Hermitean biform can be transformed into a r-Hermitean biform b y^ 0 via multiplication by /i G H*, and vice versa. Let 6i = hbhe a Tj-biform. Since 6 is a r-biform, 6i is a biform with respect to the anti-automorphism C '-^ h^h^^. If this were equal to T;, then, up to an irrelevant real factor, we had h = i. But this again implied 6i(t),j;) = i6(t),j;) = i6(?,t)) = ii6(j;, t))i, hence 6i(t),j;) = —Ti(6i(j;, t))). However, this can only be Tj-Hermitean if and only if 6 = 6i = 0. D 1.9.6 Quadrics Because of Equation (1), the projective classification of polar maps immediately leads to a corresponding classification of quadrics. AUowing for some incorrectness, we will call two quadrics projectively equivalent if the polar maps defining them are related this way. It is easy to prove that equivalence of the polar maps implies equivalence of the quadrics as sets; more precisely: If Fo, Fi are two equivalent polar maps, and if <; is a polarity realizing their equivalence, i.e. Fi = gFog^^, then for the associated quadrics Qo,Qi we have the relation Qi = g{Qo). The converse of this statement, however, is, in general, not true. Consider, for example, the projective plane over the field Q of rational numbers. For every prime number p G Z the quadrics determined by the equations ( x ° ) 2 + p ( x i ) 2 = 0 a n d {xy

-p{x^f

=0

are equal. In fact, they only contain the point satisfying x'-' = x^ = 0, as can easily be seen using the prime decomposition in Q. The polar maps defining them, however, cannot be equivalent: Any polarity over the rational numbers realizing the equivalence would also be an equivalence over the real numbers, but this contradicted the Theorem of Inertia. Considering the equations over the real numbers, their solution sets are obviously not projectively equivalent: One consists of a single point, and the other is the union of two lines. Over the complex numbers the defining polar maps, and hence their solution sets as well, projectively equivalent.

102

1 Projective Geometry

R e m a r k . In the case that the scalar domain K is an algebraically closed field, e.g. the field C of complex numbers, and if the polar maps and quadrics un question are defined by a symmetric bilinear form, then the converse holds in general: //, under the assumptions above, the quadrics Q, Q corresponding to the symmetric bilinear forms b, b are equal, then the b and b are proportional. To prove this we show that the quadrics are different, if the forms are not proportional. So take a point x = [p] with 6(j;, p) = 1 (cf. (3)). If 6(j;, p) = 0, then X G Q \ Q, i.e. the quadrics are different. Hence, normalizing 6 by a suitable factor K G if * we may assume that 6(p, p) = 1 as weU. Since the forms are supposed to be not proportional, there is a vector t) such that 6(t), t)) y^ 6(t),t)). Now consider the line joining x to the corresponding point y = [t)]. It has the parametrization z(t) = [3(t)] w i t h 3 ( t ) = p + t)t. Its intersection with the quadric Q is determined by the roots ti,t2 of the quadratic equation

Hiit), iit)) = t^ + 2tK?, t)) + Kt), t)) = 0, for which tit2 = b{t), t)). Considering the intersection QOxVy we analogously obtain two parameter values ti, t2 with tit2 = Kt), t))- These roots exist, since the scalar domain is an algebraically closed field. Because of 6(t), t)) ^ 6(t), t)), they have to be different. Hence the intersections with the line x V y, and therefore the quadrics, cannot be equal. • The affine classification of quadrics can be derived by first distinguishing a hyperplane at infinity and then discussing the various position possibilities for the quadric with respect to this hyperplane. Conversely, one might start from the affine classification (cf. Proposition 1.5.9.11 for the case if = R) and then, after having passed to homogeneous coordinates, decide, which among the affine types are projectively equivalent. Example 2. Let if = R. We denote by Qr^i the quadric corresponding to the normal form Fr^i. Then the equation of the quadric Q^.o is r-l

^(x*)2 =0, 0 < r < n + l ;

(25)

i=0

hence Q^.o = Ker Frfi. Note that we never consider the nopoint o as an element of any quadric. In particular, Q„+i,o = 0 is the empty set; this is expressed by calling it the empty quadric. The set of base points for the related coordinate system a^ = [ofc], k = r,... ,n, span the kernel KerFrfi = a^ V . . . Va„, which thus, for r < n, always lies in the coordinate hyperplane x° = 0. Since this projective (n — r)-plane is now defined by the quadratic equation (25), it is usually called a double counted projective (n — r)-plane. The reason for this term is the following: Consider the family of quadrics defined by

1.9 Polarities and Quadrics

103

depending on the parameter t G R. For t ^ 0 each of these quadrics "decomposes" into the pair of hyperplanes X

0

•tx^ = 0 , x ° + t x ^ = 0 .

Letting then t tend to zero, both the hyperplanes move towards one and the same hyperplane (x°)^ = 0, which, for this reason, has to be counted twice. Let us now return to equation (25). As in Section 5 we choose the hyperplane x° = 0 as the hyperplane at infinity for an affine space A " C P " and consider, as usual, only the part AQrj := A " n Qrj as the affine quadric. In the inhomogeneous coordinates, the affine points x° = 1 from (25) satisfy the equation for AQr^: r-l

j2{xr = -1. This, obviously, has no real solution, so that for all r the affine quadric AQrfi is the empty set: The kernel KerF^.o completely lies in the hyperplane at infinity. A similar consideration shows: The quadric Q„+i,i has the equation n

-ixy + J2{x^f=0,

(26)

i=l

its section with the hyperplane at infinity x'-' = 0 is the empty quadric Qn,o, and hence it is the hyperellipsoid AQn^i^i = Q„+i,i, whose equation has the following normal form in inhomogeneous affine coordinates n

^ ( x * ) 2 = 1,

(cf. Proposition L5.9.11)

(27)

i=i

Interpreting the x* as the orthonormal coordinate of an n-dimensional Euclidean space (27) becomes the equation of a hypersphere with radius 1 and the origin of the coordinate system as its center; for n = 3 Figure 1.9 shows an ellipsoid, which is projectively, even affinely, equivalent to it. It is not difficult to prove that for n > 1 the hyperellipsoids are the only non-empty real quadrics for which there is a hyperplane not intersecting them. Hence the hyperellipsoids are the only real projective quadrics that can be represented as closed surfaces in a Euclidean space; any other non-empty quadric intersects the hyperplane at infinity, if it is considered as affine or Euclidean. For topological reasons they are, however, always compact subsets of the ambient projective space. Let now Q = Qn+i,i be a non-degenerate quadric of the n-dimensional projective space; the degenerate quadrics may always be viewed as cones over non-degenerate ones, cf. Example 4 below. We suppose that its index / is greater than one and write its equation in the form

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1 Projective Geometry

Fig. 1.9. An Ellipsoid

Here |p| denotes the Euclidean norm in the vector space V " ^ for which the x^,i = 0,... ,n, are orthonormal coordinates. Since these coordinates are, at the same time, also homogeneous point coordinates for the associated projective space, and as the nopoint does not belong to the quadric, for arbitrary r > 0 the solution set for this equation is the product of two spheres of dimensions / — 1 and n — / of radius r with the origin as their center lying in orthogonal coordinate planes: n — lf

S^,-\r) X Sr\r) I ( ? O J ? I ) I ? O — (X

, • • • ,X

), P]^ — (X'

= >a;"),|?ol = l ? i l = r } -

Since each point x = [(POJ?I)] ^ Q has precisely two representatives in this solution set, differing in the factor ± 1 , we stiU have to identify opposite vectors in the solution set; assuming r = 1 leads to: The real, non-degenerate projective quadric Qn+i,i C P " of index I > I is bijectively represented by the quotient set l-l Q n+l,l s- X S"-'-/{±l}, I > 1. Applying straightforward topological or differential geometric considerations the bijection in question can be proved to be a homeomorphism or diffeomorphism with respect to the canonically induced structures. In the case / = 1 already considered, the first factor is 5*° = {1, —1}; choosing, say, 1 as the representative for pg the representative of the point is already uniquely determined, and we again obtain the representation of the quadric as the hypersphere (27). Fixing a homogeneous coordinate system the points x G P " of the real projective space are bijectively represented by unit vectors whose

1.9 Polarities and Quadrics

105

first coordinate is positive; this way, the projective space itself is represented as the unit hypersphere of the associated vector space with identified opposite points (cf. also Example 2.2.1): P " = S ' " / { ± 1 } , (if = R ) . Since now, in the representation of Q just described, already the first vector Po has to be non-zero, we conclude for the representation of the real nondegenerate quadrics in general Qn+i,i = - P ' " ' X S^-\

1 < / < (n + l ) / 2 .

For n = 3 we obtain apart from the ellipsoid (/ = 1) also the hyperholoid QA^2, whose affine image (cf. Fig. 2.18) has to be completed by an eUipse in the plane at infinity x^ = 0. Since the projective line itself is homeomorphic to a circle, the above representation amounts to the homeomorphy of the hyperboloid with the product S^ x S^, which is also called a torus. Figure 2.53 shows a torus embedded into the Euclidean space E as the surface with the parametrization described in (2.7.56). Obviously, the hyperboloid itself is not embedded in this way into the Euclidean (or projective) space; note t h a t each plane, hence t h a t at infinity as well, intersects the hyperboloid in a non-empty set. • Exercise 5. Prove for K = R and n = 2: li Q G A^ is one of the afRne quadrics ellipse, hyperbola, or parabola, then there is a unique projective quadric Q
^i/(?'t)) = Zl^'^'=-In fact, according to Exercise 5.4 we may also identify the vector space W" := £ ( 0 i , . . . , 0 n ) of the hyperplane at infinity with the vector space of the afline geometry A". Prove that for p / 0 the polar F{x) of the point x := [Oo + p] is the hyperplane with Hessian normal form (cf. (1.6.3.37))

106

1 Projective Geometry z = [ao+3]Gf(.)^(3,^) =

^ .

For p tending to 0, the point x tends to ao, and Fix) converges to the hyperplane at infinity F{ao) = H. Example (Equation (1, i, 0 , . . . , vertices of

3 . Obviously, for K = C the quadric Q „ + i corresponding to F „ + i (25) with r = n + 1) is not empty; e.g., the point with coordinates 0) belongs to Qn+i- Again denoting by Uj = [Oj], j = 0 , . . . , n, the a polar simplex for F „ + i we obtain Cp := [a2p + a2p+i i] G Q „ + i , p = 0,...

,k := [{n - l ) / 2 ] ,

(28)

M'^ :=coV...VCfc c g „ + i . Note t h a t the restriction of the bilinear form associated with F„^i vanishes identically.

(29) to

M'' •

D e f i n i t i o n 3. Let 6 be a a-Herniitean, skew Herniitean, or alternating biform over the vector space V with the skew field K as scalar domain. A subspace W C V is called isotropic (with respect to b) if there exists p G W, p 7^ o, such t h a t for aU t) G Vt^ the equation 6(p, t)) = 0 holds; W is called totally isotropic if the restriction of b vanishes, b\W x W = 0. D C o r o l l a r y 15. Let char K ^ 2, and let b he a a-biform determining the polar map F of the projective geometry ^ " . Then a projective subspace A is contained in the quadric Qp corresponding to F if and only if the restriction of b to it vanishes, bj^ = 0, i.e., if the subspace W
we have •

C o r o l l a r y 16. Let char K ^ 2. If A is a subspace contained in the quadric QF of the polar map F, then A V K e r F also lies in Qp; in particular, KerFcQpP r o o f . If W is the vector space corresponding to A, then W is totally isotropic. Since the defect subspace (6) is also totaUy isotropic, and, in addition, b{Wi„ W) = 0, the sum W + Wf, again is totally isotropic. As the defect subspace corresponds to the kernel of F, this implies the assertion of Corollary 16. • E x a m p l e 4. Let Qp be a quadric with Ker F ^ o. Then K e r F is called the vertex space of the quadric. For char i f ^ 2, according to (16) we have X V K e r F C QF for each x G QF- For this reason, quadrics corresponding to degenerate polar maps are also called projective cones, and the maximal projective subspace contained in QF is the generator of the cone. •

1.9 Polarities and Quadrics

107

Exercise 7. Let char A" / 2, let QF be the projective cone determined by the (7-biform b, and let A be a subspace complementary to KeiF. a) Prove: QF

=

y

a; V Ker F,

xeQpnA (xWKeiF)

A {y V K e r F ) =Ker_F for x,y e QF n A,x

y^ y.

b) The restriction (of. Proposition 7.7) F\A is a polarity described by the biform b\W X W , where W denotes the vector space associated with A. - c) If A i , A2 both are complementary subspaces to KeiF, then the polarities -F|Ai, F\A2 are projectively equivalent. Using the elementary definition of cone sections (Exercise 1.5.9.3) this leads to a new proof for the first part of Exercise 5.— Hint. Recall, in this connection, (7.33) and Proposition 8.2. 1.9.7 P o l a r M a p s of P r o j e c t i v e L i n e s As a preparation for the study of tangents and tangent subspaces of a quadric, we first want to discuss polar maps F of a projective line ^ . For the rank r of F the only possible values are r = 1, 2 by (8). T h e first case is the subject of the following simple proposition: P r o p o s i t i o n 17. Let F be a polar map of the projective line ^ over the skew field K, char K ^ 2. Then the rank r of F is equal to 1 if and only if all points X ^ K e r F have the same image F{x) = a; in this case, QF = K e r F = a. P r o o f . Let 6 be the a-Hermitean form defining F. Now F has rank 1 if and only if K e r F just consists of a single point, say a = KeiF. Obviously, a G QFAccording to Proposition 3, we may suppose t h a t the matrix of h has diagonal form; hence the equation for Q F is cr(x°)/3oa;° = 0 with /3o ^ 0. This implies x° = 0. Thus QF = ai = [oi] is the first base point of the corresponding projective frame. Hence QF contains just one element, and this imphes a = ai. The coordinates of the image point F{x), x ^ a, can be computed using the annihilator of the covector with components Mo = cri^x )/3o, Ml = 0, yielding y° = 0, y^ = 1. Thus F{x) = a. The converse immediately follows from the definition of the rank. D T h e possibilities occurring in the case r = 2 again depend b o t h on the field K and on the type of the quadric. According to CoroUary 8.3, F is now bijective and thus defines an involution of P .

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1 Projective Geometry

E x a m p l e 5. Let K be a skew field with char if ^ 2. By Proposition 3, any polarity of ^ can be described by a a-biform 6(y, t)) = a(x°)/3oy° + <j{x})f3iy' = 0.

(30)

Because of 6(aj, Ck) = f3i ^ 0, i = 0,1, the base points Oj = [Oj] do not lie on the quadric Qp- For every point x G Q F we may thus take the coordinates to be (x°,x^) = (1,^) with ^ G if*, and then write ^ = ^{x). This imphes a; G Q F ^ ^ a(e)/3ie = -Po-

(31)

If if is a field, and a = id^, then (31) is equivalent to

e = -fhifh-

(32)

So there are only two possibilities, Qp may either be empty or else contain two points. This coincides with the result of Exercise 4.13 b); in fact, in this case F is, at the same time, a projectivity defined by the linear isomorphism y

-pix\y^=pox°.

(33)

Relation (30) immediately implies (even in the case of an arbitrary skew field K) that the points of the quadric Qp are nothing but the fixed points of the involution F. For if = R both possibilities occur according to Proposition 4: QF = 0 for i^ = i^2,o, and QF consists of two points for F = i^2,i- If K is algebraicaUy closed, e.g. if = C, and a = id^, then it is just the second case that occurs. • E x a m p l e 6. Consider the Hermitean normal forms F from Proposition 7 for if = C , H , n = l , r = 2. The equation of this quadric is ez°z° + z^z^ = 0 , e = ± l .

(34)

For e = + 1 equation (34) has no solution in P . I n case e = — 1, each solution of (34) satisfies x° 7^ 0,x^ ^ 0; hence we can take the coordinates of the solution point to be (2;°, z^) = (1, ^); ^ G if*. Then ee = \i? = 1 (e = - 1 ) ,

(35)

i.e., the set of solutions is the unit circle for if = C, and the unit hypersphere S*^ in case if = H. Let now x G P^ be a variable pre-image point with coordinates (1,C), C ^ K*. For the homogeneous coordinates of the image point y = F{x) we then obtain from (30):

ey° + ey' = 0. Because of ^ G if*, necessarily y° ^ 0; again setting (y°,y^) = (1,??) this yields

1.9 Polarities and Quadrics ri=-eC'

= -ei/\i\''{ieK*).

109 (36)

In case e = — 1 this is nothing but the formula for the inversion (or the map hy reciprocal radii) in the unit circle S*^ C C or in the unit hypersphere S*^ C H, respectively. For e = + 1 this inversion still has to be composed with the point reflection in 0. In each case these involutions interchange the points 0,00 eK. a Example 7. Let now K = H. Consider the involution J2 on the quaternionic projective hue P ^ , cf. (23). The associated quadric Q has the equation x°ix°+x^ix^ =0.

(37)

Its solutions are all points x with coordinates (1,C) satisfying ^ = j c o s a + ksina, a G R,

(38)

i.e., the unit circle in the j,k-plane of H. The image y = F{x) of the point X e P^ with coordinates (1,C), C ^ H*, has coordinates (1,??) satisfying V = i^im'-

(39)

In H = R** this is a reflection in the j , k-plane composed with the involution in the unit hypersphere S*^ C H; again the points 0 and 00 of H are interchanged hy F. a 1.9.8 Tangents and Tangent Subspaces In this section we want to describe the possible positions of a hue relative to a quadric. This will lead to the notions of tangents and, more generally, tangent subspaces of a quadric. Example 8. Let again char if ^ 2, let F denote a polar map, and let QF be the associated quadric in P " . For a; = [p] G QF consider the line h = x\/ y, where y = [t)] ^ x is an arbitrary point in P " . Each point z e h, z ^ x, can be represented in the form z = [fS^ + t)] with uniquely determined ^ G if. Because of 6(p, p) = 0 we have: z G QF C\h if and only if z = x or if the following equation is satisfied:

^(e)K?,t)) + Kt),?)e+Kt),t)) = o.

(40)

Next we will distinguish the following cases: a) i\{F\h) = 0: This holds if and only if 6(p, t)) = 6(t), t)) = 0; by Corollary 15 it is thus equivalent to a; V y C QFb) Yk{F\h) = 1: Then, according to Proposition 17 and Example 4, Qpifi = QF n h consists of a single point, hence QF C\ H = {x}. This is the case if and only if 6(p, t)) = 0 as weU as 6(t), t)) 7^ 0 in (40). If x is singular.

110

1 Projective Geometry

i.e. X e KevF, then 6(p, t)) = 0 for all y G P " . Thus {x\/y)nQp = {x} for every y G P"'\QF- If a; is a regular point of QF, then 6(p, t)) = 0 defines the polar F{x) of a;, and for the points y (, F{x), y y^ x, we now have x\/ y C F{x). In this case as weU as in case a) the line h = x \/ y is caUed a tangent for the quadric QF- More generally, we say that a projective subspace B G ^ " is tangent to the quadric Q F at a regular point x G Q F if a; G B C F{X). If a; G QF is a regular point of the quadric, then TXQF '•= F{x) is the tangent hyperplane of the quadric QF at the point x. The totality of tangent subspaces at x forms the bundle F{x)/x. For the singular points x G Kev F tangent subspaces will not be defined. c) Yk{F\h) = 2: In this case h is caUed a secant or secant line of QF- In fact, equality holds in (40) if and only if 6(j;, t)) ^ 0. Hence x has to be a regular point of QF- Moreover, the rank condition together with (40) implies that QF H h contains at least one more point z ^ x. Since 6 is a-Hermitean, the equation e=-6(t),j:)-i6(t),t))/2

(41)

determines a solution for (40). (Because of char if 7^ 2, 2 7^ 0 belongs to the center Z{K*)) As Examples 6, 7 show, there may well be further solutions; if, however, if is a field, and 6 is bilinear, then (41) determines the unique point y G QF H h different from a;, cf. Example 5. • The following proposition contains a geometric interpretation of isotropic subspaces: Proposition 18. Let char if ^ 2, and let F he a polar map of the projective geometry ^ " described by a a-Hermitean biform b. Moreover, consider a projective subspace o ^ A = -KiyV) with associated vector space W. Then W is isotropic if and only if Ker(F\A) ^ o. Here, Ker(i^|A) = ( A A K e r i ^ ) U B ( A ) ,

(42)

where B(A) denotes the set of those regular points of the quadric QF at which A is tangent to QFP r o o f . The defect subspace oi b\W x W is the vector space associated with Ker(i^|A), which immediately implies the first assertion. Moreover, X G Ker(i^|A) if and only if a; G A C i^(a;).

(43)

For this, there are just two mutually excluding possibilities: Either i^(a;) = P " , i.e. x G AAKeri^, or Dimi^(a;) = n— 1, and then x belongs to B{A). D Corollary 19. Under the assumptions of Proposition 18 , let F be a polarityThen W ^ o is isotropic if and only if A = 'K{W) is tangent to QF in at least one point x; the set B{A) of points of contact is a projective subspace of A, whose vector space is the defect subspace of h\W x W. D

1.9 Polarities and Quadrics

111

Exercise 8. Consider the hyperboloid and show that there are tangent subspaces A C P" for which B{A) / QF n A; hence, apart from the points of contact, other points of QF niay belong to A as well (cf. Figure 2.1). 1.9.9 D u a l i z a t i o n : C o q u a d r i c s Although we defined correlations and, more generally, correlative maps as monotonously decreasing maps between projective geometries, we nevertheless preferred to use points as the basic elements, e.g. in the definition of quadrics. The notion which is, in this sense, dual to t h a t of a correlative m a p now starts from hyperplanes as the basic elements; to each hyperplane U G P ' " a point X = F{U) G <5™ or the nopoint o are assigned, where F is determined by a
= [a(u)] G QT-

(44)

T h e pencil of hyperplanes incident with x is determined by T{X) = x , and we thus obtain an ^-correspondence between hyperplanes U G P ' " , D=me g'™ via D i F{U] ^=^ B(t),u) := (t)|a(u)) = 0.

(45)

Now, the (T-biform B is linear in f G W' and a-linear in u G V'. Again, let char i f 7^ 2, Q™ = «P", and let B be cr-Hermitean over V x V\ cf. (2). Then B(u,u)=0

(46)

defines the set Q'p := {U = [u] G P ' " | t / L F{U)), which will be caUed the coquadric associated with F. The traditional names for Qp and Q'p, respectively, are curve (n = 2), surface (n = 3), or hypersurface (n arbitrary) of second order or second class. Dual to the notion of the tangent hyperplane F{x) at a regular point x G QF is the notion of the point of contact F{U) ^ o at the regular hyperplane U G Q'p. If F is a polarity, then for the pairs (a;, X ) of pole and polar, X = F(x), we immediately have X I F{x)

-^^ X I F{X),

(47)

and hence Q'F = F{QF),

QF = F{Q'P):

The coquadric Q'p (quadric QF) is the set of tangent contact) of the quadric QF (coquadric Q'p).

(48)

hyperplanes

(points of

Exercise 9. Let F be a polarity, and let (uj) = {a{x'^)){aij) be its matrix representation in homogeneous coordinates. Show that, in the same coordinate system, equation (46) for the coquadric Q'F corresponding to F is described by n

Y^ Uid'aiuj) i,j = 0

= 0 with (c*^) := {aij)-\

(49)

112

1 Projective Geometry

1.10 Restrictions and Extensions of Scalars In order to further investigate the dependence of the quadric on the scalar domain, particularly obvious in Examples 9.2, 9.3, it will be useful to consider how the projective geometry changes under its variation. In this section, let L always be a skew field, and let if C L be a skew subfield. Then L is also a right vector space over K; we will always assume r := dim^ L < oo. Generalizing Definition II.8.5.4 L is then called a finite extension of K of degree r. For the cases we will mainly deal with here the degrees are: dimpt C = dime H = 2, dimR. H = 4. Fix a basis (a^), p = 1 , . . . ,r, for L such that ai = 1 is the unit element of L, and let V be a vector space over L. Restricting the scalars to if, more precisely, performing scalar multiplication in the vector space V only with elements p e K, {^, p) e V X KI—> ^ij, e V, V becomes a vector space over K to be denoted by V^x- Decomposing the scalars X e L with respect to the basis (a^) of L over K the scalar operations of L on V are transformed into corresponding ones of if on V\x- Applying this to the vector coordinates of V over L the following is easily verified (cf. Exercise 1.4.4.8) Lemma 1. Let V be an n-dimensional right vector space over L, and let (b;) be a basis for V. Then (biap), I = I,... ,n, p = 1 , . . . , r, is a basis for the vector space V\K over K obtained by restricting the scalars, hence AmiK V\K = dim^ V • dim^ L.

(1) D

1.10.1 Hopf Fibrations The vector spaces V" over L, V"^, and L over K occurring in (1) determine projective geometries. We will study the projective point spaces P"^^{V), p " ' ' - i ( F | ^ ) , P'-^L^K), and the canonical maps (cf. (1.2))

TTK:

F \ { O } ^ P " ^ - 1 ( F '\K) |

Projectively, restricting the scalars results in blowing up each point x G pn-ij-^-j ^Q ^ projective subspace 6^^{x) C P"^^^{V\x) isomorphic to P^^^{L\K)More precisely, we have

1.10 Restrictions and Extensions of Scalars

113

P r o p o s i t i o n 2. Let V be an n-dimensional right vector space over the skew field L, let P"^ (V) be the corresponding projective space, and let P"^^ (^\K) be the projective space associated with the restriction V\K to the skew sub field K C L, r = diniK L < oo. Then there is a uniquely determined surjective map

e : P " ' ' - 1 ( F | K ) -^ P"-\V)

(2)

satisfying the following equation involving the canonical maps TT = 6 O TTx-

(3)

The preimage 9^^{x) is a projective (r — l)-plane in P"^^ (V^x)- If i^i); I = I,... ,n, is a basis of V over L, and Ui is the domain of definition for the inhomogeneous projective coordinates (Ci), c/. Lemma 2.6, then n

xeUi^^

s{x) := bi + ^

bji{{x)

(4)

determines a section s for the canonical map n, i.e. TTOS = idfy^, and the map

^••y = Me e-\u,) ^

{e{y), [t)/s(^(y))]K) € t/i x P'-\L\K)

(5)

is a bijection (similarly for the other domains Uj). P r o o f . For each p G V \ {0} we have 7rK(j;) = pif C pL = 7r(p), and hence 9{iK) := pL is surjective and uniquely determined. Obviously, pL = t)L if and only if there is A; G L* such that t) = pA;. Decomposing k with respect to the basis (ttp) of L we obtain for the complete preimage of a; = pL G P " ^ ^ ( V ) : ^^HWL)

= M K V [pa2]K V ... V [iar\K-

(6)

Since the vectors (pa^), p = 1 , . . . , r, are linearly independent over K for each p 7^ 0, the corresponding points span an (r — l)-plane in P™'^^{V\j(). To render (5) meaningful, recall the definition of the ratio of two vectors from 1.4.3. By the definition of 9, y = [t)]^ G 9^^{x), if t) = s{x)k for some k G L*; here k is uniquely determined by y up to a right factor K G K*. Hence

Ws{9{y))]K =

[k]KeP'-'{L K)

is uniquely determined by y. This readily implies that ^ is a bijection. D The situation described in Proposition 2 is the projective analog of a topological fibration: P™'^^{V\K) is represented as a disjoint union of the "fibres" (6), parametrized by a; = [pj^ varying in the space P " ^ (V). The latter is covered by "chart domains" Uj whose preimages 9^^{Uj) are direct products as described by (5), where ^ maps the fibres 9^^{x) onto the fibres p^^{z), z = {9{x), [k]K), of the product on the right-hand side. It is easily verified that the second projection from the direct product onto the fibres.

114

1 Projective Geometry

P2 o

P''-\L K)

is a projectivity. Using inhomogeneous coordinates to introduce a topology or the structure of a diflFerentiable manifold on the spaces over K = R, C, H, all the maps occurring in (3)-(5) become open and continuous or arbitrarily often differentiable, respectively, and 9 turns out to be a fibration even in the topological sense. Example 1. The fibrations 0 are closely related to the fibrations of spheres described in 1935 by H. Hopf [55]. To see this we define the covering maps A : S*^ -^ P^. Look at S*^ as the unit sphere in R'^+-^ and interpret P ^ as the set of hues through its center, then A : t) G 5 ^ ^

A(t)) :=

M R G P^

(7)

is a double covering: In fact, the preimages of any point y = [t)]R € P R are ±t) G S*^, and each projective line has the corresponding principal circle in S*^ as its preimage. Hence for if = R C L = C the map 6*0 : = 6 ' o A : S-^"-! —> P ^ - ^

(8)

is a fibration of the sphere S*^"^^ by circles S*^. In particular, for n = 2 the projective line is homeomorphic to the Riemann sphere, PQ = C = S'^ (cf. Example 1.3), and so we obtain the Hopf fibration: The sphere S^ is fibred into circles S^ over the sphere S*^, 00 : S"" ^S^,

e-\x)=S\

(9)

Analogously, embedding R C H leads to the maps

eo = eo\ with A: s-^"-! ^ P t ' \ s •• -PR"^ ^ - P ^ ^

(lo)

where 6 and 6o are the fibrations by projective spaces P R and spheres S*^, respectively; in fact, by Proposition 2 we have

e-\x) = Pl, e-\x) = A-i o e-\x) = s\

(ii)

For n = 2 we use H = S*** to obtain a fibration of the sphere S*^ by threedimensional spheres S^ over the sphere S"*, 0^:S'^S\

e-\x)=S\

(12)

Replacing the quaternions by the Cayley numbersor octonions as they are also called, one obtains a further Hopf fibration (cf. N. Steenrod [100], §11.20) ^„: 5 1 5 ^ 5 8 ,

e-\x)

= Sr

Finally, the embedding C C H leads to the series of fibrations

(13)

1.10 Restrictions and Extensions of Scalars e : P^^-^

^

P'^K

115 (14)

By Exercise II.8.9.9 we have H | c — C^, and hence the fibres are

e-\x) = Ph = s\

(15) D

Exercise 1. Prove under the assumptions of Proposition 2: a) For every three colhnear points ao, 01,02 G P"^^{V) there always are three colhnear points bo,bi,b2 e P"^^^{V\K) with 9{bi) = ai. - b) For every four colhnear points Ui e P"^^{V), i = 0 , . . . , 3, there are four collinear points bi e P"''^^{V\K) such that 0{bi) = ai if and only if A" n CR(ao, o i , 02, 03) / 0. For the subsequent sections the following considerations will be useful: If V, W are vector spaces over the skew field L, and P{V),P{W) are the corresponding projective point spaces, then a m a p f : V \ {0} ^ W \ {0} generates a unique m a p F : P{V) -^ P{W) with F(7r(p)) = 7r(/(p)) if and only if for all p G V \ {0} and each X e L* the vectors /(pA) and /(p) are linearly independent. For this to hold, it is sufficient t h a t there is a function h : L* —> L* satisfying /(pA) = /(p)/i(A) for all p G F \ {0} and each X e L* (or /(pA) = /i(A)/(p), in case V is a right and Vt^ is a left vector space, etc.). Exercise 2. Prove under the assumptions and in the notations of Proposition 2: If /i : L* ^ L* is a map with h{K*) C K*, and f '• V \ {0} -^ V \ {0} is a map with /(pA) = /(p)/i(A), then the maps F : PIV^K) ^ - P ( ^ | K ) , F : P(y) -^ P(y) generated by / are well-defined, and, moreover, 9 o F = F o 9. 1.10.2 C o m p l e x S t r u c t u r e s T h e restriction of a complex vector space V to the real numbers is also called its realification. This leads to the notion of a complex structure on a real vector space and to the representation of the algebra of complex linear maps as a subalgebra of the real endomorphism algebra. In particular, we obtain a representation of the complex linear group GL{n, C) as a subgroup of the real linear group GL{2n, R ) . In the sequel we will describe some consequences for the projective properties of the fibration 6: P 2 " " ^ ( F | R , ) ^ P " - i ( F ) . In V scalar multiplication by i = A/—1 defines a C-linear and all the more R-linear isomorphism, the complex structure: /(p) = p - i , /2 = - i d - ^ .

(16)

Since / has no real eigenvalues, it generates an equally denoted fixed-point free involution / G PL{P^ ), which hence has the normal form (4.42). In

116

1 Projective Geometry

fact, if Oi, / = 1 , . . . , n, is a basis of V, then the decomposition of the complex coordinates into real and imaginary part, n

n

1=1

1=1

2n v=l

with On+i ••=0,11 = / ( O ; ) , i-e. / ( a „ + ; ) = - 0 ; ,

(17)

yields a basis for V-^, in which / has this normal form. If, conversely, W " is a 2n-dimensional real vector space, and / G GL(W^^", R ) is a hnear automorphism of W with P = — i>^\Y' then {hz)

eW

xC\—>

p := PC + I{T)ri G W^ for z = C + i??

(18)

defines the structure of an n-dimensional complex vector space on W; we will denote it by V " , and so, obviously, Vt^^" = V\^. For this reason, an R-linear m a p / on Vt^ satisfying P = — i>^\Y ^^ called a complex structure (cf. Definition II.8.9.4) on W\ according to the result of Exercise 4.15 all complex structures are similar. Of course, any complex linear m a p a '.V ^ V is real linear as weU. Expressing it in a basis (Oj/), v = l , . . . , 2 n , of V\^ satisfying (17) we obtain for the real and imaginary part, using the obvious notation: a{i) = (A + i B)(y + i t)) = (Ay - St)) + i(At) + By);

(19)

here A, B e M „ ( R ) are real square matrices, and p, t) G R " , cf. Exercise 1.5.4.18. Exercise 3. Show with the notations just introduced: a) Restricting the scalars from C to R defines a homomorphic embedding of the real endomorphism algebras c: (Endc(F"))|R—>Endu.(W^").

(20)

b) A linear map b e EndR(W^") belongs to ( E n d c ( F " ) ) | R if and only if it commutes with the complex structure / , i.e., it satisfies / o 6 = 6 o / , and this again holds if and only if its matrix (/3^) has the following block structure with respect to a basis (17) (cf. Exercise 1.5.4.18) C:=(/3^)= M " ^

I withA,BeM„(R).

(21)

c) The determinant of any matrix of the form (21) is non-negative. (Note that in every basis (17) / has the matrix ^

" Q " I with 1„ = (S)) e M „ ( R ) .

(22)

Hence we may suppose that the matrix of the complex endomorphism a has Jordan normal form (cf. Proposition 1.5.8.4 and Corollary 1.5.8.2); use the special

1.10 Restrictions and Extensions of Scalars

117

form the matrices A,B from (19) have in this case, insert it into (21) and apply the Laplace expansion theorem to compute the determinant of C recursively.) - d) det(C) = |det(6)p. - e) After realification of V" the complex linear group GL{V") becomes a subgroup of the real linear group GL{W^"). Show that the subgroup of projectivities generated by all g G GL(V") acts transitively on the real projective space P{W^"). Exercise 4 . Let / be a complex structure on the real vector space W^". Denote by f)i e PL{P^'^^) the projective involution generated by / . Again, let V" be the complex vector space determined by / , and let 9 : P ^ ^ ^ -^ P^^ be the fibration from Proposition 2. Let Pa '• -PR*^^ ~^ PR'^^ denote the projective map generated by the linear map a e EndR(W^"). Prove that there exists a map pa • P^^ -^ P^^ satisfying 9 o pa = Pa ° 9 if and only if pa o I = I o pa] here Pa is projective or anti-projective. 1.10.3 Q u a t e r n i o n i c S t r u c t u r e s Restricting in (20) the homomorphism c to the linear group GL(V"', C ) the matrices in (21) are all invertible, hence we obtain a real representation of GL(V", C ) as a subgroup ofGL(2n, R ) . Now we want to apply the restriction of scalars from H to C in a similar way to produce a complex representation of the quaternionic linear group GL{n, H ) in GL{2n, C ) . E x a m p l e 2. If t / " is an n-dimensional right vector space over the quaternions H , then by restricting the scalars to C (cf. § 1.2.3 and Exercise II.8.9.9) we obtain a 2n-dimensional vector space V ^ " = U?^ over C together with the fibration (14). To see this we write the quaternions in the form (/ = 2 ; + J W ; G H , with z, w G C

and apply this representation, e.g., to the coordinates of a point. Note t h a t the dilation dj : p i-^ pj is neither H - nor C-linear; because of z j = iz for z G C, it rather defines a conjugate Unear m a p J(p) := pj with J(pA) = J(p)A

(A G C ) , J2 = - idy

.

(23)

T h u s J determines a fixed-point free, involutive colUneation on P ^ ^ ^ ( V ) , which has the normal form (4.43) in a suitable basis of V ^ " . Starting from an arbitrary basis (o;), / = 1 , . . . , n, of the quaternionic vector space t / " such a basis is constructed as a„+; := a;j = J ( o ; ) , i-e. J ( a „ + ; ) = - O ; .

(24)

If, conversely, J is an involutive conjugate linear m a p on the complex vector space V ^ " satisfying (23), then (p, ( j r ) G F x H i — > p(jr := p2; + J(p)w; G F for (jT = z + j w G H (z, w G C) (25)

118

1 Projective Geometry

defines the structure of an n-dimensional, right, quaternionic vector space over V, t h a t win again be denoted by t / " ; here t/"^ = V " . For this reason, J is caUed a quaternionic structure on V " . Exercise 4.15 immediately imphes t h a t all quaternionic structures are similar: For two quaternionic structures J i , J2 there is always c G GL{V'^", C) such t h a t J2 = c o J o c^^. Since each H-linear m a p a is also C-linear, decomposing the coordinates into their complex components with respect to a basis (24) we obtain (cf. Exercise II.8.9.9): « ( 3 ) = ( ^ + J B)(? + J t)) = (^? - Bt)) + j(Bj: + At)).

(26)

Here A denotes the complex conjugate matrix of A.

D

Exercise 5. Show in the notations of Example 2 that the canonical embedding I. : EndH(C/") -^

(Endc(F2"))|H,

(27)

is a homomorphism of real algebras. (Note that according to Proposition II.7.2.5 EndH(C^") is only a Z(H)-algebra.) A C-linear map b e E n d c ( F 2 " ) ) belongs to EndH(C^") if and only if it commutes with the quaternionic structure J, i.e. J ob = boJ.

(28)

This again holds if and only if its matrix has the block structure (/3M)= ( 5 " / )

withA,BeM„(C).

(29)

with respect to a basis (24). Exercise 6. Show that every complex matrix of the form (29) has real determinant. If (29) is the matrix of an H-linear map 6, then the norm N{b) := Nc{b) := det(/3^) does not depend on the choice of the basis (Oj). (See also Exercise 8.) Exercise 7. Let (Oi) be a basis of the quaternionic vector space U". Splitting the vector coordinates into their real components: p = Ojg = aiXo + aiixi + aiix2 + aikXi

(30)

(sum convention, I = l,...,n) we obtain the basis (Oi, 0; i, Oij, Oik) for the real vector space W^" = U?^^. Now we combine the coordinates of p into blocks Ta '•= i^a) e R-", a = 0, . . . , 3 . Then each R-bnear map a e E n d R ( W " ) can be represented by a block matrix: (t)J = (A„^)(?^), ?^ e R",A^p

e M„(R).

Show that a is H-linear if and only if its block matrix has the following special form: 'Ao -Ai

-A2

-A:

(^-) H1; t -t -I I • , ^ 3 -A2

Ai

Ao

(31)

1.10 Restrictions and Extensions of Scalars

119

Here the blocks of (31) depend on the quaternionic coordinate matrix of a via torn) = A o + i A i + j A 2 + kA3 e M „ ( H ) .

(32)

One obtains essentially the same block matrix structure starting from (29) and passing to real coordinates. Exercise 8. Let V" be an n-dimensional right vector space over H and consider a e EndH(T^"). Prove: a) There is a basis (Os), s = 1,.. . ,n, of V" such that a has a Jordan matrix (1.5.8.29) with respect to (Os) in which the diagonal entries AK = ^K. + irjK. are complex numbers with r?^ > 0. - b) Two such maps a,b are H-similar, i.e., there is a c e GL{V", H) with b = coaoc^^ if and only if they have identical Jordan matrices with respect to suitable bases. - c) (cf. Exercise 6) N{a) > 0.

(33)

(Hint. Consider a first as a C-linear map on V"^. If W C V"^ is a complex subspace in which a\W is described by a Jordan box, then examine J(W).) 1.10.4 Projective E x t e n s i o n s Restricting the scalar domain of the vector space V determining a projective geometry from LtoKcL does not change the structure of the Abelian group [V, +] of the vector space; by Proposition 2 the restricted scalar multiplication generates a new projective geometry with scalars K as well as the surjective m a p 9 from the point space of the if-geometry onto the point space of the L-geometry, cf. (2). Frequently, one wants, however, to embed a projective if-point space P^ into a projective L-point space P^, i.e., to find an injective m a p L : P ^ -^ P^ preserving as much as possible of the projective structure. In this situation, one might make good use of the richer and often simpler possibilities for the solution of algebraic equations over L to draw conclusions concerning the if-geometry. Particularly interesting, and frequently applied in algebraic geometry, is the transition from the real to the complex. T h e simplifications taking place here are, e.g., evident in the classification of polar maps, cf. Propositions 9.4 and 9.5. Often, extending the scalars proceeds quite directly: Describe the objects of interest in homogeneous or inhomogeneous coordinates with respect to K, and then just allow these coordinates to take values in all of L. To provide a conceptual foundation for this straightforward procedure we again start from the situation described in the beginning of this section. D e f i n i t i o n 1. Let i f C L be a skew subfield. Restricting the scalars each (e.g. right) vector space V over L becomes a vector space over K. A if-subspace W C V | x is caUed a K-form of V li W is a minimal if-subspace generating V, i.e. if ZhiW) = V^. If this holds, then V is caUed an L-extension of W. D ^ In the case R C C instead of R-form the term R-structure is frequently used; to avoid confusion with the complex and quaternionic structures defined above, and for the sake of a unified notion we prefer the name R-form here as well.

120

1 Projective Geometry

L e m m a 3. / / Vi, V2 are right L-extensions of the K-vector space W, then Vi = V2; set V := Vi. Each K-basis of the K-structure W is also an L-basis of the extension V. In particular, diniK W = dim^ V. P r o o f . By definition each of the vector spaces Vi, i = l,2,is the L-hnear span of W. This imphes Vi = V2- If ( a i , . . . , Or) are if-hnearly independent vectors of W, then they are L-hnearly independent in V as weU: Assume, e.g. that Or is an L-hnear combination of the others, — 2_^ (ip^p, ^p £ L.

Consider the subspace Wi in W = Wi © arK complementary to arK, Wi = [oi,..., Or-i]K- Then the L-hnear span of Wi is obviously equal to V, and hence W cannot be a minimal L-subspace generating the space V. D This already justifies the method for finite-dimensional vector spaces mentioned above: If (Oj), « = 1 , . . . ,n, is a if-basis for W, then each element of the L-extension V has a unique basis representation n p = / ^ OjAj, Aj G L. i=i Conversely, if (bj), « = 1 , . . . , n, is an arbitrary L-basis of V, then the if-hnear span W := 2K{{bi), « = 1 , . . . ,n) is a K-iorm of V. Returning to the bases we immediately obtain: Corollary 4. Under the assumptions of Definition 1 let d i m V = n < 00. Then there exists a K-form W C V. The L-linear group GL(V, L) acts transitively on the set of all K-forms on V. The latter is thus a homogeneous space isomorphic to GL{n,L)/GL{n,K). P r o o f . It suffices to note that the isotropy group of a if-form W consists of all L-linear transformations h G GL(V, L) with h{W) = W. Since the L-linear transformations are if-linear as well, the assertion h G GL{W, K) foUows. Fixing an arbitrary if-basis of W as an L-basis for V, the embedding GL{W, K) C GL{V, L) is realized by the group of all the L-hnear transformations whose matrix belongs to GL{n, K). D Corollary 5. Let a G Aut(L) be an automorphism that preserves K, u^K) = K. If a : Wi -^ W^ is a K-linear or K-a-linear map, respectively, then there is a uniquely determined linear or L-a-linear map a : Vi -^ V2 of the L-extensions Vi, V2 ofWi or W2, respectively, extending a: a\Wi = a. Expressed in terms of the category of linear maps the assignment ai-^ a forms a covariant functor from the category of right vector spaces over K to the category of right vector spaces over L. Here

1.10 Restrictions and Extensions of Scalars

121

(a + 6) = a + 6, aa, = aa (a G Z{L) Pi K). P r o o f . In the finite-dimensional case we consider if-bases (Oj), (ba) of Wi and W2, respectively, and also allow for elements A* G L in the basis representation of a, a(p) = a(ajA*) = bo,a"o-(A*). For infinite-dimensional vector spaces one proceeds similarly making use of the existence of bases (cf. Exercise 1.4.4.7). • We call a the linear (or a-Unear) extension of the map a. Analogous considerations work for left vector spaces. If the parities of the vector spaces are different, then the automorphisms are to be replaced by anti-automorphisms. E x a m p l e 3. Consider the inclusion of fields R C C and the complex plane V = C^. Then W := (1, 0 ) R e (0, l ) R is an R-form on V. The real subspace U := (1, 0)R© (i, 0)R only spans the complex line (1,0)C, so it is no R-form on V. The R-linear map a from W = R^ to itself defined by a : {y') = {a^){x^), (x^) G R^, (a}) G M 2 ( R ) , has the linear extension a G Endc(V^) determined by simply allowing (x^) G C^ in the definition, and the conjugate linear extension described by

a: (a:^)GF^(y^):=(4)(x^'). D

The extension of vector spaces and a-linear maps has immediate consequences in projective geometry: Corollary 6. Let K be a skew sub field of the skew field L. For every finitedimensional right projective space P " = P ( V F " + \ i f ) there is a unique right projective extension P^ = P ( V " ^ , L) corresponding to the L-extension V of the K-vector space W; the point space P " is canonically embedded into

PI by t : a ; = j;if G P " I — > L{X) := fL e PI.

The L-extensions of the projective subspace B

(34)

:= 60 V . . . V 6^ C P " is

B ^ = (60 V . . . V bk)L = i-ibo) V . . . V L{bk).

(35)

/ /
f og = fog.

(36) D

122

1 Projective Geometry

We leave it to the reader to formulate corresponding conclusions for maps between different projective spaces, for left projective geometries, and for projective geometries of different parity. Corollary 6 allows to extend the projective geometries over a skew field K functorially to projective geometries over every skew field L containing K. According to Corollary 6 in particular the projectivities g G PL{W"^^) are extended as projectivities g G PL{V"^^), so t h a t PL{W"^^) naturally appears as a subgroup of PL{V"^^). Obviously, the action of PLCW^'^ ) over P ^ is not transitive; in fact, it leaves P " invariant. The properties invariant under this subgroup describe the position of configurations in the projective geometry ^{V) with respect to the embedding defined by (34); in the foUowing Sections 5 and 6 we wiU discuss this in greater detail. Furthermore, consider the if-restriction V\x of V; the notions referring t o it are marked by the label "if". Obviously, this restriction is not equal to the original if-vector space W; in fact, d i m ^ V^x = d i m ^ W • d i m ^ L. T h e subspaces of V^x are called K-subspaces of V; obviously V itself is a if-subspace of V, for whose dimension (1) holds. The projective subspaces corresponding to t h e m will also be called K-subspaces. E.g., the image of the embedding P^ := i{P^) C P^ defined in (34) is an n-dimensional projective if-subspace t h a t will be called a projective K-form on P^. In particular, the points X G P ^ ai'^ caUed K-points in P^- A frame (OJ; e) for P ^ is called a K-frame of P ^ . Exercise 9. Let P^ C P J be a projective A'-form on P J . Using the notation for projective A'-subspaces A,B C P ^ just fixed prove: (AVKB)K = ( A L V L B L ) n P ^ ,

(37)

{AAKB)K

(38)

XLKA

= {ALALBL)nP"K,

•<=^ X e P x

and XLLAL-

(39)

Exercise 10. Each A'-basis (Oi) for T^"+^ determines homogeneous coordinates on P x and on P J . Prove that a point x e P £ lies in PJ^ if and only if the ratios of its homogeneous coordinates with respect to a A'-basis (Oi) satisfy x* • (a;^)^^ G K, i,j = 0,...,n. Hence n

P'k = {x = [J2aix%\x' e K, {x') / 0}.

(40)

C o r o l l a r y 7. Under the assumptions of Corollary 6 each projectivity f on P ^ has an extension f to P ^ , f\P^ = f • If, conversely, F is a projectivity satisfying FiPl)=Pl (41) on P 2 , then F\P^

is a collineation.

•

1.10 Restrictions and Extensions of Scalars

123

E x a m p l e 4. The following example shows t h a t (in the notations of Corollary 7) F\P^ is not necessarily a projectivity, if it satisfies (41): Consider the usual embeddings K=CcL = ll, C = £ R ( { l , i } ) . Then the m a p a with '*(('/*)) '•= 0<1^)^ s = 0 , . . . , n, is a linear m a p of the right vector space H " + ^ . Because of [a((z«))] = [(jz«)] = [(z-j)] = [(z-)], the projectivity on P^ defined by F{[f]) = [a(p)] determines on the C-form P c C P H which corresponds to the standard basis nothing but the antiprojectivity generated by conjugation. D Nevertheless, we have C o r o l l a r y 8. If L is a field, and F is a projectivity then -F|-Px «5«*^ is a projectivity.

on P^

satisfying

(41),

To prove this, it suffices to note t h a t F leaves the cross ratio invariant, which takes only values in K for throws from P^, and to apply Proposition 4.6. D 1.10.5 T h e P r o j e c t i v e K - G e o m e t r y of P ^ According to Corollary 4 all projective if-forms on P^ are projectively equivalent. Now we again fix one of these forms and denote by PLn,K C PL{P^) the subgroup of all projectivities on P ^ which are extensions of if-projectivities on P ^ ; according to Corollary 8, in the case of a field L this group is the isotropy group of P ^ for the action of PL{P2) described in Corollary 4. Obviously, we have the isomorphy PLn,K

= PL{Pl).

(42)

T h e invariants of the transformation group [PLn,K, P2] describe the geometric properties of figures in P ^ with respect to the projective if-form P^, in short, the projective K-geometry of P ^ . E x a m p l e 5. For the applications particularly important is the case K = H, L = C, i.e the real geometry of complex projective space. Example 4.7 immediately shows t h a t the notion of a real structure introduced there essentiaUy amounts to an R-form: Staudt chains are the same as R-forms on P ^ (cf. the following exercise). • Exercise 1 1 . Let J denote a real structure on the complex projective space P c E^rid at the same time the conjugate linear map of the associated vector T^"+^ generating it, cf. Example 4.7. Prove: a) The Staudt chain corresponding to J is an R-form on Pc, and each R-form on P^ is a Staudt chain. - b) If P R C P g is an R-form, then the basis (bj) of T^"+^ is an R-basis if and only if J can be represented in the normal form (4.46) with respect to (bj). - c) Pin,R = { / e P L ( P £ ) | / o J = J o / } .

(43)

124

1 Projective Geometry

E x a m p l e 6. Consider the extension from C to H . Let P ^ C P^ be a C-form on quaternionic projective space, and let (bg), s = 0 , . . . , n , be a C-basis of the associated vector spaces W = Wc^^ C F H + ^ = V. In the decomposition x" = z" + j w**, z",-)!}" e C, the equations w" = 0, s = 0,. . . , n,

(44)

characterize the subset P^ C P^; the (z^) are just the homogeneous coordinates of the points from P ^ . W i t h respect to P ^ C P H the involutive projectivity J on P^ with coordinate representation J : (x/) i-^ (i x**) (cf. Example 4.8, in particular (4.52)) has analogous properties to the real structure J from Exercise 11: a) We have Pl = {xeP'^\J{x) = x}; (45) the involutions J with normal form (4.52) and the C-forms P ^ C P ^ bijectively correspond to one another. b) For a given P ^ C P H , the (bs) form a C-basis of P c if and only if J has the normal form J{bs) = bsi, s = 0,...,n, (46) with respect to (bg). As Example 4 shows, PLnc is no longer the isotropy group G of PLnC under the action of PL{P^). Making use of the continuity of projectivities on P^ the following is easy to show: c) T h e isotropy group G of a C-structure P ^ C P H is G={fePL{P^)\foJ=Jof}.

(47)

T h e m a p f E G i-^ / l - P c ^ ^ u t P ^ is an isomorphism from G onto the group of coUineations on P ^ consisting of projectivities and anti-projectivities. • Exercise 12. Prove the statements in Example 6. Hint. Take the situation described in Example 4 into account. E x a m p l e 7. Now we indicate a scheme generalizing Examples 5 and 6. Note first t h a t for each automorphism a G Aut L the set K^ of fixed elements, Ka := {C & -^|o"(C) = C}) is a skew subfield of the skew field L. Suppose t h a t the following condition is satisfied: B) Let L be a skew field, char L y^ 2, and let a G Aut L be an involutive automorphism, a^ = id^, a ^ id^. Set K = K^. In general, K is not commutative. In Example 5, cr[z) = I; is the conjugation, and in Example 6
= -i{z +iw)i

= z -iw.

(48)

Let then V " + ^ be a right vector space over L, and let (Oj) be an arbitrary basis for V. Setting

1.10 Restrictions and Extensions of Scalars n

125

n

J(^0ix*):=^0ia(x*) i=0

(49)

i=0

defines a a-linear involution on V, and this again determines an equally denoted collineation J G Aut{P"{V)). We call J a a-conjugation if, for a suitably chosen basis, J is represented in the form (49). If the related automorphism is clear from the context, J will simply be called the conjugation. Two elements A, B (points, vectors, subspaces, . . . ) are called conjugate, if J{A) = B holds. D P r o p o s i t i o n 9. Under Assumption B), K determines a K-form on P^ := P^(V) by PI moreover, for each K-form satisfying (50).

= K^,

:= {x e P^V)\J{x) P^

C P^

each a-conjugation

= x};

J

(50)

there is a unique a-conjugation

J

P r o o f . Let (Os) be a basis for V in which (49) holds. Then the K-iorm (40) determined by this basis obviously belongs to the fixed-point set of J . If, conversely, J{x) = x, then for the vector {x") G L"+^ given by the homogeneous coordinates of x with respect to the basis [Os] there is q e L* with a{x'') = x^q. This implies a'^{x'') = x" = a{x'')a{q) = x^qa{q). Since at least one x** 7^ 0, we obtain qa{q) = 1, (51) and each coordinate xf ^Q oi x satisfies {x/)^^a{xf) L e m m a 10. Assume solution 1^0 ^ L of

= q. We prove

that (51) holds for q ^ L. Under Assumption

r V(e) = q generates

all solutions

B) any

(52)

setting ^ = K^O, K € K*.

P r o o f . From
^(eeo"') = ^(e)^(eo)-^ = e'/^(
126

1 Projective Geometry

Exercise 1 3 . Prove assuming that B) is satisfied: a) There exists a q & L* with (7(g) = —q. - b) If go G L* is a fixed element such that (7(go) = —go, then every x E L can be uniquely represented in the form x = a + go/3 with a,l3 E K, and, moreover, diniK L = 2. - c) The map a E K i-^ 'r{a) := qoaq^^ e L is an automorphism of K; T is involutive if and only if go belongs to the center Z{K) of K. - d) Prove by means of an example that there not necessarily is a g G L* satisfying at simultaneously (7(g) = —g and a{q) = q^^. Hint. Consider the conjugation in Q[V^3]. 1.10.6 JC-Classification of P r o j e c t i v e L - S u b s p a c e s Now we will t u r n to the projective if-geometry of the L-subspaces of Definition 1 easily implies: L e m m a 1 1 . In the assumptions and notations of Definition holds: If A C P^ is a projective L-subspace, then A n P^ K-subspace, and DiniK A n P ^ < DiniL A.

P^-

1 the following is a projective (53)

In (53) equality holds if and only if {AnPl)L

= A.

(54) D

P r o p o s i t i o n 12. Assuming that B) holds consider the projective extension P^ C P^, where P^ is defined by the a-conjugation according to (50). Then (54) holds for a projective L-subspace A C P 2 ' / '^'^'^ only if J{A) = A. P r o o f . Using a if-basis (Oj), i = 0,...,l = D i m x A n P ^ , to describe A n P ^ equation (54) immediately imphes / = Dim^ A, i.e. equality in (53), and J{A) = A. Conversely, assume t h a t J{A) = A holds. Thus we again find a if-basis (Oj), « = 0 , . . . , / , for ADP^. Suppose t h a t k := Dim^ A > I. Then B:={An

PDL

CA,1

= D i m i B < k,

and there is a point b e A\ B. Consequently, J{b) G J{A \ B) = A\B; particular, we have b ^ J{b). Choose a vector b with b = [b]. Obviously, J(b + J(b)) = b + J(b), J(b - J(b)) = - ( b - J(b)),

in

(55)

and hence

a:=[b + Jib)] eAnPl,

a' := [b - J(b)]

eAnP^.

Since (Oj) is a if-basis for this if-subspace. Exercise 13 imphes

0 := b + J ( b ) = J2 a i 7 \ a' := b - J ( b ) = ^

aa"q,

(56)

1.10 Restrictions and Extensions of Scalars

127

7*, 7'* G K and q e L* with aiq) = -q.

(57)

From this we immediately conclude I

2b = ^ 0 i ( f + 7"'/),

(58)

0

and because of charL 7^ 2 we obtain 2b 7^ 0 and b = [b] e B. But this contradicts the choice of b. Hence / = k, and Lemma 11 implies (54). D From Proposition 12 the classification of L-subspaces in the projective if-geometry on P^ can easily be derived: P r o p o s i t i o n 13. Maintaining the assumptions of Proposition 12 consider two projective subspaces A, A\ C P\. Then there is a projectivity f G PLn^K with f{A) = Ai if and only if D'miL A = BiiriL Ai and Dim^ A D P ^ = Dim^ Ai D P ^ .

(59)

P r o o f . Because of f{P^) = P^, the necessity of condition (59) is clear. To prove the converse, we adapt a if-basis to the space A. First we choose a basis Oo,..., Od G W^ for the space A n P ^ ; here W = 1^"+^ denotes the if-vector space corresponding to the if-structure P^ . Now A n P ^ = J{A) n P ^ = (A A J{A)) n P ^ , and Proposition 12 implies d = D i m K A n P ^ = DimLAA J(A),

(60)

since A A J{A) is J-invariant. Let k = Dim^ A; complete the basis Oo,..., a^ of A A J{A) by the vectors bd+i,..., bfc G F " + \ PI = P " ( F ) , so that they together form a basis for A. Then J{A) = [oo] V . . . V M V [J(bd+i)] V . . . V [J(bfc)], and A V J{A) is spanned by the 2k — d+l vectors (Oa), a = 0 , . . . , d, (bj,), (J(tij/)), i/ = (i+ 1 , . . . , A;. By the dimension formula we have DimL A V J{A) = 2k- Dim^ A A J(A) = 2k - d,

(61)

and hence the vectors just introduced are linearly independent. Now we fix an arbitrary q e L* satisfying equation (57) (cf. Exercise 13), and define a, := b, + J ( b , ) , a, := (b, - J{b,))q-\

(62)

Because of J(Oj/) = a^ and J(Oj/) = CXu the a^, a^ he in Vt^"+^, and, conversely, we can express the bj,, J{bv) linearly by them (charL ^ 2):

128

1 Projective Geometry b^ = {a^ + a^q)/2,

J{b^) = {a^ - a^q)/2.

(63)

Hence the 2k — d + 1 vectors Oo, • • •, Ofc, Od+i, • • •, iifc form a if-basis for the J-invariant subspace A V J{A). We complete t h e m choosing (OA) for A = 2k — d + I,... ,n, to obtain a if-basis of V"^^. If the subspace Ai determines the same dimensions d, k as A, then, in a similar way, we can assign to it a if-basis of V t h a t has the same position properties with respect to Ai as the if-basis Oo, • • •, a„ to A. These bases determine a projectivity / G PLn,K

with / ( A ) = A i .

D

C o r o l l a r y 14. Under the assumptions of Proposition 12, let A C P^ be an L-subspace. Then the dimensions k = Dim^ A, d = D i m ^ AnP"^ satisfy the inequalities max{-l,2k-n)
k = Dim^ A,

P r o o f . T h e necessity of (64) follows from (60), D i m i A V J ( A ) < m i n ( n , 2A; + 1)

(65)

together with the dimension formula in Proposition 1.1. If, conversely, the pair {d, k) satisfies inequality (64), then, because of d < A; < n, we first find d + 1 independent points a^ € - P ^ i '^ = 0 , . . . , d. Inequality (64) implies n — d > 2{k — d); hence we find 2(k — d) independent points a^ = [Oj/], a^ = [a^] G P ^ , ly = d + I,.. .,k, such t h a t OQ, . . . , Ofc, a^^-^,..., a'^, are independent. We now again choose q ^ L* with (57) and define bi, for V = d + I,... ,k, by (63). Setting bj, : = [b^] we thus conclude t h a t A = ao V . . . V Od V bd^i V . . . V 6fc is a A;-dimensional subspace of P ^ with D i m x Ar\P''^ = d. D Exercise 14. Let L be a skew field, and let K C Z{L) be a subfield belonging to the center of L; moreover, let {xo,xi,X2,X3) be a throw in P\. Prove: a) There is a A'-line P]i C P\ with Xi e P]i, i = 0 , . . . , 3, if and only if CR(a:o, xi;x2, X3) belongs to K. - b) For every throw {xo,xi,X2,X3) in P\i there is a complex line P c C P H with Xi e P c , i = 0 , . . . , 3. 1.10.7 E x t e n s i o n s of N u l l S y s t e m s , P o l a r M a p s , a n d Q u a d r i c s In this section we want to investigate the relations between fof nuU systems, polarities, and quadrics in P ^ to those in the projective extension P^- To this end, we need a few simple preparations. To each if-form WK on the right L-vector space V " + there corresponds a uniquely determined if-form on the dual vector space V: W'j,:={ueV'\u{WK)cK};

(66)

1.10 Restrictions and Extensions of Scalars

129

Wj^ is the (n + l)-diniensional if-vector space of V arising from linearly extending the linear forms on WK to V; as already the notation indicates, it will always be identified with the dual of WK in the sequel. Obviously, the basis dual to a if-basis (Oj) for V is a if-basis of V in the dual K-ioTui W'K- We again interpret the elements of the corresponding projective spaces p/n -^ pin ^g hyperplanes in P^ or P ^ , respectively. Exercise 1 5 . Let dimx L = r < oo. Prove that for each hyperplane C/ = [u] e P j ^ the inequalities n — r< Dimx UnP^ < n — 1 hold; moreover, Dimx UnP^ = n—1 holds if and only if C7 e P'KFirst we will discuss the extension of an auto-correlative m a p F on P ^ to an auto-correlative m a p F on P^. According to Definition 7.2 F is generated by an a-linear m a p a : WK -^ ^ K I where a is an anti-automorphism of K. Using T-linear extension the foUowing is easy to show: L e m m a 15. Let the auto-correlative map F on P^ be induced by the a-linear map a : WK -^ ^'K- ^ can be extended to an auto-correlative map F on J-*2 */ dnd only if the anti-automorphism a of K has an extension to an anti-automorphism T of L. For each of these extensions T there is a unique extension F = F-j- induced by the r-linear extension a of a; with respect to a K-basis (Oj) of V (Note the sum convention!) it is determined by a(OiX*) = T ( x * ) a ( O i ) .

E x a m p l e 8. If L is commutative and F is a null system over the projective space P^ with the subfield K determined by the alternating bilinear form 6(p, t)), p, t) G W^^, then the linear extension 6L(P, t)), p, t) G V^^^, defines a null system of the same rank over the projective extension P^- In fact, the matrix of 6L with respect to a if-basis of V coincides with the matrix of b with respect to this basis. Compare also Exercise II.7.9.10. • We will now discuss extensions of polar maps in a number of examples. E x a m p l e 9. Take K = H and L = C. By Exercise 1.2.1.3 we have a = idpt, and there are two extensions of this automorphism to C: T = i d c , or T is the conjugation on C. In the first case, each polar m a p Fr^i of rank r and index / leads to a polar m a p Fr of rank r on P ^ , cf. Proposition 9.5. If, conversely, (Oj) is a basis of the complex vector space V " + , with respect to which the auto-polar m a p Fr is in normal form, then this implies for the R - s t r u c t u r e P R C P C determined by the basis fjA = OA i, A = 0 , . . . , / - 1, bjx = Ojx, i/ = / , . . . , n, t h a t -FVI-PR. has rank r, index /, and extends as Fr to P ^ . If, on the other hand, T is the conjugation, then extending a m a p of type Fr^i yields a m a p

130

1 Projective Geometry

on P Q , denoted in the same way, t h a t is determined by a Hermitean form, cf. Proposition 9.7. • E x a m p l e 10. Consider i f = R , L = H , and extend idn. as the conjugation T. Then Fr^i again extends to the maps from Proposition 9.7 denoted in the same way. On the other hand, because of (9.19) we can also extend the identity idR to H as the anti-automorphism T; (cf. (9.17) with q = i). It is easy to show t h a t the polar m a p Fr^i on P^ then extends to a m a p of type Jr on P ^ , cf. Proposition 9.14. D E x a m p l e 11. Consider the embedding C C H . By Proposition 9.5, for a = i d c there only exist the auto-polar maps of type Fr. A straightforward verification shows t h a t the anti-automorphism TJ (by (9.17) with
= ^c^h''{^,t)),

j;,t)G VF'n+1 K •

The 6" are symmetric bilinear forms over WK, which, in general, are independent and different from zero; they define s polar maps on P ^ t h a t have to be considered simultaneously. Hence, in general, investigating the if-geometry of polar maps on P ^ , and consequently also t h a t of quadrics, might be quite extensive. • Exercise 16. Consider an R-form P^ C Pc- Let -F be a polar map on P ^ of type Fr,i determined by the Hermitean form b over the complex vector space V"^ . Prove the following equation: 6(j:,t)) = a(j:,t)) +i/3(j:,t)), j:,t) e W^+\

(67)

where a(j., t)) is a symmetric and /3(j;, t)) an alternating real bilinear form on W^^^. If, conversely, a and /3 are given as symmetric and alternating bilinear forms on W^^, then there is a unique extension of the form b defined by (67) to a Hermitean

1.10 Restrictions and Extensions of Scalars

131

form over V x V. This again determines a polar map on P ^ of a certain type -FV,i (of. Proposition 9.7). What are the relations between the invariants of b, a and /3? Exercise 17. Again, let an R-form P^ C Pc be given. Assume that the polar map F on P c is defined by the symmetric bilinear form b that has the decomposition (67) on the R-form. Prove that the complex quadric QF meets the R-form in the intersection of two quadrics: QFnPS, = Q„nQ^.

(68)

Cayley-Klein Geometries

According to F. Klein's Erlanger Progranim [65], that we already described in § 1.6.5, geometry and the theory of transformation groups are closely related. The largest groups, from which we will start in this chapter, are the linear groups GL{V"^^) of finite-dimensional vector spaces over a skew field K and the projective groups PL{P") assigned to them, as they were defined in Chapter 1. In this chapter we will summarize their properties in the cases of greatest interest for applications, i.e. K = R, C, H , and wiU also add a few facts. Distinguishing an object F in one of the projective geometries determined by these groups as the absolute the subgroup Gp of all projective transformations leaving F fixed, i.e. the isotropy group of F (cf. Appendix A.3), is a finer projective geometry. It studies all the properties remaining invariant when the projective action is restricted to the subgroup Gp. Since this group is a subgroup of the projective group, in general there are more preserved properties than the projective ones; hence they form what is called a supergeometry oithe projective geometry, cf. O. Giering [44]. A. Cayley [29], [30] and F. Klein [60], [61], see also [62], Bd. 1, studied these geometries in the particular case that the polarity F belongs to a real projective geometry and that invariant metrics exist. The frequently quoted lectures by F. Klein [63], [64] contain detailed discussions of these geometries as apphcations of his Erlanger Progranim. In this connection the terms Cayley-Klein spaces, geometries, and metrics are often used. We will not specify these notions here, even the extent of which is not agreed upon in the literature. The book by O. Giering [44] already mentioned presents such an approach. It also contains a detailed bibliography extending up to about 1980, including many comments in the text and more special results than we intend to present here.

2.1 The Classical Groups The name "classical groups" is a rather historic-literary description of a class of linear or matrix groups arising not only in geometry but playing a fundamental

134

2 Cayley-Klein Geometries

part also in many other mathematical branches ranging from algebra, number theory, and analysis to theoretical physics. Although this term appears in the titles of two monographs from different periods of time, first probably in the year 1939 on H. Weyl's ground-breaking book [110], and then again in 1971 on J. Dieudonne's report on results [36], the notion classical group itself is not defined in either of them. The same is true of the very clearly written lecture notes [26] by P. J. Cameron, which emphasize the classical groups and the associated projective geometries over finite fields. T h e book [47] by L. Grove focusses on the algebraic aspects of the theory of classical groups. As in these texts we will use the term here in the same somewhat vague sense. In Section 2.1 we start by describing these "classical" groups, whose geometries wiU then be investigated in the subsequent sections.^

2.1.1 T h e L i n e a r a n d t h e P r o j e c t i v e

Groups

We will even refer to the linear groups and the projective ones derived from t h e m as "classical", although, as a rule, they are not simple Lie groups. They were already introduced in the first chapter, since they determine the automorphism groups of projective geometries. The classical groups considered below typically are subgroups of these linear or the projective groups. As before, the group of linear automorphisms of an n-dimensional vector space V " over the skew field K is denoted by GL{V^). It is isomorphic to the group of invertible square matrices of order n: GL{V'')'::^GL{n,K).

(1)

For a field K the group GL{n, K) C Mn{K) consists of the matrices whose determinant is different from zero, cf. Corollary 1.5.4.7. For linear endomorphisms a of the vector space V" the determinant of the matrix (a*) for a does not depend on the chosen basis; it is called the norm N(a) of the endomorphism, cf. Proposition 1.5.7.2, Definition 1.5.7.2. Based on a generalization of the determinant to matrices over skew fields by J. Dieudonne [35], there is a characterization similar to the one above this case as well, cf. E. Artin [4], § IV. 1. Since the only non-commutative skew field we consider here is t h a t of the quaternions, the norm definition contained in Exercise 1.10.6 is sufficient for our purposes. For the norm homomorphism a G GL{V")

I—> N{a) e K* {K s. field or i f = H )

(2)

^ According to the classification of complex and real simple Lie algebras, going back to W. Killing and E. Cartan, there are finitely many infinite series of simple Lie algebras, which are called classical, and finitely many so-called exceptional algebras, cf. J. Tits [105], [49], [84]. The terminology we use here is in accordance with this scheme. As already mentioned in the introduction, the finitely many simple groups corresponding to the exceptional algebras and their geometries will not be discussed in this book.

2.1 The Classical Groups t h e kernel Ker N is just the special linear SL{n,

K) = SL{V")

135

group

:= {a e GL{V")\N{a)

= 1},

(3)

t h a t will also be viewed as a classical one. T h e same holds true for the projective groups PLn=GL{n+l,K)/Zn+u where Zn+i := {iS'j)k\k

G ZiK*)}

(4)

denotes t h e center of GL{n + 1, K), cf. Exercise n.7.4.7 and Definition 1.4.3. For the fields i f = R , C we already described the connection between the projective and t h e special linear groups in Example 1.4.5; in t h e notations introduced there we have PLniR.)

= SL{n+l,R.),

P L „ ( R ) ^ \SL\in PLniC)

n = 2k,

+ 1, R ) / { ± / „ + i } , n = 2k + l,

= SLin+l,C)/Kn+i,

(5) (6) (7)

where Kn+i = {{Si)k\k

= e2-i'/"+i^ / = 0 , . . . , n }

denotes t h e group of t h e ( n + l ) - s t roots of unity. In t h e case of t h e quaternions we obtain a similar representation in a straightforward way: According to Exercise 1.10.6 the norm N{g) is positive for each g G GL{n, H ) . Hence g can uniquely be written in t h e form g = gop, N{go) = 1, p > 0, N{g) = (?^. This implies P L „ ( H ) = SL{:n + 1, H ) / { ± / „ + i } .

(8)

E x e r c i s e 1. Show that the the normal subgroups occurring in the denominators of (6)-(8) are nothing but the centers of the numerators. E x e r c i s e 2. Let T^|c denote the complex restriction of the right quaternionic vector space T^". Show that the group 5 L ( n , H) is isomorphic to the subgroup of all linear maps g G SLiY\Q) commuting with the quaternionic structure J of T^|c (cf Exercise 1.10.5). In addition, find natural embeddings of SL{n, C) and SL(n, R) into 5 L ( n , H ) that, with corresponding identifications, lead to the following chain of subgroups SL{n, R) C SL{n, C) C SL{n, H) C SL{2n, C). Remark. T h e subgroup of SL{2n, C) isomorphic to the group SL{n, is often denoted as SU*{2n) , cf. S. Helgason [49], S. 445.

H)

136

2 Cayley-Klein Geometries

2.1.2 T h e P r o j e c t i v e I s o t r o p y G r o u p of a C o r r e l a t i o n To define the remaining classical groups we will now proceed as follows: In the projective geometry ^ " we distinguish a symmetric correlation F, i.e. a non-degenerate null system or a polarity, which will be called the absolute correlation; its isotropy group for the action of the projective group P i „ : PGn{F):={gePLn\goFog-'

= F},

(9)

cf. Definition 1.8.2, is called the projective group of the correlation F. Let now b = bp he any a-biform defining the correlation F. According to Proposition 1.7.6 it is uniquely determined up to a proportionality factor K G K* and non-degenerate by Corollary 1.8.3. Its isotropy group for the action of the hnear group GL{V) described by (1.8.12) with K = 1 is denoted by UGm{b), m = dim V; hence g e UGm{b)

^ ^ g e

GL{V^)

and b{g^, gt)) = 6(y, t)) for ah y, t) G F ™ . (10) T h e observations in Section 1.8, cf. (1.8.12), imply

L e m m a 1. Let b be a a-biform defining the absolute correlation F, and let g G PLn be a projectivity induced by the linear transformation a G GLCV"^ ). Then g G PGn(F) if and only if 6(aj;, at)) = K(a)6(j;, t)) for all p, t) G F"+"^;

(11)

here K(a) G Z{K)* is a scalar depending only on a satisfying The canonical map p : GLCV"^ ) —> PLn thus yields the

a{K{a)) = K(a). inclusion

p{UGn+l{b))CPGn{F).

(12)

P r o o f . Relation (1.8.12) immediately implies assertion (11) with K{a) G K*. Since b is non-degenerate, there are p, t) G V " + ^ such t h a t 6(j;, t)) = 1. By (11) we have for arbitrary A G i f 6(aj;A, at)) = K(a)6(j;A, t)) =

K{a)a{X)

= o-(A)6(aj;, at)) = o-(A)K(a)6(j;, t)) =

a{X)K{a).

Since a is surjective, K(a) belongs to the center of K. Similarly, b{at), ap) = K{a)b{t), p) = K{a) = a(b(af, at))) = a(K(a)).

a Obviously, the set of all a G GL{V"^^) satisfying (11) is a subgroup; it is called the conformal group of the biform b and is denoted by C t / G „ + i ( 6 ) . Inserting a = 0,2 o ai into (11) we obtain

2.1 The Classical Groups C o r o l l a r y 2. The map K : C t / G „ + i ( 6 ) -^ Z{K)* In particular, for a field K it satisfies

is a group

K(a)"+i=a(]V(a))]V(a).

137

homomorphism.

(13)

P r o o f . Choose a basis, replace (11) by the corresponding matrix equation, {a{a1)){hki){(J^^) = n{a){h^l

(14)

cf. (1.8.13), and compute the determinant for the matrix (ajj) of a, N{a) = det(ajj). This immediately leads to formula (13). • To describe the closely related groups P G „ ( F ) , UGn+i{b), CUGn+i{b) in detail requires to have a classification of biforms over the skew field K at ones disposal. For i f = R , C, H this task was accomplished in Section 1.9 for the polarities and (in the case of an arbitrary K) in Section 1.8 for the null systems. In the sequel we restrict ourselves to these important cases and refer for the general theory to J. Dieudonne [36]. 2.1.3 T h e Symplectic Groups Let F : ^ " ^ ^ ^ " ^ be a non-degenerate null system, and consider a biform b : V x V ^ K determining F. Then, according to Proposition 1.8.5, K has to be a field, and b is bihnear as weU as alternating; moreover, V has even dimension, dim V = 2n. For fixed K and n, all these null systems F are equivalent. In a suitably chosen basis the matrix of b has the form (1.8.19) with r = n. Frequently, a somewhat different normal form is preferred: Applying the coordinate transformation iC

—

iC

.

iC

—

iC

1 ^ —

_L. . . . . 77'.

I _L0)

in the new coordinates the matrix of b has the form

A vector space V " with a distinguished non-degenerate, alternating, bilinear form b is called symplectic just like the bases (or coordinates) in which b has normal form (16). If (cu^) is a symplectic cobasis, i.e. a basis dual to a symplectic one, then, in it, the alternating 2-form b is expressed as b = u;^ A u;"+i

+ u;2 / \ ^n+2

+ . . . + ^ " A U;^'',

(17)

and vice versa (cf. Example II.8.3.2). T h e transformations a G Sp{V " ) := UG2n{b) are also caUed symplectic, and the group Sp{V " ) is the symplectic group. Since K{a) = 1, for a G UGm{b) relation (13) implies: a{N{a))N{a)

= 1,

a G UGm{b),

K a field;

(18)

138

2 Cayley-Klein Geometries

thus for bilinear b and a G UGm{b) we always have ]V(a)^ = 1 (cf. Exercise 3). More precisely, symplectic transformations satisfy N{a) = 1

ae

SpiV"^").

(19)

For n = 1, i.e. in a two-dimensional vector space, the alternating 2-form (17) coincides with the area form determining the equi-afhne geometry, expressed as a scalar multiple of the determinant. So the symplectic group Sp{V'^) is equal to the special linear group SL(V ), and the symplectic geometry on V'^ coincides with its equi-afhne geometry. Formulas (6), (7) show t h a t the geometry determined by the symplectic group on the real or complex projective line presents nothing new as compared to projective geometry; it is for the projective space P^ t h a t the symplectic geometry first becomes interesting. Exercise 3 . Prove (19) under the assumption char A" = 0 or char A" > n. Hint. Consider the alternating 2n-form b" := f\" b. (19) also holds without the assumption above; the proof, however, is more difficult, cf. E. Artin [4], Proposition 3.25. Exercise 4 . Let [\^^",6] be a symplectic vector space. Prove: A transformation g e GL{V^") is symplectic if and only if its matrix with respect to a symplectic basis has the block structure (a}) = ( ^ ^ ) with A, B,C,De

M„{K),

A'C = C'A, B'D = D'B, A'D-C'B

satisfying = I„.

(20)

These matrices are, of course, also called symplectic. They form a subgroup of GL{2n, K) isomorphic to SpiV'^'"') denoted by Spin, K) and again called the symplectic group. T h e projective space p ^ n - i ^ Piy'^^'-) corresponding to a symplectic vector space where the null system F determined by an alternating biform h is distinguished is called projective symplectic; the projectivities preserving F form the projective symplectic group PSp^ := PG2n-i{F). Now we consider the cases i f = R , C separately. E x a m p l e 1. Let p ^ n - i ^^^ ^ ^^^j projective symplectic space. By (6), considering only the transformations a with N{af = 1 we already obtain all projectivities, which by (13) implies K(a)^" = 1. T h e m a p Sg with the coordinate representation So-. y' = x " + \ y"+* = x \ * = l , . . . , n ,

(21)

satisfies n{so) = — 1 . Since, by CoroUary 2, K : |S'Z(|(2n, R ) —> { ± 1 } is a homomorphism as well, we conclude PSp^{n)=p{Sp{n,n))yjp{so{Sp{n,n)).

(22)

2.1 The Classical Groups

139

Each real, projective symplectic transformation can thus be described by a symplectic transformation a G Sp{V ") or a transformation of the form sooa, a G Sp{V " ) . As only the dilations belong to Keip, the homomorphism theorem implies

P ^ p J R ) = {Sp{V'-) U

So{Sp{V'n)/{±hn}. a

E x a m p l e 2. Consider the complex projective symplectic space p 2 n - i ^ p(^v^ny gj^^^^ j ^ ^ ^^^j^ dilation d^, ij, e C*, K(d^) = /i2 by (11), we may always suppose that the symplectic projectivity g = p{a) G PSp^ is generated by a symplectic transformation a G S'p(V^"); in fact, p(a) = p{a o d^) and K(a o d^) = K{a)i^ = 1 always has a solution. Hence we have

PSp^iC) = p{Sp{V^")) =

Sp{n,C)/{±l2n},

since only the elements ±/2„ belong to Sp{n, C) n K2n (cf. (7)).

•

2.1.4 T h e O r t h o g o n a l G r o u p s Now we want to discuss the polarities F. For if = R, C, H they were classified in Section 1.9. The results of this classification are shown in Table 2.1 at the end of Section 2.1.6. We wiU discuss the cases one by one. E x a m p l e 3. Let V" be a real vector space, and let 6 be a distinguished symmetric, non-degenerate, bilinear form of index / on V. Then [V^",6] is called a pseudo-Euclidean vector space and the corresponding isotropy group is called the pseudo-orthogonal group 0{l,n — I) := t/G„(6). Since —b is invariant if 6 is, we have 0{l,n-l)

=

0{n-l,l),

and we may assume 0 < / < n/2. Obviously, for / = 0 we just have a Euclidean vector space and the orthogonal group 0{n) = 0 ( 0 , n) (cf. Corollary 1.6.2.4). For the theory of relativity the Minkowski space is of fundamental importance, i.e. the four-dimensional pseudo-Euclidean vector space of index 1. The associated group 0(1,3) is called the Lorentz group. Since a = idpt, we conclude from (18) ]V(a)2 = 1,

aeO{l,n-l).

(23)

In order to determine the isotropy group PO{l,n — I) := PGn-i{Fnj) of the corresponding polarity -Fn,i, again by (5) and (6) we may start from the

140

2 Cayley-Klein Geometries

linear transformations a with N{af = 1; according to (13) these transformations satisfy K{a)" = 1. If n 7^ n — /, i.e. / 7^ n/2, then by the Theorem of Inertia, Proposition 1.5.9.6, in fact K(a) = 1, and, moreover, PO{l,n-l)=p{0{l,n-l))

= 0{l,n-l)/{±In},

I y^ n/2.

(24)

Next we consider the case 0(n, n), i.e. F2„^„, and, introducing coordinates in which b has normal form, we see that by (21) the map SQ again has the factor K{SO) = —1. As in the case of the symplectic group this imphes PO{n,n) = p{0{n,n)yj So{0{n,n))) and PO{n, n) ^ {0{n, n) U So{0{n, n)))/{±hn}-

. ^ ^ ' D

Example 4. The complex Euclidean vector space [V", h] is defined in analogy to Example 3; this time, let 6 be a non-degenerate, symmetric, bilinear form. Its isotropy group is denoted by 0{n, C) := t/G„(6) and called the complex orthogonal group. As in Examples 2, 3 we conclude for the isotropy group of the polarity PO{n, C) := P G „ _ i ( F „ ) : PO(n,C) = 0(n,C)/{±/„}.

Proposition 1.9.3, see also Proposition 1.5.9.6 and Corollary 1.5.9.3, implies that for any non-degenerate symmetric bilinear form 6 and if = R, C we can always find bases (ej) of V " in which the orthogonality relations h{ti,tj)

= SijCi,

i,j

= 1,.

..,n,

hold; here Cj = —1 for if = R, « = 1 , . . . , /, and Cj = 1 else. Such bases are called pseudo-orthonormal or orthonormal, respectively, if / = 0 or if = C. A linear transformation a belongs to the isotropy group t/G„(6) if its matrix (a^) is pseudo-orthogonal with respect to a pseudo-orthonormal basis, i.e., if the orthogonality relations n

^Cka^a^ fc=i

= CiSij,

i,j = l,...,n,

(26)

hold. The groups of pseudo-orthogonal matrices isomorphic to the isotropy groups are denoted like these. Again, relation (13) imphes that for pseudoorthogonal matrices det(a^)^ = 1. The subgroups of matrices (or linear transformations, respectively) whose determinant (norm) is equal to 1 are called special pseudo-orthogonal groups and denoted by SO (I, n — I). For / = 0 and if = R it is called the special orthogonal group SO{n) or the special complex orthogonal group SO{n, C).

2.1 The Classical Groups

141

2.1.5 T h e U n i t a r y G r o u p s Let now K = C or K = H, and consider a non-degenerate Hermitean form b of index /, cf. Lemma 1.9.8. T h e related isotropy groups are called pseudounitary, or unitary in the case / = 0. For K = C they are denoted by U(l, n—l), U{n) := t / ( 0 , n ) , and for i f = H by Sp{l,n/), Sp{n) := Sp{0,n). As in Example 3 we may suppose 0 < / < n / 2 . The notion of a pseudo-orthonormal basis defined above is also applied to pseudo-unitary groups. T h e matrices of pseudo-unitary transformations with respect to these bases are again called pseudo-unitary. They are characterized by orthogonality relations analogous to (26): n

^Cka^a^ = CiSij, i,j = l,...,n. fc=i For i f = C we compute the determinant or use (13) to obtain |]V(a)| = | d e t ( a } ) | = l.

(27)

(28)

Note t h a t , e.g., a = idy •e^'^ is always pseudo-unitary, whereas, in general, we have N{a) = e^'^'-v ^ i. T h e kernel of the norm homomorphism {K = C ) , N -.ae

U{l,n-

I) I—> N{a) e S^ = {z e C\ \z\ = 1},

is called the special pseudo-unitary SU{l,n-

group

I) := {a G U{l,n-

the special unitary group is defined as SU{n)

l)\ N{a) = 1}, := SU(0,n)

(29)

(cf. (1.6.2.33)).

For the quaternionic pseudo-unitary groups Sp(l, n — I) we now prove the following result describing complex realizations of these groups: P r o p o s i t i o n 3. Let V = V^ be an n-dimensional, right vector space over H , let b be a non-degenerate, T-Hermitean biform of index I over V, and let T{I) = q be the conjugation in H (cf. Proposition 1.9.7). Moreover, denote by V\c the complex restriction of V. Decompose 6(j;, t)) into its two complex components according to 6(j:,t)) = a(j:,t))+j/3(j:,t))

(?,t)GF),

a(j;, t)),/3(j;, t)) G C . Then: 1) a is non-degenerate and Hermitean of index 21 over V\Q,; 2) [3 is a non-degenerate, alternating, bilinear form over V\Q,; 3) the group Sp{l, n — l) is isomorphic to the group of complex linear formations Sp{l, n-l) = U{21, 2(n - /)) n Sp{n, C ) ,

(30)

trans(31)

where t / ( 2 / , 2(n — /)) is the group isomorphic to U{2l,2{n — I)), consisting of the transformations leaving the Hermitean form a invariant whose matrix

142 (aij)

2 Cayley-Klein Geometries is the following

block matrix

in a symplectic

basis (with respect to (3)

(Oj,0„+j), i = 1,-••,"-•• f-Ii (ajj) =

Ki^n-i

0 0 0 0 I„-i 0 0 0 0 - / ^ 0

V 0

0

0

\ (32)

In-lJ

P r o o f . Since 6 is a T-biforni, we have 6(t), y) = a(t), y) + j /3(t), ?) = 6fe t)) = «(?, t)) + j /3(?, t)). Now j - / 3 ( ? , t ) ) = / 3 f e t ) ) - j = -j-/3(?,t))Comparing the components shows t h a t a is Hermitean, and /3 is bihnear as weU as alternating. So let (Oj), « = 1 , . . . , n, be a pseudo-orthogonal basis with respect to 6, i.e., suppose t h a t b{ak,ai) = ekSki,

k,l = l,...,n.

(33)

Then (ofc,0„+fc), k = l , . . . , n , with o„+fc := afcCftj is a basis for Vc- A straightforward computation shows t h a t this basis is symplectic with respect to /3, and t h a t the biform a has the matrix (32) in it. This again implies t h a t each H-linear transformation a leaving b invariant preserves a as well as /3 as a C-linear transformation, hence a G tJ{2l, 2{n — l))r\Sp{n, C ) . T h e following exercise shows t h a t each transformation a G t / ( 2 / , 2 ( n — /)) n Sp{n, C) von V | c is H-linear. This implies (31). • Exercise 5. Complete the computations left out in the proof of Proposition 3. Hint. Represent the matrix of the C-linear map a with respect to the basis (flfc), fc = 1 . . . , 2n, defined in the proof as a block matrix of the form CDJ'

A,B,C,DeM„{C).

Then derive the conditions to be satisfied by a e tj(2l,2{n — I)) as well as a e Sp{n,C), and conclude from them two expressions for a^^ as block matrices of the same type. Their comparison shows a(pj) = a(p) j . This imphes that a is H-Iinear. Proposition 3 shows the close relation of Sp{l, n — I) with the symplectic group Sp(n, C ) ; moreover, it justifies the notation. In particular, Sp{n)

= Sp{n, C) n t / ( 2 n ) ,

(34)

where the basis (Ofc), k = 1 , . . . , 2n, defined above is symplectic and, at the same time, orthonormal. T h e following exercise formulates a result similar to Proposition 3 for the real restriction of a complex pseudo-unitary vector space.

2.1 The Classical Groups

143

Exercise 6. Let V = V" be an n-dimensional complex vector space, and let T^|R be its real restriction; moreover, let 6 be a Hermitean biform over V of index I. Prove: a) The decomposition of b(j, t)) into its real and imaginary parts, 6(?,t)) = a(?,t)) + i/9(?,t))

?,t)eF,

defines a non-degenerate symmetric bilinear form a of index 21 as well as a nondegenerate alternating bilinear form /3 over T^|R. - b) The pseudo-unitary group U(l,n — I) is isomorphic to the group of real linear transformations, U{l,n-l)

^ d{2l,2{n

- I)) n

Sp{n,'R),

where 0{2l, 2(n — I)) denotes the group isomorphic to the pseudo-orthogonal group 0{2l, 2{n — l)) of real linear transformations which leave the bilinear form a invariant and whose matrix has the form (32) with respect to a symplectic basis (Oj, On+j), j = 1 , . . . ,n, of T^|R. Hint. If (Oj), j = 1 , . . . ,n, is a pseudo-orthonormal basis of V, then define fln+j := ttj i fij and proceed as in the proofs of Proposition 3, Exercise 5. Exercise 7. Consider the action of the unitary group U(ji) on the Hermitean vector space V". Let G2n,fc(R-), 0 < fc < 2n, denote the Grafimann manifold of k-dimensional real subspaces of the real restriction T^|R of V. Prove: a) G2n,fc(R.) is invariant under the action of U(n). - b) For fc = 1 and fc = 2n — 1 the action of U(n) on G2n,fc(R-) is transitive, and for 1 < fc < 2n — 1 it is not transitive. E x a m p l e 5. Now consider the Hermitean polarities Hnj of P " ^ (if) for K = C, H . Because of (7) and (8), we may again start from SL{n,K) in order to determine their isotropy groups: PU{1,n-l):= PSp{l,n-l):=

PGn-i{Hn,i) PGn-i{Hn,i)

for K = C, for K = H;

by (13) we obtain K"(a) = 1, and, applying once again the Theorem of Inertia (Lemma 1.9.8), for / ^ n/2 we even arrive at K{a) = 1. This leads to PU{1, n-l) PSp{l,

n-l):=

= SU{1, n - l)/Kn

ioi K = C,l^

Sp{l, n - l)/{±In}

n/2,

for K = ii,l^

n/2.

For / = n / 2 use (21) and construct from Sg a transformation with K{SO) = —1, this yields a result similar to (25). • 2.1.6 T h e Q u a t e r n i o n i c S k e w H e r m i t e a n P o l a r i t i e s Next we want to study the polarities i7„ := J „ of the quaternionic projective geometry ^ " ^ ^ ( H ) , cf. Proposition 1.9.14; each of t h e m is described by a skew Hermitean biform (1.9.22) with r = n. Once more we decompose this biform b according to (30) into b o t h its complex components a, [3 and consider the complex restriction V^^. Since 6 is skew Hermitean with respect to H

144

2 Cayley-Klein Geometries

(cf. 1.9.16), the biform a is complex skew Hermitean, and [3 is symmetric as well as C-bilinear. Let (bj), j = 1 , . . . , n , be a basis of V " with respect to which the matrix of 6 has the normal form (1.9.22). Then (b;, b„+i), / = 1 , . . . , n, with bn+i '•= b; j is a basis of V f c ! denoting by x'=e+iC^\

l = l,...,n, e',r+'eC,

(35)

the quaternionic coordinates of the vector p with respect t o (b;), the i^'', k = l , . . . , 2 n , are its complex coordinates. In these, the biforms a,/3 are expressed as follows:

«(?,t)) = i E ( ^ " V - r + V + ' ) ,

(36)

n

/3(y,t)) = - i ^ ( e V + ' + r + V ) -

(37)

1=1

This shows t h a t both biforms are non-degenerate. Hence f3 is complex Euclidean. Changing coordinates in V\c according to C = (C + i C

)/v2, C

= (?

-1?

)i/v2

we arrive at normal forms for both biforms: n

«(?,t))=E(^"'"^'^"-^"'V"+'),

(38)

n

1=1

T h e subgroup of complex orthogonal transformations leaving a invariant is denoted by 0 * ( 2 n ) ; set, moreover, SO*{2n)

:= SO{2n,

C) n 0*(2n).

(40)

Exercise 8. Prove that SO* (2n) is a subgroup of SU* (2n) (cf. the remark following Exercise 2). Exercise 9. Prove that the subgroup of GL(V?(-.) leaving a invariant is isomorphic to the subgroup U(n,n). Let now a G GL{V") be an H-linear transformation satisfying (11). Lemma 1 then implies K(a) G R*, and we may always suppose K = ± 1 . Left multiplication by the imaginary unit k G H*, i.e. applying the m a p

2.1 The Classical Groups n

n

S o : j ; = ^ b ; x ' i — > So(p) = ^ 1=1

145

b; k - x ' ,

(41)

1=1

yields an element SQ G SLCV"") satisfying (14) with K{SO) = — 1 . Since we only have to consider elements a G SL{V") by (8), for the isotropy group of Hn a result similar to (25) holds: PGn-i{Hn)

= {SO*{2n)

U So{SO*{2n)))/{±hn}-

(42)

Exercise 10. Determine the matrix for the map So in the complex restriction T^I^ defined by (41), show So G 5 L ( F " ) , and K{SO) = - 1 . Exercise 11. Let V" be a vector space over the skew field K, and let 6 be a non-degenerate cr-Hermitean or skew cr-Hermitean biform over V. Moreover, let a e GL(V") be a linear transformation preserving orthogonality: 6(p,t)) = 0 always implies 6(a(p),a(t))) = 0. Prove that a e

2 . 1 . 7 SL{n,

CU„{b).

K)

is G e n e r a t e d b y T r a n s v e c t i o n s .

To prove the invariance of a geometric property under all the transformations from a certain group G it is often useful to know a system of generators for the group consisting of particularly simple transformations. In fact, it then suffices to verify the invariance for these generating transformations. In this section we will prove t h a t the special linear group over a field K is generated by transvections. A transvection is understood to be a projectivity / G PLn for which there is a hyperplane H C P " t h a t / leaves point-wise fixed, i.e. f\H = i d j ^ , and which has no additional fixed points. Obviously, this definition also makes sense for projective spaces over skew fields. Considering, as in Exercise 1.5.3, affine space as a subset of projective space the translations are just the restrictions of transvections, whose fixed-point set is the improper hyperplane, to affine space. Note t h a t now, in projective geometry, different transvections have different hyperplanes as their fixed-point sets, in contrast to the translations of the projectively embedded affine geometry, whose fixed-point set is the distinguished improper hyperplane we keep fixed. The basis representations of transvections have particularly simple matrices, if the points aj = [Oj] G H, j = 1 , . . . , n, of the frame are fixed points themselves. Let B denote the matrix of the transvection 6 in a corresponding basis with ao = [ao] ^ H, then, just hke for the translations, we have

B = f ^ J ) with OT^oGif". At first, the conditions in the definition only lead to the form

(43)

146

2 Cayley-Klein Geometries K polarity biform b

UG„{b) 0{l,n-l),

R Fn,i

bilinear, symmetric, of index I

C Fn

bilinear, symmetric

0(n,C)

C H„,i

Hermitean of index I

U{l,n-l), Uin) :=C7(0,n)

H Hn,l

Hermitean of index I

Sp{l,n-l), Spin) : = 5 p ( 0 , n )

H Hn

quaternionic skew Hermitean

oln) :=O(0,n)

0*(2n)

The normal forms for the non-degenerate biforms above are

Fn,i : -Y^x"y"

+ Y^

x''y\0
r. = l+l

Q!=l

n

Fn :

^x^j/^ I

n

H=

Q!=l

l+1

Hn : 2 J x^ iy^j=i

Table 2 . 1 . Polarities, biforms, and their isotropy groups

a

0

0 /„A

with a G if*, O G i f " ,

XeZ*.

Since, by Definition 1.4.3 for the projective groups (cf. (1.4.22)), the kernel of the canonical m a p GL{n + 1, K) -^ PLn is the center of GL{n + 1, K), i.e. Z{n+l)

=

{In+iX\XeZ*},

we may multiply the matrix by A^^, or suppose without loss of generality t h a t A = 1. Now it is easy to check t h a t the corresponding projectivity has no fixed point outside of H if and only if a = 1 and 0 7^ 0 both hold. The following exercise characterizes the linear transvections. Exercise 12. A linear transformation b e GL{V"^^) if it has the form 6(p) = id^(p) + 0w(p),

is a transvection if and only

2.1 The Classical Groups

147

where w G T^' is a linear form, w / 0, satisfying w(o) = 0. This again is the case if and only if rk(6 — idy) = 1 and (6 — idy)^ = 0. T h e isomorphies (5)-(8) show t h a t for the scalar domains i f = R , C, H the projective groups essentially are quotients of special linear groups. Hence the following proposition is of particular interest for projective geometries over these scalar domains: P r o p o s i t i o n 4. Let K be a field. Then the special linear group SL(n+ 1, K) and the group of projectivities PSL(n+ 1, K) C PLn corresponding to it are by transvections. P r o o f . We prove this by induction, starting with n = 1. Let G denote the subgroup of projectivities generated by the transvections. According to (43) each transvection can be generated by a linear transvection with determinant one. Hence we have G C PSL{n + l , i f ) . The transvections leaving, say, the point oo of a projective scale on the line invariant are the translations represented in this scale as x i-^ x + a, and the group they generate acts transitively on the complement of oo. Thus G acts doubly transitively on the projective line P , i.e., we can always find an element h E G transforming the points x,y with x ^ y into the points h{x) = 0, h{y) = 1. (We identify the points with the corresponding elements of a fixed projective scale.) Now take an arbitrary g G PSL{2,K), g ^ e. We then find xi G P^ such t h a t g{xi) = oo. If xi ^ oo, then there is a transvection t\ with the " hyperplane " H = {0} and ti(oo) = x i , so t h a t for gi = goti the point 31(00) = 00 is fixed. Hence if g^i can be represented as the product of transvections, this is also true of g; so we may suppose (7(00) = 00. Moreover, we find a transvection to with fixed point 00 mapping yo = g{0) to 0. Thus go = to o g satisfies go{0) = 0. W h a t we now still have to prove amounts to showing t h a t a projectivity g G PSL{2, K) with g{Q) = 0 and g{oo) = 00 can be represented as a product of transvections. Such a transformation is generated by a linear transformation with a diagonal matrix, and the equation below displays a decomposition into transvections: a O \ _ / l l W 1 O W l -a-i \ / 1 0 a-ij ^ V 0 1 j V a - 1 1 / V 0 1 j\a-a:^

0 I

This proves the statement for n = 1. Suppose t h a t the assertion is already proved for n — 1, and take g G PSL{n + 1) C PLn- Since the group G generated by the transvections acts transitively on the projective space P " , we may suppose t h a t g has a fixed point, which we choose as the vertex a „ + i = [o„+i] of a coordinate simplex. In a corresponding basis the matrix B oi g then has the form \c

a ^

where c = {7,i+i, • • •, 7^+1} and d e t ( C ) = a. It is easy to show t h a t each matrix of the form

148

2 Cayley-Klein Geometries 'In 0

generates a transvection in P " . Hence it suffices to show that

is a product of transvections. Considering a decomposition analogous to (44) one sees that the transformation with the matrix A = is a product of four trans vections. Muhiplying Bi by A from the right leads to the matrix

with det(Ci) = 1. By the induction hypothesis we can represent Ci as the product of finitely many transvection matrices TQ, of order n. Extending these as T — to matrices of order n + 1 we obtain a representation for BiA as a finite product of transvection matrices, which completes the proof. • >From Proposition 4 we easily obtain the following statement concerning generators for the projective groups in the cases of most interest for us, if = R , C : Corollary 5. The projective groups PL2k(R'), PLn(C) are generated by their transvections. To obtain a system of generators for the groups PZ(2fc+i(R-) of real projectivities in odd dimensions it suffices to take all transvections and add a single transformation reversing the orientation with determinant —1. P r o o f . For if = R, C the assertion immediately follows from Formulas (5)-(7) together with Proposition 4. D

2.2 Vector Spaces with Scalar P r o d u c t In the preceding section we determined the isotropy groups of biforms for the natural action of the linear groups and at the same time introduced the socaUed classical groups. The geometries of these groups have some properties in common which we want to study in this section; below we will discuss some of the geometries important for later applications in detail.

2.2 Vector Spaces with Scalar Product 2 . 2 . 1 V e c t o r , P r o j e c t i v e , a n d Affine

149

Geometries

A linear group is understood to be a subgroup G of the general linear group GL{V) of a finite-dimensional vector space V over a skew field K, or, having fixed a basis, the corresponding matrix group isomorphic to it. Its vector geometry is determined by the transformation group [G, V], where, as a subgroup of GL(V), G acts linearly on V. The contents of this geometry is formed by the G-invariant properties of the space V itself and of all the spaces functorially related to it: set products, tensor spaces etc. In particular, this includes the projective geometry [G,^(V)], where the action of G is defined by the projectivities corresponding to the elements of G. To study the ajfine geometry of the group G we start from Formula (1.5.8) describing the general affine group of the affine space; taking not any element B G GL{n, K) but only those from the linear group G C GL{n, K) uniquely determines a subgroup GA C 2l(n) of the affine group. T h e transformation group [GA, A " ] for this affine action of GA over the affine space A " then determines the affine geometry corresponding to [G, V]. T h e group GA may also be defined as the subgroup of the affine group 2l(n) generated by the union of the group of translations T ( A " ) with the group G , ^ ; in the case of a field K it is the semi-direct product of T ( A " ) with the group G , ^ , cf. Exercise 1.5.1. > F r o m these rather general observations together with what we discussed in Sections 1.4-6 as well as in the preceding chapter it is obvious t h a t the vectorial point of view, at least methodically, dominates both the other two; thus we will emphasize it in this section and fix a corresponding terminology. D e f i n i t i o n 1. A vector space with scalar product is a pair [V, (,)] with the following properties: 1. V is a finite-dimensional right vector space over the skew field K. 2. T h e scalar product (,) is a a-biform over V, (p, t)) = 6(j;, t)), where a is an involutive anti-automorphism of K. 3. T h e scalar product is non-degenerate, i.e., for each p G V , p 7^ o, the linear form (p, •) over V is different from 0. 4. T h e scalar product may be an alternating or a symmetric bilinear form, or (T-Hermitean. 5. If (,) is not alternating, then suppose t h a t char i f ^ 2.

Let G C GL(V) denote the group of automorphisms of the vector space with scalar product, i.e. the linear automorphisms g satisfying (grp, gt)) = (p, t)) for all p, t) G V . As described in the introductory paragraph, this group G then also determines the associated projective and affine geometries, respectively. According to this definition we will speak of a projective or affine space with scalar product whenever a scalar product is distinguished in the underlying vector space. Both, the geometry and the underlying space are called symplectic if the scalar product is alternating, otherwise they are called Hermitean.

150

2 Cayley-Klein Geometries

If i f is a field and the scalar product is bilinear and symmetric, then the geometry is called orthogonal; hence this is a special case of a Hermitean geometry. Talking about the index of a Hermitean form always refers to the cases i f = R , C or H ; this notion is not to be confused with t h a t of the index of a space with scalar product (see Definition 3 below). For i f = R and a scalar product with positive index /, the corresponding geometry is called pseudoEuclidean, for i f = C, H and / > 0 these spaces are called pseudo-unitary. The Euclidean and unitary spaces discussed in Chapter 1.6 are included as the case / = 0. 2.2.2 S u b s p a c e s Throughout this section we wiU suppose t h a t V is a vector space with scalar product (,). The notions isotropic and totally isotropic introduced for subspaces W CV in Definition 1.9.3 wiU always refer to the scalar product (,). In particular, a vector p G V is caUed isotropic if p 7^ o and (p, p) = 0. Likewise, orthogonality is always taken with respect to this scalar product: T h e vectors p, t) G V are caUed orthogonal if (p, t)) = 0; subspaces U,W C V are ort/io^onaHf their vectors ^ eU,t) eW are orthogonal (cf. Definition 1.6.1.1). Definition 1.6.2.16 of the subspace W cV orthogonal to a subspace W C V applies: W^ : = { t ) G F | ( p , t ) ) = 0 for ah p G VF}. We will not, however, call it the orthogonal complement, since, in general, the condition Vt^ n Vt^ = {0} is not satisfied. Instead, the following is an immediate consequence of the definitions P r o p o s i t i o n 1. A subspace W C V is isotropic if and only if the intersection W nW is non-trivial; together with W its orthogonal subspace W is also isotropic. • C o r o l l a r y 2. IfW
(1)

{W^)^=W,

(2)

{u + w)^ = u^nw^, (unw)^ = u^ + w^, C7 C W implies W ^ C C/^.

(3) (4) (5)

2.2 Vector Spaces with Scalar Product

151

2.2.3 E. W i t t ' s T h e o r e m Now we want to formulate a fundamental proposition by E. W i t t concerning the extension of isomorphisms from subspaces to isomorphisms of vector spaces with scalar product. To this end we first define the notion of a morphism for the class of pairs [W, b] consisting of a vector space over a fixed skew field K together with a a-biform 6, where
ihtjeW)

(6)

is called a morphism of these pairs and an isomorphism if cp is bijective. If W, W are subspaces of vector spaces with scalar product, then we will call t h e m isomorphic if there exists an isomorphism cp: [W,{,)\W

X W] -^ [W,{,)\W

X W].

Obviously, for fixed a and K the class of pairs [W, 6], b alternating or aHermitean, respectively, with the morphisms defined above form a category. Exercise 2. In the notations and assumptions of Definition 2, assume that the map ip : W ^ W is a morphism, and that b is non-degenerate. Prove that ip is injective, and [W,b] is isomorphic to [ip(W),b\ip(W) x ip(W)]. In the following subsections we will prove E. Witt's

Theorem :

P r o p o s i t i o n 3. (E. W i t t ) . Let V, V be finite-dimensional vector spaces with scalar products over a skew field K, which are isomorphic. Let, moreover, cp : W ^ W be an isomorphism from the subspace W C V onto the subspace W C V. Then there is an isomorphism 'ip : V ^ V of the vector spaces with scalar product such that cp = 'ip\W. 2 . 2 . 4 P r o p e r t i e s of I s o t r o p i c S u b s p a c e s To prove this proposition we will need a few facts concerning isotropic subspaces t h a t will also frequently be apphed elsewhere. L e m m a 4. Let D G V \ {o} be an isotropic vector in the vector space V with scalar product {,). Then there always is an isotropic vector 6 G V such that (D, 6) = 1; D, 6 are linearly independent.

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2 Cayley-Klein Geometries

P r o o f . Since D 7^ o and the scalar product is non-degenerate, we find a vector tn G V with (D, tn) ^ 0. As (D, tn) is hnear in tn, we may normahze tn to have (D, tn) = 1. If the scalar product is a-Hermitean, we try to find 6 in the form 6 = D^ + tn, ^ G if. Now for a-Hermitean scalar products we have o-((tti, tn)) = (tn, tn), so the vector 6 with ^ = - ( t n , tti)/2 is a solution; condition 5 in Definition 1 guarantees its existence. In a symplectic space each vector tn 7^ 0 is isotropic; so we may set 6 := tn. If D and 6 were hnearly dependent, then 6 = DA, hence (D, 6) = (D, D)A = 0 contradicting (D, 6) = (D, tn) = 1. • Exercise 3 . Let T^", n = 2m, be an n-dimensional symplectic vector space. Show that for every k < m there is a totally isotropic subspace of dimension k. P r o p o s i t i o n 5. Let W"^ C V^ be a totally isotropic subspace of the vector space V with scalar product, and let (ba), a = 1 , . . . , m, be a basis of W"^. Then there is a sequence (ba), a = 1 , . . . , m, of isotropic vectors in the space V such that {b/s, ba) = Sai3, {b/s, ba) = 0 , and, moreover,

the

(a, /3 = 1 , . . . , m ) ,

sequence bi,bi,...,b™,b™

is linearly

(7)

independent.

P r o o f . For m = 1 the assertion follows from Lemma 4. Suppose t h a t Proposition 5 is already proved for aU totaUy isotropic subspaces Wo with dim Vt^o < rn and consider the hnear span Wo := £ ( b i , . . . , bm-i)- By the induction hypothesis we find corresponding vectors b i , . . . , bm-i with the properties stated in Proposition 5 for m — 1 instead of m as weU as ba instead of ba- Set t/2™-2:=£(bi,...,b™_i,bi,...,b™_i).

Looking at the scalar products of these basis vectors it is obvious t h a t U is not isotropic. Hence U is also not isotropic, and, moreover, V = U®U^.

(8)

> F r o m (8) we obtain the decomposition bm = " „ + D™, where Um & U has the representation m—1

m—1

/ ^ ba^a a=l

+

/ ^ ba^cia=l

Forming the scalar product of this equation from the left with b/3, the relation {bi3, bm) = {bi3,Um) = 0 Icads to the equality A^ = 0, hence

J2 ^"Ka=l

(9)

2.2 Vector Spaces with Scalar Product

153

Thus we have D™ 7^ o and (b™, bm) = (Om, Om) = 0; in fact, bm and Um are isotropic as elements of W. Relation (9) together with the linear independence of (ba), a = I,... ,'m, implies t h a t Dm, is different from zero and hence also isotropic. According to Lemma 4 there is an isotropic vector bm ^ U with (Om, bm) = 1, since U it not isotropic. Because of {bm, bm) = (Om, bm) = 1, the space A := 2{bm, bm) is not isotropic, and hence the same holds for A . As b i , . . . , bm-i lie in A , by the induction hypothesis there are vectors b i , . . . , bm-i € A with the properties stated in Proposition 5 (for m — 1 instead of m ) . It is easily verified t h a t the sequence (7) formed by these vectors together with bm, bm meets all the requirements. • C o r o l l a r y 6. Let V" be a vector space with scalar product, and let W he a totally isotropic suhspace. Then k < n/2.

C V^ D

Exercise 4 . Let V^ be a real pseudo-Euclidean or a complex pseudo-unitary vector space with scalar product of index I < n/2. Prove that for each k < I there is a totally isotropic subspace of dimension k in V, and that there is no totally isotropic subspace of any dimension k > I. D e f i n i t i o n 3. Let V" be a vector space with scalar product. T h e maximum of the dimensions of totally isotropic subspaces W C V" is caUed the index

a

ofV".

This definition is not in accordance with t h a t of the index for a symmetric bilinear form (in the case i f = R ) or a Hermitean form {K = C , H ) , cf. Definition L5.9.7 and Section 1.9. If the form is non-degenerate, then its index Ig is related to the index / of a vector space with scalar product just defined via / = min(/o, n-— ^o) (10) (cf. Exercise 4). T h e index introduced in Definition 3 makes sense for an arbitrary skew field K and any scalar product. Exercise 3, e.g., implies t h a t every symplectic vector space V ™ has index m. Let U C V he an arbitrary subspace. Then by Definition 1.9.2 the subspace U C\U is the defect subspace of U. Hence for the defect d e f t / : = d e f ( ( , ) | t / x U) we have deft/ = deft/^ = d i m ( t / n t / ^ ) . C o r o l l a r y 7. Let U C V be an arbitrary subspace. Then the dimension every minimal non-isotropic suhspace W containing U satisfies dim VF = dim t / + d e f t / . P r o o f . First, the foUowing is straightforward.

(11) of

(12)

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2 Cayley-Klein Geometries

Lemma 8. If Ui is a subspace of U complementary to the defect subspace Uo, then U = U„ (BUI is a decomposition of U into orthogonal subspaces, and UI it not isotropic. • Therefore, the subspace t/^ C V is not isotropic as well, and Uo C U^ . According to Proposition 5 there is a non-isotropic subspace Wi with Uo C Wi C t/^ , namely the subspace spanned by the sequence (7) with m = def U, bi,... ,bm a basis of Uo- Hence W := Wi © t / i D t/ is a non-isotropic subspace satisfying (12). By Proposition 5 we may perform this construction within each non-isotropic subspace containing U. In the case of a minimal subspace W containing U the result has to be W itself, proving the assertion. D 2.2.5 The Proof of E. Witt's Theorem Now we turn to the proof of E. Witt's Theorem and start by proving Lemma 9. Let V, V be vector spaces with scalar products over K, both symplectic or both a-Hermitean. Moreover, let U be an arbitrary subspace, and let W D U be a minimal, non-isotropic subspace containing U. Then every infective morphism cp : U ^ V can be extended to an infective morphism P r o o f . We start from the decomposition constructed in the proof of CoroUary 7: where we have to set Wi=Uo(B

Uo and Uo = £ ( 6 i , . . . , 6™), m = deft/.

The image t/:=^(t/) = ^(t/„)©^(t/i) is isomorphic to U, since (f is injective. Consequently, (Ca) := {Lp{ba)), a = 1,.. .,m, is a basis for the defect subspace Uo '•= f{Uo) of U, and Ui •.=_Lp{Ui) is a non-isotropic subspace of U complementary and orthogonal to Uo- Hence C/^ C V is not isotropic, and Uo is a totally isotropic subsp_ace of C/^ . By Proposition 5 we find a vector sequence Ci, Ci,..., Cm, Cm, in V analogous to (7) with corresponding properties. It is easy to see that ^c is extended to an injective morphism tp : W ^ V by setting il)\U := LP, il){ba) := Ca, a = 1,.. . , m . If both spaces are symplectic, then

2.2 Vector Spaces with Scalar Product

155

(Ca, Ca) = - ( C a , Ca) = - 1 = ( 6 a , &„),

whereas, because of o-(l) = 1, in the a-Hermitean case (Ca, Ca) = Cr((Ca, £„)) = 1 = {ba, &„). D

Note t h a t , in Lemma 9, we did not suppose isomorphy of V and V. Towards the end of the proof we impUcitly used the simple construction described in the following Exercise: Exercise 5. Let U = ^l^^^Ui be a direct sum of pairwise orthogonal subspaces Ui C V. Let, moreover, ipi : Ui ^ V he injective morphisms, for which C/ := ^l^i'i^i(Ui) is a direct sum of pairwise orthogonal subspaces in V. Show that, under these conditions, r

r

i=l

i=l

defines an isomorphism from U onto U. First we will now prove E. W i t t ' s Theorem for symplectic vector spaces. By Lemma 9 it suffices to prove the statement for non-^sotropic subspaces; note t h a t W and tp{W) are isomorphic. Since V and V may be_supposed to be isomorphic, without loss of generality we can assume V = V . In fact, p : V ^ V is an isomorphism, and hence it suffices to find an automorphism a of V mapping W into p^^{W); then tp = p o a is the isomorphism we need. W i t h these observations the symplectic case is straightforward: For any non-isotropic subspace W C V oi the symplectic vector space V, W = f{W) is also not isotropic. According to Proposition 1 the same holds for the orthogonal subspaces W

and W

. Since the dimensions of both

are equal, there is an isomorphism (p^ : W -^ W (cf. Lemma 1.8.4 and Proposition 1.8.5). Then by Exercise 5 the m a p tp = ip (B f^ composed of If and if^ provides the automorphism tp G Sp{V). Obviously, one can prove E. W i t t ' s Theorem using the same method for all the scalar products classified in Sections 2.1.4 to 2.1.6 {K = R , C, H ) , looking at the normal forms and, if necessary, applying the Theorem of Inertia for the associated a-biforms. W i t h this done we next prove E. W i t t ' s Theorem in the general case. As above we suppose V = V and proceed by induction on m := dimVt^. For m = 0 the assertion is trivial. So assume t h a t the proposition holds for all dimensions less t h a n m. In W we choose any subspace U CW oi dimension d i m t / = TO — 1. Then ipi := ip\U is an isomorphism onto U := f{U), which we can extend to an automorphism 'ipi : V ^ V by_the induction hypothesis. Set Wi := Vr^(W^)- Then we have U = i^^^U), and (po := V-f^ o ^ : W ^ Wi is an isomorphism leaving every element of the subspace U C W fixed. If we succeed in finding an extension of cpo to an automorphism -i/'o of

156

2 Cayley-Klein Geometries

V under the additional assumption that there is a subspace U"'-^ c W"' satisfying ipo\U = idjj, then the proof is complete. In fact, we can then extend (fio = "^r^ o ^ to an automorphism V'o of V, for which V'o(W^) = Wi holds. So ip = ipio tpg is an automorphism of V satisfying tp\W = ip. The next step is to prove an auxiliary statement generalizing Exercise 5: L e m m a 10. Let V be a vector space with scalar product, let Wi C V, i = 1,2, be subspaces, and let Ui : Wi ^ V be injective morphisms. Suppose, moreover, that W-i n W2 = {0} and u-i{W-i) D uaiWa) = {o}. Finally, assume that for all pj G Wi (wi(?l),W2(?2)) = (?1,?2)-

(13)

Then is an injective morphism from Wi (B W^ into V. P r o o f . Because of Wi D W2 = {o}. Definition (14) is correct. It is easy to verify that for aU p, t) G Wi © W2 the condition {u{^),u{\))) = (p, t)) is satisfied. Since u : Wi © W2 -^ ui{Wi) © U2{W2) is surjective, and as the dimensions of both spaces coincide by ui{Wi)r\U2{W2) = {o}, u is injective as weU. D Let now L^"™"^ C W^ be a subspace with f\U = idjj, and lei a eW\U be a complementary vector. Consider the map a := (f — i ^ '.W^V, where i^ denotes the embedding of W into V. Since we can represent any vector VoeWasVO=u+ai with u eU, ^ e K, the image H := a{W) is equal to £(95(0) - 0), i.e. dimH" < 1. If dimH" = 0, then ip{a) = 0, hence (p = id^;^, and V" = i d ^ is a suitable extension. Let then dimH" = 1. We consider H and show that U C H . I n fact, (^(?) - ?, t)) = (^(?), t)) - (?, t)) = (^(?), t) - ^(t)))

(15)

for p, t) G W. Inserting here p = 0 and t) e U we conclude (95(0)— 0, t)) = 0. Let now W2 C H be a subspace satisfying Vt^n Vt^2 = {0} and Vt^n Vt^2 = {o}We show that taking ui = ip, Wi = W, U2 = t\Y ^^^ assumptions of Lemma 10 are satisfied: Indeed, from Mi(pi) — p^ G H" and p2 G H" we obtain for the scalar product (MI(PI) — Pi,p2) = 0; this implies (13). Hence we can extend cp to an injective morphism cpi : W (B W2 -^ V. In order to apply this construction we will distinguish two cases: 1. Vt^ is not contained in H . Then by (15) ip{W) = W does also not lie in H . Since U C H has dimension m — 1, for every subspace W2 C H complementary to U we have dim W2 = n — m,

n = dim V.

2.2 Vector Spaces with Scalar Product

157

Similarly, W n H^ = U , i.e. W r\W2 = {o}. Hence W (BW2 = V, and by Lemma 10 we obtain an extension of cp to an automorphism 'ip of V. - 2. Consider now the case W C H . Then from (15) we conclude W C H . First we will show_that (p can be extended to an injective morphism ip : H ^V.liW = WcH, then we find a subspace W2 C H complementary to W. Otherwise, W OW = U, and hence we could find a subspace W2 complementary to W in H with W2 n W^ = {0}: To see this, it is sufficient to note that Vt^ + 14^ has dimension dim t/ + 2; if b and b are vectors spanning with U the subspace W and W, respectively, and if Ui is a subspace of H complementary to W + W, then W2 = £(b + b) + Ui is such a complementary subspace. By Lemma 10 and using the argument above we obtain the extension 0 oi f to H we wanted to construct. It remains to extend (p to an automorphism V" of V. In the second case W + W C H implies (p{a) — a £ HnH ; hence H is an isotropic hyperplane. Thus, V is the smallest non-isotropic subspace containing H , and by Lemma 9 we can extend (p to an automorphism '0 of V. • 2.2.6 Transitivity Results The importance of E. Witt's Theorem lies in the wealth of geometric applications to actions of the classical groups on subspaces; the classification of subspaces with respect to this action is reduced to the classification of biforms on them, which is known in the essential cases. In particular, this leads to interesting transitivity results: Corollary 11. The automorphism group G C GL(V) of a vector space with scalar product (,) acts transitively on the isomorphy classes of subspaces W dV with the induced hiform 6 = (, )\W x W (cf. Definition 2). D As a consequence this almost aUows to completely describe the orbit decomposition of the Grassmann manifold Gn^k of A;-dimensional subspaces of an n-dimensional vector space with scalar product, see the examples below. Example 1. Let V " be an n-dimensional pseudo-Euclidean vector space with a scalar product of index /. By the Theorem of Inertia, Proposition 1.5.9.6, the isomorphy class of the bilinear form h-^ := {,)\W xW is determined by its rank r^ together with its index l^] we may, however, replace the rank by the defect d-^ = dim Vt^ — r-^. By CoroUary 11 the pseudo-orthogonal group 0{l, n — I) acts transitively on the set Gn^k,s,d of A;-dimensional subspaces with index s and defect d. In the case of a Euclidean vector space, i.e. if (,) is positive definite, we always have l^ = d^ = 0; hence the orthogonal group 0{n) acts transitively on the Grassmann manifold Gn^k of A;-dimensional subspaces in V". D Example 2. Let [A", V^,K, (,)] be an affine geometry over a vector space with scalar product. The isomorphy type of a k-plane H C A " is understood

158

2 Cayley-Klein Geometries

to be the isomorphy type of the vector space W corresponding to H with the induced biform b-^. Since the group of translations T(A) acts transitively over the A;-planes parallel to a fixed A;-plane, Corollary 11 implies: The subgroup GA C 2l(n) of the afRne group generated by the automorphism group G C GL{V) together with the group of translations acts transitively on each of the sets of A;-planes of the same isomorphy type. • Corollary 12. The automorphism group G of a vector space with scalar product acts transitively over the set of k-dimensional, totally isotropic subspaces. ~ k

IfW\W are two such subspaces, and if (ba), (Ca), a = I, • • • ,k, are arbitrary bases for W and W, respectively, then there is a transformation g €z G with g{ba) = Ca, a = I,... ,k. P r o o f . Obviously, the linear map cp : W ^ V determined by (p{ba) = CQ, is a linear morphism, which can thus be extended to an automorphism g €z G by E. Witt's Theorem. D Example 3. The symplectic group Sp{V'^"), and hence the projective symplectic group PSp^ as well, acts transitively on the projective symplectic space p2n-i^ • Example 4. Let Q C P " be any quadric of the projective space P " determined by the non-degenerate a-biform 6. By (,) := 6 we define a scalar product on the associated vector space V. Note that 6, and hence also the scalar product, are determined by the quadric only up to a factor k G K*; choosing a fixed value for this factor is called a gauge; the automorphism group G of \V, (,)] does not depend on the choice of this gauge factor. The results of this section imply: 1. The group PG of projectivities on P " preserving Q acts transitively on Q. - 2. The dimension of a projective A;-plane A G Q satisfies Dim A < (n — l)/2. - 3. Let m be the maximal dimension of a A;-plane in Q. Then m + 1 coincides with the index / of the scalar product; for this reason, / is also called the index of the quadric. For every k < I the group PG acts transitively on the set Qk of projective A;-planes contained in Q. D Example 5. As discussed above, Euclidean geometry is a specialization of affine geometry, cf. 1.6.5. Taking if = R and choosing the scalar product positive definite it is a special case of Example 2. From the projective point of view it looks as follows: Distinguish a hyperplane H^ as the absolute in real projective space (often also called the hyperplane at infinity). Then the resulting geometry is that of the affine space A " = P " \ H""^^, cf. Section 1.5. There we already know that the points x G H"^^ correspond to the directions in A". By Exercise 1.5.4 the vectors in affine geometry may be interpreted as the elements of the vector space W" corresponding to the absolute hyperplane H" . Prescribing now in H" an absolute polarity, the

2.2 Vector Spaces with Scalar Product

159

correspondence (,) := b associated with it (Definition 1.7.4) defines orthogonality: two directions a; = [p], y = [t)] are orthogonal if and only if (p, t)) = 0. If the scalar product has index 0, then it defines the empty quadric Q = 0 in H"^^; fixing a gauge, e.g. distinguishing a vector p G W", p 7^ o, as a unit vector completes the projective view on Euclidean geometry, cf. also Exercise 1.9.6. Choosing instead of the positive definite scalar product one of index / leads, in a similar way, to the whole series of pseudo-Euclidean geometries, and in the case K = C one obtains the complex-Euclidean geometry this way. Just as in Example 2, a variety of transitivity statements foUow. • 2.2.7 N e u t r a l Vector Spaces We now again refer to Proposition 5 and call a vector space with scalar product neutral if it may be represented as the direct sum of two totally isotropic subspaces (cf. N. Bourbaki [19], §IX.4.2; E. Artin [4] calls such spaces hyperbolic, a terminology in conflict with the frequently used notion of hyperbolic space in the geometry of constant negative sectional curvature, cf. Section 6 below). T h e term neutral will also be applied to the associated scalar product. Obviously, every symplectic vector space is neutral. In the following exercises the reader can find some applications of this notion. Exercise 6. Let V" be an n-dimensional neutral vector space, and consider a decomposition of V into the direct sum of two totally isotropic subspaces, V = Wi e W2. Prove: a) The map <^:pe W 2 ^ ^ < ^ ( p ) : = ( p , . ) | W i e W i

(16)

is (7-linear and injective. - b) Both subspaces have equal dimension, dimWi =dimW2,

(17)

and ip is a cr-linear bijection. - c) If (p^,), a = 1,... ,k, is a basis for Wi, then there exists a uniquely determined basis (t)^), /3 = 1 , . . . ,fc,for W2 such that (?a,t)/3)=<5«/3, a,(3 = l,...,k.

(18)

Exercise 7. If T^i,T^2 are two neutral, cr-Hermitean vector spaces of equal finite dimension over the skew field K, then they are isomorphic (the analogue of Proposition 1.8.5). Each pseudo-Euclidean vector space T^^™ of index rn is neutral; the same holds for vector spaces with the Hermitean scalar products H2m,m in the cases /f = C,H. Exercise 8. The vector space V" with scalar product has index k if and only if it contains a maximal neutral subspace W
160

2 Cayley-Klein Geometries

Exercise 9. Let V" be an n-dimensional vector space witli scalar product. Prove: a) Tliere is a decomposition V" =Wi®W2®

U, dim W i = dim Wa = index V",

(19)

such that Wi,W2 are totally isotropic, and U = (Wi © W a ) ^ does not contain any isotropic vector. Such a decomposition is called a Witt decomposition . - b) If V" = Wi © W2 © C/ is a second Witt decomposition, and if, moreover, (pQ,), (PQ,), a = 1,... ,k, are bases of Wi and Wi, respectively, then there is an automorphism g in the automorphism group G such that fl(?J = L , a=l,...,k,

g{W2) = W2,g{U)

= U.

(20)

- c) If Wi,Wi, i = 1,2, are two pairs of totally isotropic subspaces of maximal dimension with Wi n W2 = Wi n W2 = {o}, then there always is a g G G [ F " , (,)] with g{Wi) = Wi,i= 1, 2. Exercise 10. Let V" be a vector space with scalar product (,). Prove that for two isomorphic subspaces U,U C V" the corresponding orthogonal subspaces U^,U are also isomorphic. Exercise 1 1 . Let [V", (,)] be a neutral space, and let p^,, t)p, a,/3 = 1 , . . . ,fc= n/2, be a basis adapted to a Witt decomposition V" =Wi(B W2 having property (18). Prove: a) If g is an automorphism of V" satisfying fflWi = id^y , then, with respect to this basis, g is represented by a block matrix of the form ^^f]eGL{2k,K),

(21)

where, for symplectic spaces, the matrix A = A' is symmetric, whereas, for aHermitean spaces, the matrix A is skew cr-Hermitean, i.e. A' = -a{A).

(22)

- b) Conversely, each matrix with the above properties defines an automorphism g of F " withg\Wi =id|y^. According t o Exercise 7, neutral a-Hermitean vector spaces of equal dimension over the same skew field are isomorphic. This leaves the task to determine the automorphism groups 0 „ of these vector spaces with scalar product. We only want to derive a simple result in t h a t direction for later use. P r o p o s i t i o n 13. Let V be a two-dimensional neutral vector space with symmetric bilinear scalar product over a field K. Then the special orthogonal group ^02 :={(7G02|det(7=l} (23) is isomorphic to the multiplicative group K* of K, and the orthogonal group O2 is the semi-direct product of a subgroup isomorphic to Z2 with the normal subgroup SO^-

2.2 Vector Spaces with Scalar Product

V

161

P r o o f . Because of char i f 7^ 2 we may choose an isotropic basis 31,32 ^OT such t h a t (31,32) = 1/2- For g e O2 we start from the ansatz 9ii = 3 i « + 32/3> 9h =ii1

+ h^

for the basis representation, and make use of the invariance of the scalar product to derive the conditions a/3 = 7(5 = 0, a(5 + /37 = 1.

(24)

Hence, if /3 = 0, then S y^ 0 and 7 = 0; this implies the isomorphy ^-

(oa^i) '^^*^2^"^^*-

(25)

For /3 7^ 0 we similarly obtain a = 0, 7 = /3^^, i.e.

- (;.?,>(°J)("'o'°)^ T h u s O2 = S02Us(S02), where s is the transformation interchanging 3^ with 32; it is easy to check t h a t O2 is the semi-direct product of SO2 = K e r d e t with { i d y , s } = Z2. • Exercise 12. Prove under the assumptions of Proposition 13: O2 is Abelian if and only if K is isomorphic to the field Z3 with 3 elements. 2.2.8 Tensors and V o l u m e Functions In this section we want to extend the biforms to tensor spaces; moreover, we wiU discuss volume functions on vector spaces with scalar product. To this end, suppose t h a t the scalar domain is a field. Exercise 1 3 . Let A" be a field, and let b be an alternating or cr-Hermitean biform over the vector space V". Prove: a) The biform b is non-degenerate on V", if for each basis (Oi) of V" det(6(0i,0i))/O. (27) - b) The bilinear or cr-linear and linear extensions p

6®''(Pi(x)...(x)Pp,t)i(x)...(x)t)p):=]^6(p„,t)„),

peN,

(28)

0
(29)

1

or 6('''(yiA...APp,t)iA...At)p):=det(6(p„,t)^)),

respectively, define bibnear or cr-Hermitean biforms on the tensor spaces ^''V", ffV" (cf. also Example II.8.3.4); the biforms 6®", b^"^ are non-degenerate if and only if b is non-degenerate. - c) If 6 is cr-Hermitean, then this also holds for b^'' and b'^'''. - d) For an alternating biform b, 6®'' and w''' are alternating, if p is odd, and they are symmetric, if p is even.

162

2 Cayley-Klein Geometries

As in Exercise 13, let V^ be a vector space with scalar product over the field K, and denote by t / G „ its automorphism group. In general, according to (1-13) we only know that a{N{a))N{a) = 1 for a G UGn- Hence, in order to achieve invariance for a volume we have to confine the action to the special unitary group SUGn := {a G UGn\N{a) = 1}. (30) Let [.,...,.] be a volume function, and let (Oj) be a unit basis, i.e., suppose that [ o i , . . . , o„] = 1 and [pi, . . . , ? „ ] = det(C^), if pj, = ttj^l is the basis representation of pj,, k = 1 , . . . , n. We prove Lemma 14. Let [V^, {,)] be a vector space with scalar product over a field K. ForkeN and b« G F " , K = 1 , . . . , A;, det((b«,bA)) = 0

(31)

if and only if the sequence (&«), K = I,... ,k, is linearly dependent, or if the subspace spanned by it is isotropic. P r o o f . Linear dependence of the b\ implies linear dependence of the columns in the matrix ((bfj, bx)) leading to (31). Let thus the (bfj) be linearly independent; then the assertion is a consequence of the assertion in Exercise 13 (withn = A;, F ' ^ = £ ( b i , . . . , bfc), b= (,)). D Lemma 15. With the same assumptions as in Lemma 14 let (OK), K = 1,... ,k, be a basis of W C V"; let [.,...,.] be the volume function on W satisfying [ o i , . . . , Ofc] = 1 (cf. Proposition 1.4.7.5). Suppose, moreover, lx = 0..ileW\n,X=l,...,k.

(32)

Then det((y«, y^)) = a • [p^,..., y j • a([pi,..., p j )

(33)

a :=det((0K,0A)) =cr(a).

(34)

with D

Remarks. 1. If W'' is isotropic, then a = 0, and (33) reduces to det((p^,p;^)) = 0 . - 2 . The matrix {{an, ax)) formed by the scalar products of the basis vectors is called the Gram matrix. For a = id^ and a = 1, a condition always satisfied by orthonormal and symplectic bases, this is nothing but Formula 1.(6.3.8) for the Gram determinant. - 3. Consider if = R and a pseudo-orthonormal basis (Oa). Then a = ( — 1)', where / is the index of W'', and hence [Ti,---,Tk? = |det((p^,p^))|, cf. 11(8.3.47), where the absolute value on the right-hand side is missing.

2.2 Vector Spaces with Scalar Product

163

C o r o l l a r y 16. If k = n in Lemma (15) , i.e., if [.,...,.] is a volume function on V", then the constant a does not depend on the choice of the unit basis (Oi), [Oi, . . ., 0„] = 1. P r o o f . For a second unit basis (bj) of V" we conclude from (33) and (34) that b := det((bj, bj)) = a.

a 2.2.9 T h e G e n e r a l V e c t o r P r o d u c t It turns out t h a t the vector product and its properties as described in Proposition 1.6.3.2 (cf. also Exercise II.8.3.14) can be generahzed to a large extent: P r o p o s i t i o n 17. Let V^ be a vector space over a field K with the a-biform (,) as scalar product (a = i d ^ for (,) bilinear). Let, moreover, [.,...,.] be a volume function on V". Then for every (n — l)-tuple (CQ,), a = I,... ,n — I, of vectors CQ, G V^ there is a uniquely determined vector c G V^ satisfying (CT) = [ci, ••• ,Cn-i,T] for all T e V".

(35)

P r o o f . Since the scalar product can be viewed as a a-linear bijection via t) G V 1-^ (t),.) G V , and as the volume function is linear in each argument, we immediately conclude the existence of a vector c which is uniquely determined by the (n— l)-tuple (Ca) (cf. Definition 1.7.2). This vector c is caUed the vector product of the vectors CQ,; it is denoted by Ci x . . . x c „ _ i . • P r o p o s i t i o n 18. Under the assumptions has the following properties:

of Proposition

17 the vector

product

L The map n-l

(Ca)G

n^"^ClX---XC"-l«="^"

is a-linear in each argument CQ,. 2. The map is alternating. 3. Ci X . . . X c„_i = 0 if and only if the sequence (CQ,), a = I,... linearly dependent. 4. The vector product is orthogonal to each of its factors:

,n — I, is

(ci X . . . X c„_i, Ca) = 0, a = 1 , . . . , n - 1.

(36)

5. The scalar product of the vector product with itself is computed det((Ca, c^)) = a(ci x . . . x c„_i, Ci x . . . x c „ _ i ) , where a is the constant (16).

associated

with the volume function

by (37)

according to

164

2 Cayley-Klein Geometries

6. The vector product Ci x . . . x c„_i is isotropic if and only if the subspace spanned by the factors, £ ( c i , . . . , c „ _ i ) , is isotropic. 7. For any unitary transformation g G UGn the vector product transforms according to gr(Cl) X . . . X ^ ( C n - l ) = gr(Cl X . . . X C „ _ l ) ] V ( g r " ^ ) .

(38)

P r o o f . Properties 1 to 4 immediately follow from the definition. To prove (37) we apply (33) to ( c i , . . . , c„_i, c) with A; = n, c = Ci x . . . x c„_i. From (36) we obtain det((ca,c^))(c,c) =a{c,cf; here we made use of (35) with p = c and (c, c) = cr((c,c)). If c = o, then (37) is trivial by 3 and Lemma 14. According to 4, the subspace l ^ n - i ^ £ ( c i , . . . , c„_i) is isotropic for c 7^ o if and only if c is isotropic. In fact, c is orthogonal to W"^^. This implies 6, and by Lemma 14 property (37) follows from the equation just proved. To prove statement 7 we set c := ^(ci) X . . . X ^(Cn-i). Then 1.(5.7.32) implies (c,?) =

[gici),...,g{c„-i),g{g-\^))],

=

[ci,...,c„-i,g-\^)]N{g),

=

{c,g-\^))N{g),

=

{gic)Nig-'),^);

here we made use of (1.18): N{g-^)

= N{g)-^

= a{N{g))

for g G t / G „ .

(39) D

Remark. Since there exists an order in R , and, moreover, each positive number has a positive root, Assertion 5 in Proposition 1.6.3.2 is a specialization of Assertion 5 in Proposition 18 t h a t is impossible in general; for the same reason, the former Property 6 cannot be formulated in the general case, whereas the new Property 6 makes no sense for Euclidean vector spaces. Exercise 14. In the notations and assumptions of Proposition 17, let the d, i = 1,.. . ,n, he orthogonal. Prove: a) If (Ci) is a unit basis, [Ci,... , Cn] = 1, then = ]^(Ci,Ci). b) Show by means of an example that the converse of a) does not hold in general. c) If (ba), a = 1 , . . . ,n — 1, is an orthogonal sequence of vectors with (ba, ba) / 0, then b„ := b/(b, b) with b := bi X . . . X b „ - i (40) completes the sequence (ba) to a unit basis of V". - d) Generahzing (37) we have for arbitrary vectors PQ, , t)^ e V"

2.2 Vector Spaces with Scalar Product (?i x - - - x ? n - i , t ) i X--- x t ) „ _ i ) = a d e t ( ( p „ , t ) ^ ) ) .

165 (41)

(Hint. Recall the linearity or cr-linearity of the functions occurring in (41) in all variables and use basis representations.) 2 . 2 . 1 0 A d j o i n t Linear M a p s Let [V, (,)], [W, (,)] be two finite-dimensional vector spaces with scalar product (not necessarily of equal type) over the same skew field K. Then with each linear m a p a : V ^ W an adjoint linear map^ a* : W ^ V is associated which is uniquely defined by the equation (a*t), p) = (t), ap) for all p G F , t) G W.

(42)

This notion generalizes the one for Euclidean or unitary vector spaces, denoted by a' in § L6.4. In these geometries the self-adjoint endomorphisms are of particular interest; they are defined by the condition a* = a for a G E n d ( V ) and t u r n out to be diagonahzable, cf. Proposition L6.4.2. Moreover, they are closely related to the symmetric and Hermitean bilinear forms, respectively. To a large extent, the classification of quadrics in Euclidean geometry can be derived from the theory of self-adjoint linear maps. Quite similarly, we will be able to reduce classification problems for some special geometries with scalar product to the classification of special linear endomorphisms in the following. For simplicity, we restrict the considerations to the case [V, (,)] = [W, (,)], i.e. a G E n d ( V ) . First we note P r o p o s i t i o n 19. Let V be a vector space with scalar product, and let a be the corresponding anti-automorphism of the scalar domain K. Then, for each linear map a G E n d ( V ) there is a uniquely determined linear map a* G E n d ( V ) satisfying (42). The map cp : a i-^ a* is an involutive a-anti-automorphism of the endomorphism algebra E n d ( V ) ; moreover, ( a + 6)* = a * + 6 * ,

(43)

(a-/)* = a*cr(7) for all 7 G Z{K),

(44)

(aob)*

(45)

=b* oa*,

(a*)* = a.

(46)

P r o o f . The proof of this proposition is a simple verification based on the related definitions. For fixed t), the left-hand side of (42) is a linear function in p. Since the scalar product is non-degenerate, it defines a bijection between V and its dual space V; hence, there is precisely one vector a*t) satisfying (42). T h u s the m a p a* is uniquely defined. The linearity of a* is easy to check. In fact, for a\\ r] e K we have Do not confuse this notion of the adjoint linear map with that of the contragredient isomorphism, which is frequently also denoted by a*, cf. Example 1.7.4.

166

2 Cayley-Klein Geometries («*(t)??),?) = (t)??,a?) =cr(??)(t),ap) =cr(r7)(a*t),p) = ((a*t))r7, p).

Recall t h a t the endomorphism algebra is an algebra only over the center of the scalar domain, which, obviously, is mapped into itself by a. Let us conclude the proof by showing (44). For 7 G Z{K) we have ((«7)*t),?) = (t), («7)?) = (t),a?)7 = 7(«*t),?) = («*t)cr(7),p) = ((a*cr(7))t),p). The proof of the remaining statements is left to the reader. D T h e scalar product determines a a-linear bijection from the vector space V onto its dual V, assigning to each t) G V a hnear form u)^{f} := (t),p). Comparing the definition of the adjoint m a p a* to Definition 1.6.1, (16), of the dual m a p shows t h a t , essentially, both describe the same (apart from the application of o-^^). a* expresses, however, the dual m a p in the vector space itself, which is possible due to the available scalar product. Having any at our disposal, we can completely do without the introduction of dual vectors, as it is frequently done in elementary presentations of Euclidean geometry. Exercise 1 5 . Prove that, in analogy to the situation in unitary vector spaces, the following holds: A linear map a e End(\^) is an automorphism of the vector space with scalar product if and only if a is invertible, and a^ = a*. Show, moreover, that Formulas (43)-(46) also hold for arbitrary linear maps a, b between possibly different vector spaces with scalar products over the same skew field K. Now we want to establish a relation between biforms and linear maps, t h a t is defined by the scalar product. In fact, projectively interpreted, the scalar product is a distinguished correlation F ; more generally, each a-biform is a correlative m a p H, cf. Proposition 1.7.6. Thus, F^^ o i7 is a coUinear m a p t h a t , except for trivial cases, is induced by a semi-linear m a p a. Expressed differently, for aU p G V we have F{[a{f}\) = H{\f\); projectively, this amounts t o saying t h a t the H-polar of x = [p] coincides with the F-polar of [a(p)]. This corresponds to the relation we now want to describe. Here we will consider the general case of a a-Hermitean scalar product; for bilinear scalar products assume t h a t i f is a field and a = ICIKL e m m a 20. Let a G E n d ( V ) he a linear map.

Then

6„(p,t)) := ( a p , t ) ) , p , t ) G F ,

(47)

is a a-biform. For each a-biform b over V xV there is a uniquely determined endomorphism a G E n d ( V ) such that (47) holds with b = ba- With respect to the standard actions of the linear group over the endomorphism algebra, lg{a) = g • a- g^^ for a G E n d ( V ) , gr G GL(V), and on the a-biforms b 1-^ gb, defined by gb{^,t)) := b{g-^^,g-^t)), p, t) G F , assigning a G E n d ( V ) 1-^ ba as well as taking the inverse are G-maps for the automorphism group G of the vector space with scalar product [V, (,)].

2.2 Vector Spaces with Scalar Product

167

P r o o f . The first assertion is easily verified. So let, conversely, 6 be a abiforni; a denotes the involutive anti-automorphism of K fixed by the scalar product. For fixed p, 6(j;, t)) is a linear function of t). Hence, there is precisely one vector a(j;) for which (47) holds with ba replaced by b and all t) eV. The linearity of a is proved as above for a*. The final assertion is immediate. T h e linear m a p a and the a-biform b = ba bijectively assigned to it via (47) are called mutually associated. The next exercise describes the relation between the matrices for ba and a. D Exercise 16. Let (cu) be a basis of the vector space V with scalar product, and let (e,,):=((0,,0,)),(e^^):=(e,,)-\

(48)

be the matrix of the scalar product and its inverse, a) Denoting by (a*) the matrix of a e Endiy) and by {Pij) the matrix of the biform defined by (47), show that (a,^) = (e'=^)(a(A,))'.

(49)

Here ijij)' denotes the transposed matrix of ijij). - b) Denote by {a*\) the matrix of the adjoint map for a e EndiV) in the above basis. Prove: (a%^) = (e'''=)(a(aD)'(e,0-

(50)

Following the terminology used by A. I. Maltzev [75] for real and complex vector spaces with scalar product we define: D e f i n i t i o n 4. Let V be a vector space with scalar product. A linear m a p a G E n d ( V ) is called symmetric if a = a*; it is called skew-symmetric if it satisfies a = —a*. Symmetric linear endomorphisms are often also called self-adjoint operators, cf. Definition L6.4.2. D Exercise 17. Prove: a) If the scalar product is bilinear and symmetric, then the map a is symmetric or skew-symmetric if and only if the biform ba associated with a has the same property. - b) If the scalar product is cr-Hermitean, then the map a is symmetric or skew-symmetric if and only if the biform ba associated with a is (7-Hermitean or skew a-Hermitean,respective\y, (cf. Definition 1.9.1); the last property means that 6a(t),j;) = -o-(6a(j;,t))) for aU j;,t) e F . - c) If the scalar product is alternating, then the biform ba associated with a is symmetric or skew-symmetric, respectively, if and only if a is skew-symmetric or symmetric. - d) The map a H^ 6a is cr-linear over the center Z{K) of the scalar domain in the following sense: ba+c = ba + be, bai_, = (T{ji)ba for all ji e

Z{K).

T h e following result is frequently used: L e m m a 2 1 . Let a he a symmetric or skew-symmetric endomorphism of the vector space V with scalar product, and let W C V be an invariant subspace for a, i.e. a{W) C W. Then the orthogonal subspace W is also invariant fora:a{W^) C W^.

168

2 Cayley-Klein Geometries P r o o f . For all ^ e W

and any vo e W we have (a(w),y) = ± ( w , a ( p ) ) = 0 ,

since a(tti) G W. Hence a(p) G W^. The following is easy to prove:

D

Lemma 22. Let V = 0-^ Vt^^ be a decomposition of the vector space with scalar product into the direct sum of mutually orthogonal subspaces. Then the restriction of the scalar product from V to each of the subspaces W^ defines the structure of a vector space with scalar product of the same type as V, i.e. bilinearly symmetric, bilinearly alternating, or a-Hermitean. If moreover, a G End(V) is a linear map leaving each of the subspaces invariant, a(Ufi) C Ufi, then a is an automorphism, a symmetric or a skew-symmetric map if and only if each restriction a\Ufi has this property. • >From the theory of unitary vector spaces we know that the symmetric maps considered there have a particularly simple structure: There is an orthonormal basis consisting of eigenvectors of the map, and all the eigenvalues are real. So, the matrix of the map in this basis is diagonal with real numbers on the principal diagonal. The classification of quadrics in Euclidean geometry as well as the classification of real quadratic or Hermitean forms under the orthogonal or unitary group, respectively, all are based on this very fact, cf. Chapter 1.6. For symmetric maps in general vector spaces with scalar product the situation is considerably more complicated. Since classification results for symmetric and skew-symmetric maps have interesting geometric applications, we now want to derive some structural properties of these maps. Reading the already cited books by A. I. Maltzev [75] is very stimulating, further results in this direction can be found there. The theory probably goes back to K. Weierstrass [109] and L. Kronecker [70], cf. F. R. Gantmacher [41], [43], [42]. The fact that all the eigenvalues of a symmetric endomorphism in a unitary space are real (cf. Proposition 1.6,4.2) arises as a special case of the following Lemma 23. Let V be a vector space with a a-Hermitean scalar product over a field K or over the skew field H of quaternions, where in the latter case a = T is the conjugation. If a is an eigenvalue of the symmetric endomorphism a of V, and if there exists a non-isotropic eigenvector p for the eigenvalue a, then a = a(a). If a is skew-symmetric, then, under the same assumptions, a = —a{a). P r o o f . We have (ap,p) = (p, ap), since a is symmetric. The eigenvector property implies If if is a field and p is not isotropic, then this yields the assertion. For a T-Hermitean scalar product over the quaternions note that {f,f) G R, i.e..

2.2 Vector Spaces with Scalar Product

169

this number belongs to the center of H ; hence, a G R . For a skew-symmetric endomorphism, the only thing to be changed in the equation for the scalar product is the sign. D T h e following simple example shows t h a t , for indefinite pseudo-unitary scalar products there are symmetric endomorphisms having only isotropic eigenvectors; for the eigenvalues we then have a ^ a{a). E x a m p l e 6. Consider the two-dimensional pseudo-unitary vector space V of index 1 over the field C of complex numbers. If (^'), (??') are the coordinates of the vectors p, t) in the orthonormal basis (ei, 62), then their scalar product is Hence, Oi := ei + e2, 02 := ei — e2 are isotropic vectors also forming a basis. A straightforward calculation shows t h a t the endomorphism defined by a(oi) = Oi i, 0(02) = - 0 2 i is symmetric; here i denotes the imaginary unit. Obviously, this endomorphism has the required properties. Note, in addition, t h a t the eigensubspaces are not orthogonal; in fact, (oi, O2) = 2. D Exercise 18. Under the assumptions of the preceding example show the following: If the symmetric linear map b e End(\^) has the eigenvectors Oi, O2, then the corresponding eigenvalues are conjugate. T h e invariants of an endomorphism a for the action of the linear group GL{'V) are determined by its J o r d a n normal form if K contains all eigenvalues of a, i.e., if it is a splitting field for the characteristic polynomial of a, cf. Proposition 1.5.8.4. In this connection we define the notion of a root subspace Ua of an endomorphism a G E n d ( V ) for the scalar a G if: Ua '•= {? e V\ there is an s G No with (a - eay{f)

= 0}.

Here and in the sequel we set e := i d ^ . In § 1.5.8 these spaces are called nilsubspaces. It is easy to see t h a t each root subspace is invariant for a. Obviously, the root subspace satisfies Ua 7^ 0 if and only if a is an eigenvalue of a. We show P r o p o s i t i o n 24. Let V be a vector space with scalar product over a field K, and let a G E n d ( V ) be a symmetric endomorphism. If a [a) ^ [3, then the root subspaces Ua, U/3 are mutually orthogonal. ^

^ This proposition corrects Proposition 5 in Section 101, p. 250, in A. I. Maltzev [75]. There, due to a mistake in the proof, a / /3 is assumed instead of a / cr(/3). Example 6 shows that the proposition as formulated there is wrong.

170

2 Cayley-Klein Geometries

P r o o f . If a or /3 are no eigenvalues of a, then the assertion is trivial. Take now p G Ua and t) G t / ^ . So there are n a t u r a l numbers s, t G No for which (a - eayi^)

= o, (a - e/3)*(t)) = o.

We proceed by induction on the number s + 1 . For s + 1 = 1 we have s = 0 or t = 0, and the assertion is triviaUy satisfied. Assume now t h a t the claim is true for all s , t with s + t < k, and take s +t = k + 1. Set Pi := (a - ea)(p), t)i := (a - e/3)(t)), Then (a - e a ) " - i ( p i ) = 0, (a - e/3)*-i(t)i) = o, i.e., the induction hypothesis is satisfied for the vectors p^, t) and p, t)i. Hence, (?1, t)) = ((« - ea)(p), t)) = {a^, t)) - cr(a)(p, t)) = 0,

(51)

(p, t)i) = (p, (a - e/3)(t))) = (p, at)) - /3(p, t)) = 0.

(52)

Because of the symmetry of a, taking the difference yields (cr(a) —/3)(p, t)) = 0. Since by assumption a{a) ^ f3, we arrive at the assertion (p, t)) = 0. D C o r o l l a r y 25. If under the assumptions of Proposition 24 a «s an of a with a ^ o-(a), then the associated root subspace Ua is totally

eigenvalue isotropic.

P r o o f . W i t h f3 = a the assumptions of Proposition 24 are satisfied. Hence Ua is orthogonal to itself. • Almost identical proofs lead to corresponding results for skew-symmetric operators: P r o p o s i t i o n 26. Let V be a vector space with scalar product over a field K, and let a G E n d ( V ) be a skew-symmetric endomorphism. If a (a) ^ —fi, then the root suhspaces Ua,Ujs are mutually orthogonal. P r o o f . Because of the skew-symmetry of the endomorphism, we just have to add equations (51) and (52) instead of taking the difference; everything else remains unchanged. • C o r o l l a r y 27. Under the assumptions of Proposition 26 let a be an eigenvalue of a with the property a ^ —a{a). Then the associated root subspace Ua is totally isotropic. • Let us again consider symmetric endomorphisms, and define

Wo := 0 a^(j(a)

t/a, Wi := 0

t/„.

(53)

a=(j(a)

T h e fact t h a t the sums occurring in (53) are direct immediately follows from Proposition 1.5.8.4 concerning the J o r d a n normal form applied to the split

2.2 Vector Spaces with Scalar Product

171

endomorphism a\IJUci, where t h e sum runs over aU eigenvalues of a belonging to K. Recall t h a t an endomorphism a is said to be split if t h e scalar domain i f is a splitting field for the characteristic polynomial of a. Proposition 24 then immediately implies C o r o l l a r y 28. Under the assumptions of Proposition 24 the subspace WQ is orthogonal to Wi. If K is a splitting field for the characteristic polynomial of a, then V = Wo®Wi is an orthogonal decomposition into subspaces invariant for a; the of a to these subspaces are symmetric endomorphisms.

restrictions

P r o o f . The first assertion is immediate, since each summand occurring in WQ is orthogonal to every s u m m a n d in Wi. If i f is a splitting field, then the decomposition of V is just a regrouping of t h e direct decomposition into root subspaces determined by t h e J o r d a n normal form, hence itself a direct sum of mutually orthogonal subspaces, on which the scalar product consequently is non-degenerate, cf. Lemma 22. From here t h e proof is easily completed. • Let, moreover, a be a symmetric, split endomorphism. By Proposition 24, t h e root subspaces occurring in the direct sum representation of Wi are mutually orthogonal. As Example 6 shows, this does not hold for t h e direct summands in WQ. For t h e decomposition of WQ we have P r o p o s i t i o n 29. Let a he a symmetric, split endomorphism of the vector space V with scalar product over the field K. If X y^ '^W is an eigenvalue of a, then o-(A) is an eigenvalue, too. U\ © U^(^y^ is a neutral subspace; hence d i m t / A =dimt/<^(A). Choose one entry from each pair (A, (T(A)) and denote representatives by Ag. Then

the resulting

VFo= 0(t/A©t/<,(A))

set of

(54)

AeAo

is a direct orthogonal

decomposition

into neutral

subspaces.

P r o o f . According to Proposition 24 the summands occurring in t h e decomposition (54) are mutually orthogonal. Suppose t h a t o-(A) is not an eigenvalue of a, then U\ is a totally isotropic subspace orthogonal to V contradicting t h e assumption t h a t the scalar product is non-degenerate. By Lemma 22 no direct summand of (54) is isotropic, hence each one is a neutral subspace. Exercise 6 implies the assertion concerning t h e dimensions. • Similar to (53) we define for skew-symmetric endomorphisms

Mo:=

0

t/a, Mi:=

Proposition 26 immediately leads to

0

t/„.

(55)

172

2 Cayley-Klein Geometries

Corollary 30. Under the assumptions of Proposition 26 MQ is orthogonal to Ml. If K is a splitting field for the characteristic polynomial of a, then V =

Mo®Mi

is an orthogonal decomposition into subspaces with scalar product invariant for a; the restriction of a to any of these subspaces is a skew-symmetric endomorphism. • The analogue of Proposition 29 is proved almost literally repeating the preceding argument: Proposition 31. Let a be a skew-symmetric, split endomorphism of the vector space V with scalar product over the field K. If X ^ —o"(A) is an eigenvalue of a, then —o"(A) is an eigenvalue, too. U\ © U_^(^x) is a neutral subspace; hence dim U\ = dim U_a-(^x) • Choose one entry from each pair (A, —(T(A)) and denote the resulting set of representatives by Ag. Then M o = 0(t/A©t/-<,(A))

(56)

AeAo

is a direct orthogonal decomposition into neutral subspaces.

•

2.2.11 Properties of the Root Subspaces In this section we will derive some special properties of the root subspaces for symmetric or skew-symmetric endomorphisms. We will frequently refer to the following assumption: Assumption A. Let V be a vector space with scalar product over a field if, char if ^ 2, and let a G End(V) be a split endomorphism. Recall that for a split endomorphism the root subspaces U\ can be represented as the direct sum of Jordan cells A'{:

Ux = Al' © Al' ©

©Zi^--,

ki>k2>

•^

fVf

(57)

Here A\ is a subspace invariant for a with a basis (Oj), « = 1 , . . . , A;, in which a\A\ is represented by a matrix of the form / A 0 . . 0 o\ 1 A . . 0 0 (58) 0 0 . . A 0 Vo 0 . . 1

w

Finding the Jordan normal form for an endomorphism amounts to representing the vector space V as the direct sum of Jordan cells. The matrix of a in a basis adapted to these Jordan cells is quasi-diagonal; along the diagonal it

2.2 Vector Spaces with Scalar Product

173

contains Jordan boxes , i.e. matrices of the form (58). The type of a Jordan normal form, i.e. the sequences of the eigenvalues and the dimensions of the Jordan cells corresponding to the eigenvalues, are uniquely determined up to the order of the summands by the endomorphism a. Jordan cells may occur repeatedly; the number of cells of the form Z\^ is called the multiplicity of this cell in the Jordan decomposition for a. In general, the Jordan ceUs themselves are not uniquely determined; only the type of the Jordan normal form, i.e. the eigenvalues, the dimensions of the corresponding Jordan cells, and their multiplicity, are invariants of the endomorphism. Two split linear endomorphisms are similar if and only if their Jordan normal forms coincide up to the order of the summands (cf. Section 1.5.8 in [82]). Let G denote the group of automorphisms of the vector space with scalar product \V, (,)]; then G also acts on End(V) by (g, a) G G X E n d ( F ) i—> goaog-^

e End(F),

i.e. by restricting the action of the linear automorphisms to G C GLiV). Two linear maps transformed into one another under this action are called G-congruent. Hence the equality of the Jordan normal forms is a necessary, but in general not sufficient condition for the G-congruence of two linear maps. The specific properties of the Jordan normal forms for symmetric or skewsymmetric endomorphisms will be studied in greater detail now. Proposition 32. Under Assumption A, let A y^ o"(A) be an eigenvalue of the symmetric endomorphism a. For each Jordan cell Z\^ in the Jordan normal form of a there is a Jordan cell ^'^(x) of the same dimension k so that Z\^ © Z\^Q-) is a neutral sub space ofV. The multiplicities of A\ and AK^S in the Jordan decomposition of a coincide P r o o f . By Proposition 29 it suffices to prove the statement separately for each of the summands in (54); thus we may assume

The proof will proceed by induction on the dimension m = dwaUx- For m = 1 the assertion immediately follows from Proposition 29. Suppose that the proposition is already proved for all dimensions less than m, and let k be the minimum of all the dimensions of Jordan cells occurring in the Jordan decomposition of a. Let the cell Z\^ belong to Ux; if necessary, interchange A and o-(A). Then there is a decomposition (57) with k = kr- Set Ux ••= Al'eAl'®...®

A\--'.

In Exercise 6 we noted that in a decomposition of a neutral vector space into the direct sum of totaUy isotropic subspaces each of these subspaces determines a canonical a-linear bijection onto the dual of the other subspace. Consider the decomposition Ux = Ux ® A'^ and define

174

2 Cayley-Klein Geometries S := {? G t/<.(A)|(?, Ux) =0} = utn

t/,(A),

M := {y G t/.(A)|(?,Zi^) = 0} = (Z\^)^ n t/,(A). >From the result of Exercise 6 mentioned above we immediately obtain d i m B = k, d i m M = m - k, U^(^x) = M

®B.

The invariance of the subspaces tJx, ^ ^ under the action of a implies the invariance of the subspaces B, M under the map a. A Jordan decomposition for the restrictions of a to these subspaces leads to a Jordan decomposition for the restriction of a to t^(j(A)- Since k is the minimum of the dimensions of all Jordan ceUs for a, B itself has to be a Jordan cell of the form ^ ^ ( A ) ' ^^ identify B = ^ ^ ( A ) - -^* remains to show that H := A^ (B ^^fA) ^^ ^ neutral subspace of V. Let p be a vector from the defect subspace D = H C\ H . Then Since p G H, we have the representation ? = ?o + ?i with po e ^A C t/A, ?i G A'l^x) C t/<7(A), Now p also belongs to

{A\x))^

=

B^=Ux®U,^x),

since for the neutral vector space V the following relations hold t / f = t / A , t/;^(A) = t/,(^). Hence we obtain for the first component of p: Po G Z\^ n

?7A

= {o}.

Thus, because of {A\)^

= t/A e ((Zi^)^ n t/,(^]) = t/A e M ,

we have p = P i G Z \ ^ ( ^ ) n M = B n M = {o}. Therefore p = o, and H" is an a-invariant neutral subspace. Hence H is also a-invariant. Since the dimension of the root subspace of H corresponding to A is less than m, the assertion follows from the induction hypothesis. Note that, once the Jordan normal form of a is known, the proof provides an efficient procedure to determine the decomposition of U\ © t/(j(A) into the direct sum of mutually orthogonal neutral subspaces of the form A\ © ^CT(A)- Here, k runs through the dimensions of the Jordan cells occurring in U\. D The corresponding proposition for skew-symmetric endomorphisms, proved along the same lines, is

2.2 Vector Spaces with Scalar Product

175

P r o p o s i t i o n 3 3 . Under Assumption A let \ ^ ~'^{^) &e an eigenvalue of the skew-symmetric endomorphism a. For each Jordan cell A'l in a Jordan normal form of a there is a Jordan cell A'^^,^-. of equal dimension k so that Z\^ © Z\^^QN is a neutral suhspace ofV. The multiplicities in the Jordan decomposition for a coincide.

of A\

and A^_^,-y, •

T h e following two exercises again imply t h a t the unitary spaces with their positive definite scalar product are particularly fortunate exceptions; since there are no isotropic vectors, their symmetric and skew-symmetric endomorphisms have very simple structures. Exercise 19. Prove supposing that Assumption A holds: If A \ is a Jordan cell of a symmetric or skew-symmetric endomorphism with fc > 1, then the eigenvector of this cell, i.e. the vector Ofc in a corresponding Jordan basis, is isotropic. Exercise 20. Under Assumption A let a be a symmetric (or skew-symmetric) endomorphism, let A / (7(A) (or A / —(7(A)) and JJL be eigenvalues of a. Prove: If A \ , Zijj are Jordan cells for a, and if At / (7(A) (or y, / —(7(A)) or fc / /, then the subspace A \ e Zijj is isotropic. Exercise 2 1 . Under Assumption A let a be a symmetric (or skew-symmetric) endomorphism, and let A / (7(A) (or A / —(7(A)) be an eigenvalue of a. Consider one of the neutral subspaces occurring in Proposition 32 (or 33), B = A'l © '^^(A) (or B = A'1(B A'^^(X))- Prove: The restriction a\A'l uniquely determines the restriction of a to the other direct summand of B. (Hint. Apply Exercise 6c).) T h e foUowing proposition shows t h a t , for sufficiently large index of the scalar product, arbitrary endomorphisms, and hence also arbitrary J o r d a n normal forms, may occur as components of symmetric or skew-symmetric endomorphisms. Therefore, enumerative classifications will rapidly become complicated with increasing index and dimension. P r o p o s i t i o n 34. Let V = BQ © Bi be a neutral vector space represented as the direct sum of two totally isotropic subspaces. Then each linear endomorphism a of Bo can be uniquely extended to a symmetric endomorphism b of V leaving Bi invariant. Similarly, there exists a uniquely determined skew-symmetric extension with this property. P r o o f . Let (Oi),« = 1 , . . . , m, be an arbitrary basis of BQ, and let (bj) be the basis of Bi satisfying (Oi,bj) = 5ij, «, j = 1,. . . , m , t h a t is uniquely determined according to Exercise 6c). If (a*) is the matrix of a, then this relation immediately implies t h a t the restriction of the symmetric extension 6 we are looking for has the transposed (7-conjugate matrix to (a*) in the basis (bj) of Bi. Using the sum convention and setting

176

2 Cayley-Klein Geometries

the extension property implies /3| = (Ofc, b{bj)) = (a(Ofc), bj) = a{ai). For the matrix of the skew-symmetric extension we similarly obtain

(7J) = -(a(a}))'.

D

2.3 The Projective Geometry of a Polarity In this section we consider a-Hermitean and, especially, symmetric bilinear scalar products from the projective point of view. So we only discuss polarities; in order to investigate degenerate cases one could proceed as in Exercise 1.9.7, i.e. consider polarities on a cone section. Here we will always work in a right vector space V"+^ over a skew field K, char if ^ 2, with a a-Hermitean scalar product (,). P " denotes the projective space associated with V, and F is the polarity defined by (,) determining the, possibly empty, quadric Q = QpFinally, P G „ = PGn{F) is the projective group of the polarity F. Obviously, these assumptions will later have to be specialized; e.g., in the second part of this section we will consider non-degenerate, symmetric bilinear forms. 2.3.1 T h e Quadric of a Polarity For the projective interpretation of the notions related to the scalar product we recall that the subspace W orthogonal to any subspace W with respect to (,) is the one assigned to it by means of the polarity (cf. (1.7.22)): F: VF G *P I—> F{W) := W^ G *p. The totally isotropic subspaces are those contained in the quadric, E C Q (Corollary 1.9.15); by Corollary 2.6 their projective dimension satisfies the inequality Dim£; < ( n - l ) / 2 . (1) In Section 1.9 we pointed out that an isotropic subspace intersect the quadric Q in its defect subspace Wo = WnW^ (Corollary 1.9.19). E. Witt's Theorem immediately implies Corollary 1. Two points a,b ^ P " are PGn-equivalent resented by vectors o, b G V satisfying a=[a],b

= [b],{a,a) = {b,b).

Dually, two hyperplanes A, B C P " are PGn-equivalent sented by equations

if they can be rep(2) if they can be repre-

2.3 The Projective Geometry of a Polarity A:

(o,j:)=0,

177

B : (b,j:)=0

with PGn-equivalent poles A = [a], B = [b] whose representing vectors satisfy (2). In particular, P G „ acts transitively on the points as well as on the tangent hyperplanes ofQ. • In the next Section the orbits of points, or, more generaUy, of point sequences, wiU be examined more closely. For now we only note the following E x a m p l e 1. Let i f = R , and let F = Fn+i^i be a polarity of index /, where 0 < / < ( n + l ) / 2 . ForO < / < ( n + l ) / 2 there are three orbits of P O ( / , n + 1 - / ) inP": Q = {a; = M G P " | ( y , y ) = 0 } , /(g):={a; = [y]GP"|(y,y)<0}, A(g):={a; = [y]GP"|(y,y)>0}. The set I{Q) is called the inner region, and A(Q) the outer region of the quadric. Because of (pA, pA) = (p, p)A^, with a suitable normalization we can always represent the point a; by a vector p satisfying ( p , p ) = 0 , (p,p) = - l

or (p,p) = l.

(3)

In the case / = 0 we have Q = I{Q) = 0, and A{Q) = P " ; the group PO{n + 1) acts transitively on P " . The geometry determined by this action on the projective space is called elliptic geometry; we will discuss it in greater detail below. If / = 1, then the point x belongs to I{Q) if and only if its polar F{x) does not intersect the quadric Q. For all / such t h a t 0 < / < (n + l ) / 2 a point X belongs to A{Q) if and only if its polar F{x) intersects the quadric Q in a quadric F{x) n Q of the same index /, whereas the polar F{x) of a point X G I{Q) meets the quadric Q in a quadric of smaller index. To see this, it suffices to adapt any pseudo-orthonormal basis to the subspaces x and x^ = F{x) and then apply the Theorem of Inertia. T h e case / = (n + l ) / 2 holds a kind of special position, cf. Example 1.3 and Exercise 1. M. Stary [99] takes the properties of polars in real projective geometry mentioned here as the starting point for a definition of the inner and the outer region for a quadric in the case of an arbitrary field K with char K ^2. D Exercise 1. Let Q C P " (R) be a non-degenerate quadric. Prove that there is a projectivity a e PO{l,n + 1 — /) transforming the outer region A{Q) into the inner region I{Q) and conversely if and only if the index I of the polarity determining Q satisfies I = (p, -\-1)/2. Exercise 2. Consider the pseudo-Euclidean vector space V = T^"+^ of index I, 0 < I < (n + l ) / 2 . Prove that the associated quadric Qi C P " ( R ) can be represented as the orbit space Qi ^ (S'-^ X S"-')/Z2,

178

2 Cayley-Klein Geometries

where Z2 = { ± i d | ^ } and S''^^, S"^^ are hyperspheres in two complementary subspaces of V. (Hint. Write the equation of the quadric in the form I

J2\x^f= a=l

n+ l

J2 i-'i''

(4)

k=l+l

where the a;*, i = 1 , . . . ,n + l, are the homogeneous coordinates of the point x e Q.) In particular, Qi fs S"^^ is a hypersphere of the projective space P " ( R ) . Simple differential-geometric arguments imply that the non-empty, non-degenerate quadrics Qi are compact, connected hypersurfaces in P " ( R ) . Exercise 3 . Prove that under the assumptions of Exercise 2 the inner and the outer region of a quadric Qi are open, connected subsets of P " ( R ) . Exercise 4 . Let V" be the n-dimensional pseudo-Euclidean vector space of index I. By E„^k,s,d we denote the set of fc-dimensional subspaces H''" C V" on which the scalar product has index s and defect d, 0 < s, 0 < d, s + d < fc. Set p := k — s — d. Prove that En^k,s,d is non-empty if and only if s + d < / and p -\- d < n — I. E x a m p l e 2. Let P " ( C ) be the complex orthogonal projective space with the quadric Q determined by a bilinear scalar product (,). Since each point X G P " \ Q can be represented as x = [f], ^ e V with (p, p) = 1, the group G := PO{n + 1, C ) acts transitively b o t h on Q and on P " \ Q. Under the action of G the G r a £ m a n n manifold P„^k of A;-planes decomposes into finitely many orbits, which are characterized by the rank r of (, )|W^ x W on the corresponding {k + l)-dimensional subspaces W C V (or, equivalently, by their defect d = k — r + 1). D Exercise 5. Prove under the assumptions of Example 2: For every d e N such that 0 < d < (n + l ) / 2 and d < fc + 1, there are subspaces W'=+^ C F " + ^ with defect d. Discuss all the possible positions of a fc-plane relative to the quadric Q for dimensions n < 4. Exercise 6. Find a homeomorphism from the Grafimann manifold G2(R"^^) of oriented two-dimensional subspaces in the (n + l)-dimensional real vector space onto the quadric Q C P " ( C ) from Example 2. Hint. Consider the complex vector space C"+^ of P " ( C ) as the complexification of the real Euclidean vector space R"+^ for the bilinear extension of the scalar product, and take a positively oriented, orthonormal basis (01,02) for U^ C R"+^. Justify the definition of the homeomorphism ^{U^) := [Oi + O2 i]c and prove the claim. E x a m p l e 3. The Hermitean scalar products of index / in the projective geometries over C and H form a picture t h a t is , in many ways, similar to the pseudo-orthogonal geometry. Again, the equation of the quadric Qi can be written in the form (4), where now, of course, one has to take the norm square with respect to C or H , respectively. For / = 0 the quadric Qo is again

2.3 The Projective Geometry of a Polarity

179

empty, and the projective unitary group PU(n+ 1) or PSp(n+ 1) acts transitively on the projective space P " . Otherwise, inner and outer regions can be defined for Q as in Example 1. Passing to the realifications (4) becomes an equation in the real coordinates, which already has the normal form (4), of course with K = H; for every norm square one has to write two or four real squares, respectively. Hence, in the realification this leads to special cases of the quadrics considered in Example 1. In the complex case the inverse image e-^Qi) for the Hopf fibration 9 : P^"+\'R) -^ P " ( C ) , cf. (1.10.3), is a hyperquadric Q21 C P "^ (R), whereas in the quaternionic case the quadric Qi has a real hyperquadric Q4i as its inverse image under the Hopf fibration 9 : P**"+^(R) -^ P " ( H ) . Restricting the Hopf fibrations to these quadrics leads to corresponding fibrations of Q2i over Qi{C) into one-dimensional and of Q4i over Qiiil) into three-dimensional real projective spaces, respectively. In the simplest case {l,n) = (1,1) the base Qi(C) is the circle |z°p = |.i|2, i.e. |C| = |.V^°| = 1 in the Riemann sphere S"^ = P^(C), and 9^^{Qi{C)) Q2 C P^(R) described by

is the quadric

Note that by restricting the scalars the polarity induces more involved structures on the inverse images than just the polarities corresponding to the mentioned (real) quadrics (cf. Example 1.10.12 and Exercise 1.10.16). • E x a m p l e 4. Little appears to be known concerning the quadrics associated with the quaternionic skew Hermitean polarities H„ in P " ^ ^ ( H ) . Let b denote the skew Hermitean form determining i7„. Because of 6(j;, p) G R^, we obtain from (p, p) = — i6(p, p) = 0 three quadratic equations in real coordinates; hence the quadric they define is a (4n — 7)-dimensional real submanifold of the (4n — 4)-dimensional space P " ^ (H). Passing from Formulas (1.36) and (1.37) or their equivalent versions (1.38), (1.39) to the realification of V^c, i.e. to the real coordinates, we see that this real submanifold is the intersection of three real hyperquadrics of index 2n. For n = 2 it is a circle m S^ = P (H), cf. Example 1.9.7. D 2.3.2 Efficiency We now again consider a general polarity F and will prove certain efficiency results for the actions of its projective group PGn{F). P r o p o s i t i o n 2. Let F be a polarity of P " defined by the a-biform {,), n > 2; suppose that the corresponding quadric Qp is not empty. If for some g e PGniF) glQp = idQp, then gr = idpn.

180

2 Cayley-Klein Geometries

P r o o f . We pass to the right vector space V " , m = n + 1, with the (T-Hermitean scalar product (,) and prove the foUowing statement equivalent to Proposition 2: P r o p o s i t i o n 3. / / V " , m > 2, contains isotropic vectors, and if for some g G CUGjn and any isotropic vector D G V the foUowing relation holds (;D = Dcp with cp G K*, then g = id-^^ -c for some c G

Z{K)*.

P r o o f . By Lemma 2.4, for each isotropic vector D there is an isotropic vector 6 G V hnearly independent of it and satisfying ( D , 6 ) = 1. Then the linear span W"^ := £ ( D , 6) is a non-isotropic subspace, and, moreover, V = W (BW . Choose a basis (ba), a = 1 , . . . , m — 2, in the likewise nonisotropic subspace W containing at least one non-isotropic vector, say bi. Set AQ, := {ba, ba) and Da := b a - 6 A a / 2 + D. (5) It is then easy to verify t h a t the vectors DQ, are isotropic. By assumption there are elements c, c, CQ, G K* such t h a t grD = DC, gb = be, gX3a = ^aCa,

« = 1, . . . , m — 2.

Applying these relations to Definition (5) for the DQ, we obtain grDa = gba - 5'DAa/2 + gD = {ba - D A a / 2 + D)Ca

= 9^a — DcAo,/2 + DC, a = 1 , . . . , TO — 2. Because of TO > 2, there is at least one such relation. Since g preserves orthogonality and leaves the subspace W, hence W as weU, invariant, and as the vectors D, 6, b i , . . . , bm-2 form a basis of V, by scalar multiplication with 6 we obtain the equations c = CQ,. T h e invariance of W implies gba = bo,c, i.e. g\W = id-ji^i c. The linearity of g implies c G Z{K)*. Comparing the coefficients of the vector 6 finaUy leads to cA„/2=(A„/2)c=c(A„/2). Because of Ai ^Q we obtain for a = 1 the assertion c = c and g = idy

c.

D

Exercise 7. Prove that the assertion of Proposition 2 does, in general, not hold for n = 1. P r o p o s i t i o n 4. Let K he a skew field of characteristic char i f ^ 2 , 3 , or suppose n > 1. Consider the polarity F corresponding to a a-Hermitean scalar product. If all points x G P " \ QF not belonging to the quadric of F are fixed points for the transformation g G PGn(F), then g = i d p r i .

2.3 The Projective Geometry of a Polarity

181

P r o o f . For Q = QF = 0 the assertion is trivial. Suppose now n > 1, and let v = [D] G Q. By Lemma 2.4 there is a second isotropic vector 6 with (D, 6) = 1. As a neutral subspace, W := £ ( D , 6 ) is not isotropic. Because of d i m V > 2, there is a non-isotropic vector o 7^ o within the likewise nonisotropic subspace W . Then b = D + 0 is not isotropic as weU, and by assumption there are a, 6 G K* such that g{a) = oa, g{b) = g{
(D

+ 0)6 - oa =

D6

+ 0(6 - a).

Since D is isotropic, this is also true of g{o), and hence a = 6 as well as g{o) = D6. SO the arbitrarily chosen point v = [0] e Q is also fixed implying the assertion. Suppose now n = 1, Q 7^ 0, and let D, 6 be defined as above. Then D+ := D + 6, D_ := 6 - D

(6)

is an orthogonal basis of V^ with (D + ,D+) = - ( D _ , D _ ) = 2 .

(7)

Consider the points c(t) :=

[D+

+ D_t] eP,

te

K.

For their vectors c(t) we have (c(t),c(t))=2(l-a(t)t); in fact, 2 G Z{K): 2 belongs to the ring Zl C Z{K). Because of char if 7^ 2,3, there is a to € Zl with to ^ 0, —1,1. Since for all t G Zl obviously a{t) = t, we have CQ = [c(to)] ^ Q, ^o '•= c(to) 7^ 0+, Co ^ D-. By assumption there are scalars b^,b^,c e K* for which

3(0+) = 0+^+, 3(0-) = ^-b-,gico) = Coc. On the other hand, gi^o) = 3(0+) + 5'(D-)to = D + 6+ + D_6_to = (D+ + D_to)c. Comparing coefficients we obtain c = 6+, 6_to = tgC. Now to 7^ 0 and to G Z l C -^(if) imply 6+ = 6_ = c. Since ( D + , D _ ) is a basis of V , we again arrive at <; = id-^^ c for the linear map, i.e., projectively, g = idp. D Remark. If if is a field, then it suffices to show the existence of n + 2 fixed points of g in general position, cf. Proposition 1.3.14. Formula (1.3.35) with f = g, a[= Oj, e' = e leads to g = idpn.

182

2 Cayley-Klein Geometries

2.3.3 Orthogonal Geometry. Reflections In this section we will mainly be concerned with the geometries called orthogonal by E. Artin [4]. Let thus F be a polarity defined by a bilinear scalar product (,), where the scalar domain K is necessarily a field. T h e isotropy group of the scalar product defined on an n-dimensional vector space V^ is denoted by 0 „ = 0 „ ( ( , ) ) and called the orthogonal group. The most prominent special cases are the real groups 0{n), 0{l,n — I) and the complex orthogonal group 0 ( n , C ) . For the purposes of this section we want to speciahze the notion of a reflection already introduced in Example 1.4.6. Let Vt^"^ C V " be a nonisotropic (n — l)-dimensional subspace; then its orthogonal complement W is a one-dimensional, non-isotropic subspace called the normal of W; each vector n £ W , n 7 ^ o , i s a normal vector of W. Then, according to Example 1.4.6, the direct decomposition V = W Q)W determines a reflection

which we call the orthogonal reflection in the non-isotropic (n— l)-dimensional subspace W"^^. In this section, a reflection will always understood to be such an orthogonal reflection. For a normal vector n of W, (n, n) = p y^ 0, we have sw : y = y G F I—> swiT) := y - n • 2(n, y)/p.

(8)

A straightforward calculation shows t h a t sw € 0 „ . The corresponding projective m a p will be called an orthogonal reflection as well. Obviously, the norm (determinant; cf. 1(5.7.30)) of a reflection is equal to —1, N{sw) = —1We will now prove a proposition refining the statement of Lemma II.8.10.6 (cf. E. Artin [4], Theorem 3.20): P r o p o s i t i o n 5. Let V^ be a vector space over a field K, char K ^ 2, and let (,) he a symmetric bilinear scalar product on V^. Then every orthogonal transformation g G 0 „ can be represented as a product of r < n reflections. P r o o f . For n = 0 , 1 or gr = idy the claim is trivial. So we suppose t h a t the proposition is already proved for all n a t u r a l numbers m < n, and proceed by induction in four steps. 1. Suppose t h a t there exists a non-isotropic vector o € V \ {o} satisfying g{a) = 0, and let W := {aK)^ be the subspace orthogonal to o. Since o is not isotropic, we obtain the orthogonal decomposition V = W (B aK into (/-invariant, non-isotropic subspaces. By the induction hypothesis we find r < n — 1 reflections s,, in hyperplanes U,, of W such t h a t g\W = Sr o ... o si. Then, by Exercise 2.5, we have s^ := s^ +idaK G 0 „ ; this m a p is a reflection in the subspace W^ := U^(BaK, and, moreover, g = s^o.. .osi with r < n — 1 . 2. Now suppose t h a t there exists a non-isotropic vector o G V \ {o} for which b := g{a) — o 7^ o is not isotropic. Let s be the reflection in the likewise non-isotropic subspace W := (bK)^. Because of

2.3 The Projective Geometry of a Polarity

183

(gr(o) + 0,3(0) - a) = (3(0), 3(0)) - (0,0) = 0, we have g{a) + 0 G W.

Hence

s(gr(o) - a) = -g{a) + 0, s{g{a) + 0) = g{a) + 0, and adding both equations we obtain s o g(a) = 0 for the non-isotropic vector 0 7^ 0. By Step 1 we can represent s o gr as the product of r < n reflections s^, and thus g = s o Sr o ... o si is the product of r + 1 < n reflections. 3. Next we consider the case d i m V = 2. By Steps 1 and 2 the assertion holds true, if V does not contain any isotropic vectors. So let D G V be an isotropic vector. By Lemma 2.4 we flnd an isotropic basis ( D , 6 ) for V with (D, 6) = 1. Since each isotropic vector in V has to be a multiple of D or of 6, for g G O2 there only remain the following possibilities: a) b)

30 = 6/3, gr6 = D / 3 - \ 30 = 0/3, 36 = 6/3-1.

In Case a), the vector g{o + 6/3) = D + 6/3 is not isotropic and flxed under g, and using Step 1 we obtain the assertion. In Case b), we may suppose /3 7^ 1, since otherwise g = i d y . Then the vectors 0 := 0 + 6 and 3(o)-o=o(/3-l) + 6(/3-i-l) are not isotropic, and the assertion foUows from 2. 4. We may now suppose dim V > 3, t h a t every non-isotropic vector a&V is not invariant, 3(0) 7^ 0, and t h a t 3(0) — 0 is isotropic. From this we can even conclude t h a t 3(0) — 0 is isotropic for all a&V. In fact, if 0 G V is isotropic, then the (n — l)-dimensional subspace ( o i f ) ^ is also isotropic, and has defect 1. Hence there is a non-isotropic vector 0 G (oK)^. So we have (0, a) 7^ 0 und (0 + oc, 0 + oc) = (0, a) 7^ 0 for aU c e K. According to our assumption the vectors 3(0) — 0 and 3(0 + Oc) - (0 + Oc) = 3(0) - 0 + (3(0) - 0)c are isotropic. For the scalar square of the latter vector we thus have 0 = 2(3(0) - 0,3(0) - 0 ) c + (3(D) - 0,3(0) - 0)c2.

Inserting here c = ± 1 and (3(0) - 0, 3(0) - D) = 0, and T h e just proved imphes totaUy isotropic subspace of have

adding both the resulting equations we obtain 3(0) - 0 is isotropic. t h a t the image W := {g — idT/)(V) ^ {0} is a V. Consider now a £ V and b e W . Then we

0 = (3(0) - 0,3(6) - 6) = ( 3 ( 0 ) , 3 ( 6 ) ) - ( 0 , 3 ( 6 ) ) - (3(0) - 0, b).

184

2 Cayley-Klein Geometries

From {g{a),g{b)) = (o, b) and b G W^ we conclude (3(0) - 0, b) = 0,

{a,b-gib))=0.

Since the last equation holds for aU 0 G V, we obtain g{b) = b, i.e. g\W = idvi/±. According to our assumption there is no non-isotropic fixed vector. Hence W has to be a totally isotropic subspace as well. By Corollary 2.6, d i m l ¥ < n/2, d i m l ¥ ^ < n/2, and because of dim W + dim W

= n and Vt^ C Vt^ , we have

dimW = dimW^ = n/2 und W = W^. Thus by Proposition 2.5, V has to be a neutral space, and, in particular, n = dim V = 2r is even. The automorphism g leaves each element of W = W fixed and so has the form (2.21). Consequently, N{g) = 1. This implies that for a neutral space V'^^ and every g with N{g) ^ 1 the assertion holds. So let now s be an arbitrary reflection and let g G 0 „ be an element with N{g) = 1. Then we have N{s o g) = N{s) • N{g) = —1; hence there is a representation s o ( ; = S f c 0 . . . o s i o f s o ( ; a s the product of A; < 2r reflections. Since N{s o g) = ( —1)'^ = —1, k has to be odd and thus must be less than 2r. This implies: g = s o SkO ... o si is represented as the product of at most n = 2r reflections. • Corollary 6. For every g G 0 „ the norm of its square satisfies N{g'^) = 1. The special orthogonal group SOn:={geOn\N{g) consists of all orthogonal transformations, product of an even number of reflections.

= l} which can be represented as the •

The elements g G SOn are also called proper, and those from 0 „ \ S'0„ improper motions, or rotations and rotoinversions, respectively. Corollary 7. If g ^ 0 „ can he represented as the product ofrn — r. Hence, if n is odd (or even), then each proper (improper) motion g has at least one fixed vector p 7^ 0, i.e., g has a fixed point in pn-l(^V). P r o o f . Let g = Sr o ... o si, Sp he the reflection in Vt^"^ . Then for all r

^eUr:=f]Wp

2.3 The Projective Geometry of a Polarity

185

we have g{f) = p. A straightforward induction proves dimt/fc > n — k, and U D Ur imphes dim U > n—r. Hence in case n is odd, for every proper motion we have d i m t / > 1, and for even n this holds for each improper motion. D Exercise 8. Let g G 0„ be an element that cannot be represented as the product of less than n reflections. Prove that then the first (or the last) element in any representation g = s„ o ... o si may be chosen to be an arbitrarily given reflection. Exercise 9. Let g G 0„ be an involution. Prove: a) The decomposition V" = Wi © W-1 into the eigensubspaces of the only possible eigenvalues ± 1 is orthogonal. - b) The eigensubspaces Wi,W-i are not isotropic. - c) There is no orthogonal reflection in an isotropic hyperplane. (Cf. Example 1.4.6.) Exercise 10. Prove: a) Any g e 0 „ satisfying g\W"^^ = idw for some isotropic hyperplane W C V" is itself the identity, g = idv. (Hint. Apply Lemma 2.4 to a vector D / 0 from the defect subspace of W.) - b) If for g,h E On the restrictions to a hyperplane U"^^ C V" coincide, g\U = h\U, then g = h or g = su o h; iiU is isotropic, then necessarily g = h. By Exercise 9, there are no orthogonal reflections in isotropic hyperplanes. This fact is generalized in t h e following proposition: P r o p o s i t i o n 8. Let V^ be a vector space with a symmetric, bilinear scalar product (,), and let cp : U ^ U be an isomorphism from the subspace U onto the subspace U (cf. Definition 2.2). Then there is an extension g G 0 „ of (f (its existence is guaranteed by E. Witt's Theorem) with prescribed norm N(g) = ±1 if and only if dimt/ + d e f t / < d i m F .

(9)

P r o o f . If (7i, (72 £ On b o t h are extensions with different norms, then 9 •= 92^ °9i ^ On is a m a p with norm N{g) = — 1 satisfying g\U = i d y . Hence it has to be shown t h a t idy can be extended to an improper motion if (9) holds. We decompose U = Uo (B Ui into t h e orthogonal sum of a defect subspace Uo and a complement Ui. Then t / i is non-isotropic; in Ui there is a totally isotropic subspace Uo, d i m t J o = d i m t / o , and Uo (B Uo is not isotropic (by Proposition 2.5). Obviously, dim U (BUO = dim U + def U, and U (BUO is not isotropic. If (9) holds, then there is a non-isotropic hyperplane l ^ n - i -^ U (B Uo, and t h e reflection sw in this hyperplane satisfies N{sw) = —1 as well as sw\U = i d y . Conversely, suppose now t h a t d i m t / + d e f t / = n. Then F " = Uo®U = UO®UO(BUI. Since Uo®Uo = U^ is a neutral space with Uo C U, g\U-^ is a m a p satisfying t h e conditions from Exercise 2.11. From g\Ui = idu^ we obtain

N{g)=N{g\Ui)-N{g\Ui)

= l.

186

2 Cayley-Klein Geometries

Exercise 1 1 . a) Determine the equations of the quadrics QF associated with the real auto-polar maps classified in Proposition 1.9.4. - b) Compute the maximal dimension of a projective subspace contained in QF- - c) Prove that each type of these quadrics QF C P" can be obtained as the intersection QF = P" Pi Q of ~N

~

~N

an n-plane P" C P with a suitable quadric Q in a larger projective space P . Determine, in addition, the smallest value for A^ (depending on n and the type of F) allowing such a representation. Exercise 12. Let ( a o , . . . , a „ ) be a polar simplex of the polarity F in the n-dimensional projective space P". Denoting by Si the refiection in the i-th face F{ai) of the polar simplex prove So o si o ... o s„ = idpn .

2.4 Invariants of Finite Configurations As in Section 2 of this chapter we start from a vector space with scalar product [V"+^, (,)] determining the absolute symmetric auto-correlation F of the associated projective geometry. Its projective isotropy group will again be denoted by PGn = PGn{F). According to the Erlanger P r o g r a m m by F. Klein [65], any action of PG transformation group on a set M poses the task to describe the orbits of this action by invariants. In this section we will consider some special cases of this kind of problem in the case of transformation groups arising in a n a t u r a l way from the action of P G „ over P " as the isotropy group of F: The elements of the set M are to be described by finitely many projective subspaces of P " . For this reason we call t h e m finite configurations. The action of any element g G PGn on a configuration results from the simultaneous action of g on all the components of the configuration, i.e. the subspaces defining the configuration. We start with the discussion of point sequences. Dualizing the results by means of the polarity F immediately leads t o invariants for hyperplanes, which thus do not have to be considered separately in this section. In the sections to follow, presenting the elhptic and the hyperbolic geometries, it will become clear t h a t the distance of point pairs and the angle between two hyperplanes are mutually dual notions. In these metric geometries and in the Mobius geometry we will describe the invariants of pairs of subspaces of arbitrary dimension.

2 . 4 . 1 J - ' G „ - C o n g r u e n c e of F i n i t e P o i n t S e q u e n c e s The action of P G „ on point sequences is defined for g G -PG„ by g: {xi,...,Xk)e{P")''^^g{xi,...,Xk):={gxi,...,gXk)e{P")''\

(1)

Here (B)'' denotes the A;-fold direct product of the set B. Point sequences transformed into one another by such a transformation are called congruent

2.4 Invariants of Finite Configurations

187

or, more precisely, PGn-equivalent. We will use a similar terminology also for other transformation groups. In order to not always have to exclude particular cases, we restrict the action to the obviously P G „ - i n v a r i a n t subset consisting of sequences containing only mutually different points; so, for an arbitrary set B, we consider Mfc(B) := { ( x i , . . . ,xfc) G (B)'^lx^ ^ x, for ^ ^^ v}.

(2)

Invariance of dimension and Lemma 1.1 immediately imply the following necessary condition P r o p o s i t i o n 1. The following conditions have to be satisfied by two k-tuples {Xfj), (y^) G Mk{P^) in order to be congruent: a) For all p\ with I < pi < ... < pi < k, I < I < k the dimensions of corresponding joins have to coincide, Biinxp.v

...Vxp,

=Dimyp^V...Vyp^.

(3)

b) There exist a K & Z{K)* with O-(K) = K as well as vectors p^, t)^ G V"^ such that Xj^ = [j;^],y^ = [t)^] and (t)u, t);.) = K(?U, l„)forl
k.

T h e following shows t h a t these conditions are not sufficient. E x a m p l e 1. Take i f = R , and let ^4,2 be the polarity of index 2 in P . Then there is a totally isotropic hue H^ C Q. Moreover, it is possible to find two quadruples {x^), (y^) G Mi{H^) with differing cross ratios; hence they cannot be transformed into one another by any projectivity. On the other hand, for an arbitrary pair of such quadruples conditions a) and b) from Proposition 1 are satisfied. There is a similar example formed by two isotropic lines in the three-dimensional projective symplectic space. • Requiring, in addition t h a t the projective span of the point sequences is not isotropic leads to the following sufficient condition for congruence: P r o p o s i t i o n 2. Let (aj^i), (y^) G Mk{P^) be two point sequences conditions a), b) from Proposition 1. Suppose, in addition: c) The projective span xiV .. .V Xk is not isotropic. Then there is a projective isomorphism ^ : 031 V . . . Vajfc — > y - y \ / ...\/

with the

y,.

satisfying

(4)

properties ^{xp)

= y^, p = l,...,k,

(5)

188

2 Cayley-Klein Geometries (^(?), ^(t))) = «(?, t)) /or aU^,t)exiV

...V Xk.

(6)

/ / Dim xi\/ .. .y Xk = n, or if d) There is p ^ Z{K)* such that o-(p)p = K, then the point sequences are congruent. P r o o f . Set X := xi\/ .. .\/ Xk and m := D i m X . Then, in the sequence (ajpi) there are m + 1 points generating X. The corresponding vectors, say

then form a basis for X (according to b)). Condition a) imphes that the vectors t'o : = t ) ^ „ , . . . , b ™ : = t ) ^ ^

form a basis for the space Y : = y ^ V . . . V y j , . Consider the hnear isomorphism cp : X ^ Y determined by '/^(aiu) := tipi- From Condition b) we immediately conclude that equation (6) holds. We now prove (5). Let p = pj,, t) = t)j, be two of the mutually corresponding vectors determining the point sequences according to b). Let their basis representations in X and Y, respectively, be m

m

Forming the scalar product of these basis representations with a.^ and bi, from the left, respectively, we obtain m

(fliv,?) = ^{O'l^.O'p)^^,

i/ = 0, . . . , m ,

m

i^u, t)) = J2^^^^ ^i^^y^^ i/ = 0,. .., m. Since all vectors occurring in the second system of equations correspond to those in the point sequences above them, we can apply Condition b) to the second system. Cancelling the scalar K we obtain a system of equations with the same coefficients as the first one. Since, by Condition c), the space X is not isotropic, the ranks of these systems both are equal to m + 1; hence their solutions have to coincide: (y^) = (x^). From the definition of f we obtain m

m

m

^(?) = Yl vM"^^ = Yl ^1^''^ = Yl ^i^y^ = ^12=0

12=0

12=0

Hence the projective map generated by cp (denoted by the same symbol) satisfies the asserted equations (5) and (6). In the case m = n Lemma 1.1 implies that (f belongs to PGn(F) completing the argument. For m < n we want to apply E. Witt's Theorem, Proposition 2.3. Since, in general, f is not an

2.4 Invariants of Finite Configurations

189

isomorphism of the subspaces with induced scalar product — this only holds for K = 1, cf. (6), we now need to make use of Condition d) and define V'(j;) := (p{^){p)^^. Then, because of p G Z{K)*, (6) implies: (V-W, V(t))) = a(p-i)p-i(^(y), ^(t))) = («)-!«(?, t)) = (y, t)) for all p, t) G 031 V . . . V ajfc. Hence ip is an isomorphism of the subspaces, which, by E. Witt's Theorem, can be extended to an automorphism g G PGn- As the projective maps generated by the linear maps 'ip and cp coincide, g realizes the congruence of the point sequences. • Example 2. To see that Condition d) is not superfluous in the case m < n, take if = R, n = 4, let V^ be the pseudo-orthogonal vector space of index 2, and let (ej), « = 0 , . . . , 4, be a pseudo-orthonormal basis. Consider the point pairs 031 = [eo],a;2 = [ei] and y^ = [e3],y2 = [C4]. Then Conditions a), b), c) with K = — 1 are satisfied, but d) is not. There is a projective isomorphism (f : xi \/ X2 ^ yi\' 1)2', according to the Theorem of Inertia, however, it is impossible to extend it to an isomorphism g G PG4. This is also expressed in CoroUary 3 below: • Corollary 3. For if = R, C, H let F be one of the polarities from Table 2.1. Then, in the necessary Condition b) from Proposition 1, K can be taken to be ± 1 ; for F = F„+i, F = i^„+i,;, or F = ii„+i,;, and I < (n + l ) / 2 , K can be taken to be 1; for real null systems K can be taken to be ± 1 , and for complex ones K can be taken to be I. In all the cases where K = I, Conditions a), b) (with K = I) and c) together are sufficient for to ensure congruence of the point sequences. P r o o f . The number K only depends on the transformation to be found. In the Examples 1.3-5 and in Section 2.1.6 we proved that only the values listed in CoroUary 3 have to be considered. The statement concerning the null systems foUows from the considerations in Section 2.1.3. If, finally, K = 1, then d) is triviaUy satisfied with p = I. • 2.4.2 Orbits of Points. Normalized Representatives Corollary 3.1 already contains a statement concerning the PG„-equivalence of two points included in the more general criteria proved above. By Example 2.3, the projective-symplectic group PSp^ acts transitively on the projective space p2n-i^ Similarly, the isotropy group PG„{F) of a polarity acts transitively on the quadric corresponding to F. Hence it will suffice to consider polarities, and here again only the points not belonging to Q. These are determined by non-isotropic vectors: x = [p] such that (p, p) 7^ 0. Obviously, the scalar square depends on the chosen representing vector. Substituting p 1-^ p^, ^ e K*, leads to

(?e,?e) = ^(e)(?,?)e = e(?,?)e

(7)

190

2 Cayley-Klein Geometries

Here and in what follows we will frequently use the notation a{^) = ^, famihar from the conjugation in H, if it is clear which involutive anti-automorphism a is meant. It satisfies the usual rules,

e+?? = e+j?, e?? = j?e, 0 = 0,1 = 1, (-e) = -e, (e-i) = rMeeif*), e = e

(s) (9)

Let Ka := {C & K\^ = ^} be the set of fixed elements of a. K„ is a subgroup of the additive group of K, but, in general, it is no subring. Nevertheless we have: If ^ 7^ 0 belongs to K„, then ^^^ is also in K„; if K„ C Z{K), a is called central; in this case, K„ is a subfield of Z{K), hence also of K. Denoting by Ko the prime field of if, the equation 1 = 1 immediately implies Kg C K^j. Formula (7) leads to the right action \cK*,icK^—>\i\cK

(10)

of K* over K, for which K„ is an invariant subset. As is well-known, the scalar squares (p,p) belong to K„. We consider the action of K* over K„ and fix a representative in each orbit. Let P C K„ denote the set of these representatives. Obviously, we have 0 G P ; take 1 as the representative of the orbit {AA}, A G K* and, \i \i ^ P and —yU, is not equivalent to yU, take —yU, as the representative for {—A/xA}; if /x 7^ 0 and IJT^ is not equivalent to yU, then we choose \jr^ as the representative of {AyU^^A}. Hence, either —1 = AA for a certain A G if*, or {0,1, —1} C P. Note that Condition d) from Proposition 2 just means that K is equivalent to 1 under the action (10). Example 3. Take a = idif. Then K = K^ is a field; two elements are equivalent under the action (10), if they differ by a square factor, and the orbit space is the union of zero and the quotient group K*/{K*)^. For if = R we thus have P = {0,1, —1}; this property is the basis for the definition of the inner and the outer region of a real quadric in Example 3.1. The same example also shows that not every element of P has to occur as the representative for an orbits of a scalar square; e.g., for / = 0 the number —1 does not occur. For if = C, as for any algebraically closed field, we have P = {0,1}, cf. Example 3.2. If, however, if = Q is the field of rational numbers, then Q*/(Q*)'^ is infinite; two numbers CJ '? ^ Q* are equivalent if and only if ^ry^^ is a square, i.e., if all its prime factors have even multiplicity, cf. Proposition 1.2.6.5. The difficulties we meet here are similar to those we had to face exploiting Proposition 1.9.3 for the classification of polarities. Without any particular assumptions concerning the scalar domain if satisfactory answers to the initial question of this section will be hard to obtain. D Now we return to the general case. To each point x G P " we assign the element ^{x) G P representing (p,p) for an arbitrary representative p G V"+^ with a; = [p]. A vector p G F"+^ is called normalized if (p, p) G P. Note that a normalized representative of a point x is never uniquely determined; in fact.

2.4 Invariants of Finite Configurations K

F

a

K^

P

R F„,i idR

R

C

idc

C

{0,1}

T

R

{0,1,-1}

T

R

{0,1,-1}

Ti

£(l,j,k)

{0,1}

C H H

Fn

191

Kl

{0,1,-1} {1,-1} {1,-1}

51

Table 2.2.FinNormalization for the classical polarities. Here r denotes the conjugation in C or H, respectively, Ti = — i r i , S^ C C denotes the unit circle, and 5^ C H is the unit hypersphere.

(—p, —p) = (p, p). In general, together with p the vector pA is also normalized if A belongs to the isotropy group of ^ = (p, p) G P under the action (10). In Table 2.2 the corresponding d a t a for the classical polarities are displayed; here Kl C K* denotes the isotropy group of the element 1 coinciding with t h a t of — 1. From Corollary 3.1 we conclude t h a t i^{x) = ^{y) implies the congruence of X and y. Exercise 1. Verify the data in Table 2.2. Hint. Recall Section 1.9.5, in particular Formulas (19), (20). Exercise 2. Show that there is an orthogonal basis (Oi, O2) for which (Oi, O2) = ^5ij, and A is no square using the standard scalar product in the rational vector space Q^, for which the standard basis (ei, 62) is orthonormal. Interpreting the Theorem of Inertia by saying t h a t the number /(^) of elements in an orthogonal basis belonging to a given representative ^ G P does not depend on the chosen basis. Exercise 2 shows t h a t such a statement does in general not hold. Exercise 3 . a) Taking F as one of the classical polarities from Table 2.2, and I < n / 2 , prove that two points x,y & p n - i g^j.g PG'„_i(_F)-equivalent if and only if ^{x) = C(j/). - b) Show by means of an example that this statement is wrong for I = n/2; in this case all points not belonging to the quadric are PGn-i{F) -equivalent. 2 . 4 . 3 Invariants of P o i n t P a i r s We now want to find the invariants of point pairs {xi,X2) G M 2 ( P " ) , which are important for the foundations of metric geometries. Since two of these

192

2 Cayley-Klein Geometries

points always determine a line, we can pick up the thread of Section 1.9.7 where we studied the position relations of line and quadric. A little more special than there we will here make the following assumption: Assumption A. Let V"+ he a vector space with a a-Hermitean or symmetric scalar product over a skew field of characteristic char K y^ 2, and let F be the associated polarity of the n-dimensional projective geometry; the case of an empty quadric Q = Qp is not excluded. Compare Definition 2.1; null systems are not allowed now. Denote by Un+i the isotropy group of the scalar product, i.e. the set of all linear transformations g of V"+^ satisfying (ff?,fft))= (?, t)) for ah y, t) G F"+^ (cf. (1.10)); PUn+i denotes the corresponding group of projectivities. If one of the points belongs to the quadric, then from the results of Section 1.9.7 we immediately obtain Corollary 4. Consider, under Assumption A, the pair (a;i,a;2) € M2(P") and suppose that x\ G Q. Let, moreover, ^ = ^{x^) G P. Then there only are the following, mutually excluding possibilities: a) ^ = 0 and {^1,^2) = 0 •^^ a;i V 0:2 C Q. i') C 7^ ^ d'nd (pi,p2) = 0 "^^ ^1 V 032 is a tangent of Q. c) (pi,p2) 7^ 0 •<=^ 031 V 032 «s a cftord o / Q .

The point pairs (xi,X2),{y 1,1/2) € Af2(-P"), xi,yi G Q, are congruent if they share Property a), b) or c) and, in addition, satisfy S,{x2) = C(y2)- ^''^ group PUn^i acts transitively on each of these orbits. Lf PGn(F) = P t / „ + i , then the conditions are also necessary for point pairs to be congruent. P r o o f . The first claim immediately follows from Example 1.9.8. We will show the asserted congruence of the pairs. In a), this is a special case of Corollary 2.12. In both the other cases we normafize the vectors p2j ^2 so that C = (?2J ?2) = (152 J '52)- In c), multiplying p^ by a suitable factor we can always arrange (pi,p2) = 1- Thus, the scalar products of vectors representing both pairs of the same type a), b) or c) can arranged to be equal; assigning these pairs of bases to one another defines an isomorphism of subspaces, which, by, E. Witt's Theorem, Proposition 2.3, can be extended to an automorphism of V"^^. We have K = 1, hence the corresponding projectivity belongs to PUn+i; it reahzes the congruence we were looking for. The last claim follows from the invariance of Properties a), b), c) together with the fact that it suffices to consider transformations with K = 1. D Remark. The example of the quadric Q(-F4,2) in the real three-dimensional projective space P^ shows that the condition ^(0:2) = ^(2/2) in Corollary 4 is not necessary, cf. Exercise 3 and Example 6 below.

2.4 Invariants of Finite Configurations

193

E x a m p l e 4. Let a;i,a;2 G Q be points of the quadric, xi ^ X2- Then there only are the fohowing possibihties: Case a): a;i Va;2 C Q, or Case c): a;i Va;2 is a chord of Q. By CoroUary 4, the group PUn+i acts transitively, hence P G „ as well, on each of these sets of point pairs. If, in particular, Q is a quadric of index 1 (cf. Example 2.4), then Q contains no line, and this implies: For quadrics Q of index 1 the groups P t / „ + i acts transitively on the set of point pairs {xi,X2) e Q x Q, xi ^ X2- In particular, this holds for the polarities Fn+i^i, i f = R , and Hn+i^i, i f = C, H , from Table 2.1. According to Example 2.4 the group PUn+i acts transitively on the set of hues contained in Q. Then Corollary 4, Case c) with ^ = 0, leads to C o r o l l a r y 5. Under Assumption set of chords for the quadric Q.

A the group P t / „ + i acts transitively

on the

a From now on, we will consider only point pairs (a;i,a;2) € M 2 ( P " ) with Xj ^Q, j = 1,2. E x a m p l e 5. Let (a;i,a;2), (1/1,1/2) ^ Ai2(-P"), Xj,yj ^ Q, j = 1,2, be point pairs for which a;i Va;2, yi^y2 are tangent to the quadric Q, but not contained in Q. Then the corresponding subspaces are isotropic, but not totally isotropic. Let z = Q r\ [xi V X2) be the unique point of contact of the tangent; it corresponds to the defect subspace of the associated vector subspace. We wiU show: The point pairs {xi, X2), [Vi, 1/2) are PUn^i-equivalent if and only if i{xj)=i{yj),

J - = 1,2.

(11)

Obviously, condition (11) is necessary. Let us show the converse. Consider z = [I\TXJ = [pj]. Because of xi ^ Q, the (p^,3) form a basis for £(pi, 12)- Let i{xi) = ( ? i , ? i ) , and let ?2 = ? i a + 3/3 be the basis representation of p2- Multiplying by a^^, renormalizing the isotropic vectors 3 as well as p2 we obtain ?2 = ?i + 3;

(12)

indeed, because of (p2,?2) ¥" 0 and p^ ^ p2, ^t and /3 both have to be different from zero. Note t h a t (12) even implies C = (?l,?l) = (?2,?2) = (?1,?2),

(13)

from which we conclude t h a t all pairs of points different from the point of contact of a tangent are PUn^i-equivalent. According to Assumption (11), we may find a basis (t)i,3) for the tangent Hi y 1)2 with correspondingly equal scalar squares. By E. W i t t ' s Theorem, Proposition 2.3, this imphes the assertion. D

194

2 Cayley-Klein Geometries

E x a m p l e 6. Let K = R. We consider the polarities Fn+i,i of P " . Since here P = {0,1, —1}, for / > 1 there are inner (^ = —1) and outer (S, = I) tangents. Figure 2.1 shows part of the hyperbofoid Q

{x^f-{x'f

+ (x'Y +

„3\2

ix^Y=0

(14)

in P with one inner and outer tangent, respectively, through a point z £ Q. The intersection of Q with the tangent plane F4_2(^) at the point z consists of two lines, the generators of Q through z, separating the inner from the outer tangent. Note that here we have / = (n + l)/2; hence there is a projectivity interchanging the inner region I{Q) with the outer region A(Q). So, if (aji, 032), (j/i, 2/2) € M2{P ) are points, different from the points of contact of two tangents, xi V X2, Vi V 1)2, of Q, then there always is <; G PG'i{Fifl) such that g{xj) = y •, j = 1, 2. If / < (n + l)/2, then distinguishing the inner from the outer region becomes projectively relevant: Outer tangents cannot be transformed into inner tangents by any g G PGn{Fn+i,i); this group acts transitively on the set of outer and on the set of inner tangents, separately. •

Fig. 2.1. Hyperboloid, tangent plane, inner and outer tangents.

2.4 Invariants of Finite Configurations

195

Now we t u r n t o the most interesting case t h a t the joining hnes of each point pair are not tangent, i.e. {xi V X2) A F{x-i V X2) = o. The associated vector space W'^ := £(^-^,^2) is not isotropic. Let p^ be normahzed representatives:

Since the S,j are only determined up t o renormaUzation by Xj G K^,, i.e. the isotropy group of S,j, for the scalar product we obtain with p^- = p^Aj (?j,?j) = (?j,?j), {h,h) = Ai(?i,?2)^2, Aj G K^, j = 1,2.

(15)

Choosing in each class of {K^^ x K^^ )-equivalent elements of i f a representative r/, and normalizing this representative by Ci = C(a;i), 6 = Cix2), 1] = ??((?!, ?2))

(16)

we obtain a complete system of invariants, capable of deciding upon P t / „ + i - c o n g r u e n c e for point pairs. The examples below display a closer look at some special spaces. E x a m p l e 7. Under Assumption A, let if, char i f 7^ 2, be a field, and suppose t h a t the scalar product is bilinear, i.e. a = i d x - Then we have i f | = {1, — 1} for all ^ 7^ 0; as always KQ = K*. We define Sqix,,X2):= ,Jl\]f.

ixuX2^Q).

(17) D

Obviously, Sq(a;i,a;2) is independent of the chosen representatives p^ for Xj, and invariant with respect to transformations from PGn(F), cf. Lemma 1.1. If 031, 032 are points on a tangent not belonging to Q, then Sq(a;i, 0:2) = 1, cf. (13). We prove P r o p o s i t i o n 6. Under the assumptions of Example 7, let x\,X2 ^ Q he points on a non-tangent line B. Then Sq(a;i,a;2) = ^ if and only if xi = X2For points xi, X2 different from one another, Q C\ B ^ $ if and only if -det((Pj.,Pfc)) = {li,l2? is a square in K.

- (?i,?i)(?2,?2)

196

2 Cayley-Klein Geometries

P r o o f . As the basis for the vector space associated with B we choose an orthogonal basis (oi, O2) such t h a t [oi] = xi. Take Xj := (Oj, Oj), j = 1, 2. For a vector ^2 with X2 = [^2], let P2 = a i a + 02/3 be its basis representation. Then Sq(a;i, X2) = 1 if and only if a^Xf = Ai(Aia^ + A2/3^), and this in t u r n holds if and only if AiA2/3^ = 0. As B is not tangent, i.e., the associated vector space is not isotropic, we have A1A2 ^ 0, and this implies the first assertion. T h e second is an immediate consequence of the discussion concerning the condition for y := [p^t + ^2] ^ Q with t e K: (t), t)) = (?1, ?l)i^ + 2t(?i, ?2) + (?2, ?2) = 0.

Exercise 4. Show under the assumptions of Example 7: If the pairs («!, 0:2), (j/i, 2/2) S M2(P") satisfy Sq(a:i,a:2) / 0, then there is g G PUn+i such that gxj = pj, j = 1, 2, if and only if ^{xi) = C(j/i) and Sq(xi,X2)=Sq(yi,y^).

(18)

Exercise 5. In addition to the assumptions of Example 7, suppose that K is algebraically closed. Prove for Xj, pj ^ Q, j = 1, 2, that {xi, X2) is congruent to (j/j^, j/2) if and only if (18) holds. Exercise 5 settles the special case K = C: T h e pairs of points not belonging t o Q are classified by the P G „ ( F ) - i n v a r i a n t Sq(a;i, X2). 2.4.4 Real Orthogonal G e o m e t r i e s E x a m p l e 8. In this section we will deal with the real polarities F = Fn+ij of the projective space P " over the field i f = R , and start with the case / = 0. According to Example 3.1 the group PO{n + 1) = PGn{F) acts transitively on P " . Since the scalar product is positive definite, the associated vector space is Euclidean, and, moreover, ^{x) = 1 for all x G P " . Proposition 2 immediately implies t h a t the point pairs are classified by the invariant (17). In order to have a geometric interpretation of (17), we represent the points by unit vectors Pi,p2 such t h a t (pi,p2) ^ 0- Then by Formula 1.(6.1.17), cos^(a;i,a;2) = (?i,?2) = VSq(a;i, 0:2), the angle (p(xi,X2) is uniquely determined as the angle between the corresponding unit vectors; it satisfies 0<^(a;i,a;2)< V2,

(19)

where, according to Proposition 6, (p(xi,X2) is equal to zero if and only if 031 = 032. The real projective space P " ( R ) , in which an absolute polarity F = F„+i,o is distinguished, is called the n-dimensional elliptic space (cf. Example 3.1); the elliptic geometry will be discussed in Section 5 below in greater detail. •

2.4 Invariants of Finite Configurations

197

E x a m p l e 9. Again take K = H, and let F = F„+i,i be a polarity of index / > 0. Then for the line B = xi\/ X2 determined by two points a;i 7^ 032 the following positions relative to the quadric Q, which is non-empty, are possible (cf. Example 1.9.8): a) b) c) d)

B B B B

C Q; this can only occur for / > 1. is tangent to Q: B D Q consists of a single point. D Q = 9; for this to be the case, n > 1 is necessary. DQ = {z^,z^} consists of two points z^ 7^ ^ - •

For n > 1, aU the cases actually occur, where in Case a), because of I < {n + l ) / 2 , the inequality n > 3 has to be satisfied. Cases a) and b) were discussed in CoroUary 4 and Examples 5, 6, in general. Case c) can be treated as in the preceding example: this case occurs if and only if the restriction of the scalar product to the subspace W'^ C V"+^ corresponding to B is positive or negative definite. We now discuss Case d). Since there exist two different points of the quadric on the line B, the associated vector subspace W'^ has to contain two linearly independent, isotropic vectors. Hence W'^ is pseudo-Euchdean of index 1. By CoroUary 2.11, aU these subspaces are isomorphic, and, on the other hand, each automorphism of such a subspace can be extended to an automorphism of V"+ . So, in order to determine the invariants of point pairs it suffices to consider transformations from W to itself. Let (01,02) be a pseudo-orthonormal basis of W'^: (oi, oi) = - 1 , (02, 02) = 1, (oi, 02) = 0. For the points x e B not belonging to Q we have ^{x) = ± 1 . The correspondingly normalized vectors, with basis representation p = OiCi +fl2C2,determine a pair of hyperbolas in W'^,

(?,?) = - C i + C l = ±i, with the common asymptotes (?,?) = (-Ci + C2)(Ci + C2) = o. The points of intersection with the quadric are determined by the isotropic vectors 02 ± Oi. We may always normalize the representing vectors p^ of the points Xj, j = 1, 2, so that (TpTj) = ^{Xj) = ±1 and (pi, ^2) > 0.

(20)

Hence by Proposition 2 the numbers Ci=C{xi),C2=C{x2),Sq{xuX2),

(21)

uniquely determining the scalar products (20), form a complete system of invariants for the points pairs considered here, {xi, X2) G M2(P"), a;i, 0:2 ^ Q,

198

2 Cayley-Klein Geometries

031 V 032 a chord on Q, with respect to the group PO{l,n+1 — I). Note t h a t , for I = {n + l ) / 2 there again exist transformations SQ with K = — 1 . In this case, the ^-values of corresponding points y = So{x) satisfy:

C(y) = (t),t)) = -(?,?) =

-Cix).

In general, the congruence condition reads as follows: Two point pairs, (aji, 032), (j/i, 2/2), in the set considered here are PGn(F)-congruent if and only if there is an e such that C(yi) = eC(a;i), C(y2) = ^^(xa)

and (t)i, ^2) = ^{TI, T2)

for suitably normalized representatives. In case I < ( n + l ) / 2 necessarily e = 1, and for I = {n+ l ) / 2 either e = I or e = —I. D It is common and practical to establish the relation between the function Sq and the familiar trigonometric functions from calculus. According to Proposition 6 and because of Q n {xi V 0:2) 7^ 0, we have {xi,X2f

> {xi,Xi){x2,X2).

(22)

If b o t h points lie in the inner region, or if both points belong to the outer region, then the right-hand side of (22) is positive, and hence Sq(a;i,a;2)>l; if, however, the points xi,X2 by (17),

(23)

he in different regions, then (22) is trivial, and, Sq(a;i,a;2) < 0 .

In the former case we define the hyperbolic angle (p(xi,X2)

(24) via

cosh 95(0:1, 0:2) = +v^Sq(a;i,a;2), 0.

(25)

T h e monotony of cosh implies t h a t (25) uniquely determines the number f{xi, X2) for each point pair (a;i, 0:2) with ^(a;i) = ^(0:2); according to Proposition 6 95(331, X2) = 0 if and only if xi = X2. Hence ip{xi,X2) will also have to be interpreted as a distance; in fact, for / = 1, ip{xi,X2) is the starting point for the definition of the distance in hyperbolic geometry, cf. Section 6. 2.4.5 P r o j e c t i v e O r t h o g o n a l G e o m e t r i e s for A r b i t r a r y F i e l d s Once again we return to the assumptions of Example 7. In this section we want to show t h a t , under rather general assumptions on the field K, we have the complete elementary trigonometry at our disposal. The trigonometric functions appear as matrix functions of the simplest representations for the orthogonal groups.

2.4 Invariants of Finite Configurations

199

Consider a projective line P (K) with a non-empty quadric Q = {zi, Z2}, of. Example 1.9.5. T h e associated vector space V thus carries a neutral scalar product, whose orthogonal group O2 was described in Proposition 2.13. Passing from an isotropic basis (31,32) of ^ in the normalization (31,32) = 1/2, (3i,3i) = (32,32) = 0 to the pseudo-orthonormal basis ai = 3 i - 3 2 , 12 = 3 i + 3 2 ,

(26)

- ( o i , Oi) = (02, 02) = 1, (ai, 02) = 0, we obtain the matrix representation for g(a) computation:

G SO2

by a straightforward

c{a) s{a) \ ^ s[a) c[a) J

^ . ^ j ^ , , _ ( a + a - i ) / 2 , s{a) := {a-a-^)/2, a G K*. \ ^ \ /; , \ / \ /; , (27) For the functions c ( a ) , s{a) we have c 2 ( a ) - s 2 ( a ) = 1,

(28)

c ( a i • a2) = c ( a i ) c ( a 2 ) + s ( a i ) s ( a 2 ) ,

(29)

s ( a i • a2) = s ( a i ) c ( a 2 ) + s ( a 2 ) c ( a i ) .

(30)

These relations may also be concluded from N{g{a)) = 1 together with the fact t h a t a G K* 1-^ g{a) G SO2 is an isomorphism. In this basis, the elements from O2 \ SO2 have the matrices -c{a) s{a)\ -s{a)c{a)J-

. ^

'

If there exists a homomorphism exp from the additive to the multiplicative group of K, exp:teKt—>e*=expteK*, (32) the hyperbolic trigonometric

functions

of the field K are defined by

cosht := c(e*), s i n h t := s(e*), t e K. W i t h these definitions, equations (27)-(30) become the well-known fundamental formulas of real hyperbolic trigonometry, from which the other formulas may be deduced. Inserting a = e* one attains all the possible values of the functions c ( a ) , s ( a ) only if the exponential exp is surjective. As is well-known, for i f = R the image exp R is the multiplicative group R;^ of positive real numbers; hence, all of SO{l, 1) can only be attained if a = — e x p t is also permitted:

200

2 Cayley-Klein Geometries

Obviously, for PSO{l, 1) this is irrelevant. The whole group 0(1,1) consists of four parts, its connected components, each homeomorphic to R; they correspond to the four branches of the pair of hyperbolas x^ -y^

= ±1.

For K = C the exponential is surjective. By Proposition 2.13, the whole group SO{2, C) is homeomorphic to the punctured Gaufi plane C* = C \ {0} and connected; 0 ( 2 , C) consists of two such domains defined by N{g) = ± 1 , respectively. The topological notions used here are based on the transfer of topologies from R and C, respectively, using the bijective maps above, which thereby become natural homeomorphisms. Example 10. Let P " be a projective space over a field K with the polarity F determined by a scalar product, for which there is an orthonormal basis (Cj); this situation will be called a standard scalar product on the associated vector space V"^^: {ti,tj)

= 5ij,

i,j = Q,...,n.

(34)

For this case we now want to determine the orthogonal group 0 ( 2 , K). Let 5*^ : = { O G F 2 | (o,o)

= l}

(35)

denote the unit circle in V . Furthermore, let [p, t)] be the volume function on V"^ satisfying [ei, 62] = 1 (Proposition L4.7.5). We claim that the map geSO{2,K)>—>

0 := ffei = eia + e2/3 G S'jf

(36)

is bijective; in fact, if 0 = eia + e2/3 G Sj^, then there is a unique transformation g G SO{2, K) such that 0 = gti- With respect to the basis (ei, 62) it has the matrix 9^[p

a-l3\

: ) ,

„2 , fl2

a^+/3^ = L

(37)

The group of these matrices, which is isomorphic to SO{2,K), wiU also be denoted by SO{2,K); it is Abehan. The transformations belonging to 0(2, K) \ SO{2, K) have matrices a. fi \

^2

As the oriented angle (^(0, b), 0, b G S]^, we take the unique group element 95(0, b) = g e SO{2,K) for which ga = b. Note that this is the "correct" definition for the angle even in the case if = R, since there the number 99 is only determined modulo 27r. Having fixed the unit vector ei (and thereby the orthonormal basis (ei, t^) with v{ti, 62) = 1), we may use (36) for the identification:

2.4 Invariants of Finite Configurations

201

f = vig) •'^^ V = vi^i, 9^1) • ' ^ ^ 9^1 = ei cos (p + e2 sin cp with

cosip := {ei,gei),

sinip := v{ei,gei)

{geSO{2,K)).

(38)

Hence the trigonometric functions cos 99, sin ip are defined as matrix elements on SO{2, K). By (37), we obtain the matrix representation t h a t is weU-known in the case i f = R (cos w — sm c^ \ . , 9 = Y sm If cos If )

r, • a = cos wJi = sm w.

In connection with the notion of angle it is common to write the group operation for the angles occurring in identification (38) additively: 9 = 9{f), f = f{9),

9{fi + f2) = 9{fi) • 9{f2), f{9i • 92) =
Having agreed on this, sin 95, cos 95 become functions on the group SO{2, with values in K satisfying the familiar trigonometric addition formulas: cos 95 + sin (/C = 1,

K)

(39)

cos(c^i + c^2) = COS (fi COS (f2 — sin c^i sin c^2,

(40)

sin(c^i + (^2) = sinc^i cosc^2 + sinc^2 cosc^i.

(41)

These relations are derived simply by applying the rules of matrix multiplication. Obviously, we also have cosO = l,sinO = 0 and cos(—95) = COS 95, sin(—95) = — sinc^.

(42)

Finally, for 0, b G S]^ we have cos95(0, b) = (0, b), sin95(0, b) = [a, b].

(43)

T h e group element (p{ei, ^2) will be denoted by 7r/2, or, expressed differently, cos(7r/2) = 0, sin(7r/2) = 1.

(44)

T h e addition formulas (40),(41) imply the familiar values cosmTT = ( — 1)™, sinmTT = 0,

m G Z.

(45) D

E x a m p l e 11. Now assume t h a t —1 is not a square in K, and consider the ^

^ 2

algebraic extension K := i f (i), i := V—T, of the field K. By V we denote the if-extension of the vector space V , cf. Section 1.10.4 or Example II.7.9.9. ~ 2

Extending the standard scalar product bilinearly to V

~ 2

~

the space [V , if, (,)]

202

2 Cayley-Klein Geometries

becomes a vector space with standard scalar product over K, which, moreover, is neutral. T h e isotropic subspaces are spanned by the vectors 3i := (ei i + e 2 ) / 2 , 32 := (e2 - Ci i ) / 2 .

(46)

By (2.25), with respect to this isotropic basis each transformation g G 0 ( 2 , K) has the representation g e SO{2, K)^^-,ek*

with (73i = 317, g^^ = 3 2 7 " S

(47)

where <; 1-^ 7 is a group isomorphism. Extending the transformations g G 0 ( 2 , K) linearly we obtain equally denoted orthogonal transformations g G 0{2,K). This leads to the canonical embeddings 0{2,K) C 0{2,K) and SO{2, K) C SO{2, k). From (38) and (46) we obtain for g G SO{2, K) 331 = 3 i e " ^ , 3 3 2 = 3 2 6 - " ^ ,

geSO{2,K),

(48)

this time using a formula, well-known for i f = R , as the definition: = cos (f + i sin (f G K*. e iv —

(49)

This is an isomorphism from SO{2, K) onto the subgroup S]^ = {zek\zz

= a^+(3^

= l}(zk*,

(50)

which has to be identified with S]^. Here z = a+i f5 ^^ z := a — i f5 denotes the canonical isomorphism of the extension K = K{i). Lastly, with the notations (27), we obtain c(e"^) = cos(^), s(e"^) = i s i n ( ^ ) . (51) Now we consider the pseudo-orthonormal basis (26), which is related to the orthonormal one (ei, 62) via Oi = ei i, 02 = e2.

Since (38) makes sense also for g G SO{2,k), we obtain an extension of the trigonometric functions from SO{2,K) to SO{2,k); thus, for all

geSOi2,k): gCi = ei cos -0 + 22 sin tp, ge2 = —ei sin -0 + 22 cos tp. Comparing the matrix of g resulting from this representation in the basis (oi, O2) with t h a t from (27) leads to a generahzation of (51): cosV(7) = ^^^'^

, sinV(7) = - i ^ ~ ^ ' ^

, ^ e k*.

(52)

Using the common definitions in the case i f = R , the relations (48) lead to equations for g G SO{2,K), well-known from hyperbolic trigonometry: cos95 =

=:cosh(ic^), i s i n c ^ =

=:sinh(ic^).

(53) D

2.4 Invariants of Finite Configurations

203

Exercise 6. Let Q C P"{V) be the quadric corresponding to the standard scalar product (,) of the vector space T^"+^ over a field K. a) Prove that Q / 0 if and only if —1 can be represented as a sum of n squares. - b) Find an example with Q / 0, such that —1 is not a square in K. Exercise 7. Find an example for a vector space V with standard scalar product over a field K with V ^ T ^ K, which has a neutral subspace W C V. (Hint. Try vector spaces over finite fields.) Exercise 8. Prove that there exist positive definite scalar products in the rational vector space V^ = Q^ for which there are no orthonormal bases. 2.4.6 P l a n e C o n e S e c t i o n s In this section we consider ellipses in a projective plane P over a field K with char K ^ 2. This is meant to be a quadric whose equation can be brought into the following normal form in homogeneous coordinates: - ( x ° ) 2 + (xi)2 + ( x 2 ) 2 = 0 .

(54)

Let (,) denote the associated symmetric, bilinear scalar product in the vector space V corresponding to P = P{V ) . It is easy to verify t h a t \y , (,)] is a vector space with scalar product of index 1. For i f = R the associated quadric Q can be identified with the unit circle S\.: Since each point a; G Q lies in the complement of the line x^ = 0, chosen as the absolute, i.e., in the Euclidean plane complementary to this line, we may suppose x^ = 1. Thus (54) becomes the equation of the unit circle. In general, however, the intersection of Q with the absolute line is not empty; e.g., in the complex projective plane this intersection consists of the points with coordinates (0, l , i ) , ( 0 , 1 , — i). T h e next proposition describes the central projection of the ellipse from one of its points onto the tangent at another point. This projection allows to introduce the projective structure of a line on the ellipse. P r o p o s i t i o n 7. Let Q be the ellipse with normal form (54) in the projective plane P^ = P{V^) over a field K with char K ^2, consider n^s e Q, n ^ s, and denote by Tn,Ts the tangents at n,s. Then the map f : X e Q ^^

y := (nV x) ATs f{n):=TnATs,

eTs,

(x ^

n), ^^^'

is a bijection from Q onto Ts • P r o o f . Because of Ts C\Q = {s} and n ^ s, the line n V y is defined for all y G Ts- Each hue A ^ Tn through n meets the quadric Q in precisely one further point x = Ar\Q ^n, and we have Tn C\Q = { n } , cf. Example 1.9.8. Hence / is bijective. Here we made use of the fact t h a t the scalar product has

204

2 Cayley-Klein Geometries

index 1, and t h a t , for this reason, Q cannot contain any hne (cf. CoroUary 2.2). D By means of the bijection (55) we now transfer a projective scale from Ts into the ehipse Q. Because of n , s G Q, n 7^ s, we may choose the representing isotropic vectors 5,n, s = [s\,n= [n], such t h a t (5,n) = —1/2. Then Oo := 5 + n, oi := 5 - n, 02 = ao x oi

(56)

is a pseudo-orthonormal basis for V^. Moreover, because of (5,02) = (n, 02) = 0 , 6 := [02] is the point of intersection of the tangents b = T „ A T g . The vectors Co : = 5 , Ci := 02 form a basis of the vector space corresponding to T s , which, by (1.1.4), determines a projective scale ^ on Ts satisfying

e(s) = 0, m = 00. An elementary calculation of the inverse function / ^ ^ leads to a parameter representation for the ellipse with the projective scale ^ as parameter. This provides a proof of Proposition 7 by direct computation:

e e i ^ ^ a ; ( e ) = [5 + 02e+ne']€g.

(57)

Exercise 9. Prove under the assumptions of Proposition 7: a) With respect to the basis (56), ip e SO{2,K) I—> [Oo + Oi cos (^ + 02 sin (^] e Q is a bijection that allows to identify Q with S]^ C £(01,02) (cf. Example 10). b) Using a) and the basis Co,Ci of the vector space associated with Ts defined above, the map (55) has the coordinate representation a: = [Oo + Oi cosip + O2 smip] e Q 1—> [Co + Ci5((^)] e Ts with i{^) = -, """"^ for <^ / TT, C(7r) = 00. ^^' 1 +cos(^ v ^ > sv ; c) Compute the inverse of (58): c2

1-r

.

(58) \ J

2g

Exercise 10. For A" = R relation (58) can also be written as i{}p) = tan((^/2). Consider the field Z3 and show that, in general, not every angle ^p G SO(2,K) can be bisected, i.e., there does not always exist a i/" G SO(2, K) such that ^p = 2ip. (Recall that the operation in SO(2, K) is written additively, cf. formulas (38)-(41).)

2.4 Invariants of Finite Configurations

205

Exercise 1 1 . Prove that 0 ( 2 , K) is never Abelian for a field K with char A" / 2. If, in addition, K is & finite field with m elements, then 0 ( 2 , K) has precisely 2m + 2 elements. Transfer the projective scale from the tangent Ts into the ellipse Q, which thereby receives the projective structure of a line. Here we may replace the tangent by an arbitrary line not containing the point n ; in fact, any two such lines are related to one another by the central projection with center n. Denoting by / the bijection generated by means of another point h ^ Q, the composition h := f o f^^ preserves cross ratios by construction. Hence it is a projectivity of the tangent Ts- For the one-dimensional projective geometry of hne pencils, cf. Section 1.6.3, this observation results in a proposition going back to J. Steiner , cf. W. Blaschke [14], S. 48: C o r o l l a r y 8. Under the assumptions of Proposition 7, let ni,n2 €z Q be two different points of the ellipse and denote by Ti, i = 1,2, the line pencils with centers rij. Then the maps F-i : X eQ,x^n-i,i—>ni

Va; G n , F i ( n i ) =

Tm,

F2 : X eQ,x^n2,i—>n2

Va; G T2, ^ 2 ( ^ 2 ) =

Tn^,

H = F2oF^^ are projectivities Q, respectively.

•.n^T2,

of the projective

geometries

defined in the line pencils or on •

Obviously, for the projectivity H we have F(TnJ=niVn2, F(niVn2)=Tn,.

(59)

Thus, according to Corollary 8, two hne pencils whose centers lie on an elhpse are projectively related via the points of the elhpse. T h e converse of this fact is the definition of the ellipse according to J. Steiner: P r o p o s i t i o n 9. Let P be a projective plane over a field K, char i f ^ 2. In P consider two line pencils T I , T 2 with centers ni,n2 G P , n i 7^ n2, and a projectivity H : TI ^ T2 of the line pencils such that H{ni V 722) 7^ n i V n 2 . Then the point set Q:={gAH{g)\gen} is an ellipse. P r o o f . Choose a suitable basis (00,01,02) for the vector space V^ of P^ t h a t is adapted to the above configuration. Moreover, let n i = [oi], n2 = [02]. Then the connecting line g^ := rii V n2 is described by the covector u^ in the dual basis:

206

2 Cayley-Klein Geometries

Consider the line §2 € n with ^ ( ^ 2 ) = flo ^^'^ set Qi := ff(flo) ^ '''2- Then we have g^ ^ g^'- Indeed, g^ = ^2 would imply g-^ = riiV n2 = g2 = flo ^ ^ ^ hence ff(flo) = floi contradicting the assumptions. Thus the point p := fli Ag2 is independent of n i , n 2 ; choose Oo as its representative, p = [oo]- So, the pencil Ti is determined by the covectors from [oi]^ = [u°,u^], and T2 by [02]^ = [u°,u^], where (u^^u^^u^) denotes the basis dual to (oo, ai, 02). Let H be the m a p induced by the linear m a p a : [u°,u'^] -^ [u°,u^]. Since H{go) = 9 i = " 2 V p : x^

=0,

we have a{u^) = u^a. T h e condition -ff (92) = 9 0 . 92 = « i V P : a;2 = 0, yields a{u^) = Uo/3. Letting in both pencils the unit lines u° + u^,u° + u^ correspond to one another, we obtain a in the form of an assignment by equal coordinates: a{u°a + u^fJ) = u°fJ + u^a. For the point of intersection, x = g A H{g) equations hold

= [p], 9 € n , the following

(u°a + v?l3){f) = x°a + x^ (3 = 0, (u°/3 + u^o){f)

= x°l3 + x^Q. = 0.

For /3 = 0 this yields x = g^ A g^ = n 2 . In case /3 7^ 0 we set /3 = 1 and, having eliminated a, we obtain the equation of an ellipse in the form (x°)2-xix2=0, which is brought into its normal form by the substitution x^ =y°

- y \

x^ =y° + y\

x° = y^.

By CoroUary 8, we thus reach every point of the elhpse.

•

Exercise 12. Using Steiner's definition of the ellipse prove that there is a unique ellipse passing through any five points of the plane P ^ in general position, if char A" / 2. Hint. Consider the line pencil TI,T2 through two of the points and use the three remaining ones to construct a suitable projectivity H : TI ^ T2.

2.5 Spherical and Elliptic Geometry

207

2.5 Spherical and Elliptic Geometry In this part we will investigate two classical real geometries, spherical and elliptic, from the projective point of view; so let the base field always be i f = R . Because of the practical applications of spherical geometry (n = 2, geodesy, astronomy, see e.g., the book, [13] by H.-G. Bigalke), and due to their distinguished position as geometries of constant positive curvature (cf. [113] by J. A. Wolf, or [108] by E. B. Vinberg (ed.)) b o t h geometries occupy a central position in the foundations of geometry. Locally, both geometries do not differ; globaUy, however, the sphere is a twofold covering of the projective space of the same dimension. This fact leads to close connections between the two geometries. 2.5.1 S p h e r i c a l as a C o v e r i n g of E l l i p t i c G e o m e t r y Elliptic geometry was already defined in Example 3.1 as the real projective geometry of a non-degenerate polarity F of index 0. So let V " + ^ be an (n + l)-dimensional Euclidean vector space, and let TT : V " + ^ -^ P " be the canonical m a p (1-1.2) onto the associated projective space. The Euchdean, i.e. positive definite, scalar product then defines the polarity F: F:

A G *P" I—> A ^ G *P".

By (1.24) we know t h a t PO{n + 1) = 0{n + l ) / { ± / „ + i } is the isotropy group of the polarity F, which, in the sense of F. Klein's Erlanger Programm, determines the elliptic geometry of ^ " . If n is even, n = 2TO, then det(—/„+i) = — 1, and this implies the isomorphy P O ( 2 m + l) = S ' 0 ( 2 m + l ) .

(1)

For what follows, however, it will be advisable to consider the non-effective actions of 0(n + 1) on the spaces and manifolds occurring in projective geometry. T h e orbits of the group 0{n + 1) for its linear action on the Euclidean vector space V are the origin o and the spheres of radius r: 5"(r):={yGF"+i||y|=r}.

(2)

In fact, not only 0{n+ 1), but also the special orthogonal group SO{n+ 1) b o t h act transitively on S*" (r), which is a consequence of the special case k = I for the following exercise: Exercise 1. Prove the following simple refinement of E. Witt's Theorem for Euclidean vector spaces: are oriented subspaces of the oriented Euclidean vector space T^", 0 < fc < n, then there is a map g e SO(n) with g(A''") = B^ that transforms an arbitrarily chosen, positively oriented, orthonormal basis of A into an arbitrarily chosen, positively oriented, orthonormal basis of B^ (cf. Corollary 1.6.2.1).

208

2 Cayley-Klein Geometries

In the sequel, let (Cj), j = 0 , . . . ,n, denote a fixed ortlionormal basis of the Euclidean vector space V " ^ . We identify the points of the n-sphere with their position vectors with respect to the chosen fixed origin of the Euclidean point space. As the origin of S^{r) we choose the point eor; then, with respect to the basis (ej) and in the usual representation by orthogonal matrices (uij) (cf. 1.6.2, (27)-(29)) its isotropy group 0{n) is described by aoo = 1, a,oj = ttjo = 0 for «, j = 1,. .., n.

(3)

The isotropy groups of all the spheres S^{r) in the Euchdean group E{n+1) coincide, they are all equal to the orthogonal group 0{n+1). Within the transitive transformation group [0(n + l),S'"(r)], the isotropy group of the point eor G S"{r) is equal to 0{n), so that all these spheres are even isomorphic as homogeneous spaces (cf. Example 1.1.5.3): 5*"^ = 0 ( n + l ) / 0 ( n ) . Here, 0{n + l ) / 0 ( n ) denotes the quotient of the group 0{n + 1) by its subgroup 0{n), cf. Appendix A.3 or Definition 1.3.1.1; in geometry, it is common to call it a quotient space. The radius r > 0 is just a parameter describing the relative size of the sphere with respect to a fixed scale. Spherical geometry is now understood to be the geometry of the homogeneous space [ 0 ( n + l ) , S^{r)] in the sense of F. Klein's Erlanger Programm. Until the end of this section we will use the abbreviations G„ := 0{n + 1), SGn := SO{n + 1). To have a clear picture, we consider an n-dimensional sphere as embedded into the Euclidean (n + l)-dimensional vector space by its very definition (2) (or, via a; = o + p, into the associated point space). Disregarding the actual value of the radius r, e.g. setting r = 1, all the following considerations remain valid for the transformation g r o u p [CT-J^, CT-J^/CT-J^_ i] as well. In the following, we will write S" := S"{1). We now restrict the canonical map TT : V"+^ -^ P " to a particular n-sphere: p := ^|s„(,) : y G 5"(r) ^ x = [T] e P " . (4) Obviously, p is surjective; each point a; = [p] G P " has precisely two preimages P^\x) = {?,-?}. In general, the pre-image p^^{A) = A O S"{r) of a A;-plane A C P " is the intersection of the corresponding, equally denoted (k + l)-dimensional subspace A C V"+ with the n-sphere S"{r); these intersections are called great k-spheres, k = — 1 , . . . , n. The spherical polarity, also denoted by F, is defined by orthogonality: F:AnS'"(r)i—>A^nS'"(r).

(5)

2.5 Spherical and Elliptic Geometry

209

Obviously, A n S ' " ( r ) C A itself is a A;-dimensional sphere of radius r, and the great sphere polar to it is an (n — A; — l)-sphere of radius r. One-dimensional great spheres are called great circles. In particular, the great 0-spheres are pairs of diametrically opposite or antipodal points; for purely formal reasons, because of F{S^{r)) = oD S^{r) = 0, we consider the empty set as a ( —l)-dimensional great sphere. On S*^ with the usual geographic coordinates, the 0-sphere consisting of North and South Pole is the polar of the equator. Exercise 2. Prove: a) The map 5 : A e * P " ^^S{A)

•.=

AnS"{r)

is a Gn-isomorphy from the lattice ^" of subspaces in elliptic space onto the lattice of great spheres contained in S"{r); moreover, the corresponding polarities satisfy F{S{A))

=

S{F{A)).

b) Extending the projection (4) by p{M) := [M] to arbitrary subsets M C S"{r) we obtain p{S{A)) = A, and S{p{M)) is the smallest great sphere containing M. c) The points pj G S'"(r), i = 0 , . . . ,fc,fc< n, are said to be in general position if they are not contained in any great (fc — l)-sphere. Given fc + 1 points in general position, there is a unique great fc-sphere containing them. T h e great A;-spheres play the part of A;-planes in spherical geometry; they are particular subspheres: A set S'^ C S^{r) is called a k-subsphere if there is a Euclidean (k + l)-plane H''+^ C E"+^ such t h a t S'' = S"{r) n H''+\ 0 < k < n, and if, in addition, S'^ contains at least two points. Thus, a subsphere is a great sphere if the Euclidean (k + l)-plane defining it contains the center of S''{r), i.e. the origin of the coordinate system. So, e.g., the circles of latitude for the geographic coordinates on a sphere S"^ are 1-subspheres, among which only the equator is a great circle. Some elementary properties of subspheres are stated in the following exercise: Exercise 3. Prove that each fc-subsphere S'' C S"(r) uniquely determines the (fc + l)-plane H^^^ generating it. It is a hypersphere in H^^^, whose center m is the foot of the perpendicular from the center o of S'"(r) onto H^^^, and whose Euclidean radius is equal to ^r'^ — c^, where c denotes the distance from o to H^^^, cf. Exercise 1.6.2.3. Obviously, an arbitrary Euclidean (fc + l)-plane cuts out a fc-subsphere in S^{r) if and only if its distance c > 0 to the origin o is less t h a n r; c = 0 just leads t o the great fc-spheres. E x a m p l e 1. The restriction of the canonical m a p p to an open hemisphere ^+(r,o):={yG5"(r)|(y,o)>0},

o ^^ o,

(6)

is injective. Identifying antipodal points in the great hypersphere bounding the hemisphere.

210

2 Cayley-Klein Geometries

Fig. 2.2. A cross-cap. 5"-^(r,o):={j:G5"(r)|(j:,o)=0},

o ^^ o,

(7)

defines a bijective correspondence with projective space; once a great hypersphere (7) is distinguished, it is frequently caUed the equator. By means of the bijection, or else considering p as a topological covering, topological properties of the projective space P " can be derived from those of the n-sphere. Locally, more precisely, restricted to any hemisphere, p is a homeomorphism, and, transferring the differentiable structure, even a diffeomorphism. This way, the real projective space P " becomes a compact n-dimensional differentiable manifold. For odd dimension n it is orientable, whereas it is non-orientable for even n (see, e.g., R. Sulanke. P. Wintgen [103], where an elementary criterion for orientability can also be found). The projective plane is thus a non-orientable, closed surface, that can only be embedded with self-intersections into the three-dimensional Euclidean space E^. Figure 2.2 shows a model of this surface that, apart from the self-intersections, also shows singularities; it is called a cross-cap. In Example 4 below we will come back to this representation. Another surface, also displaying singularities is the Roman Surface discovered by J. Steiner, Figure 2.3. A detailed discussion of various representations for the projective plane can be found in the book by F. Apery [3], which, in addition, contains numerous illuminating pictures, see also the anthology edited by G. Fischer [37]. Parameter representations and methods to generate these illustrations using the program Mathematica by S. Wolfram [114] are described in Alfred Gray's textbook [46]. Boy's Surface is a representation of the projective plane free of singularities with self-intersections in the Euchdean space E^, Fi gure 2.4. W. Boy defined and discussed it in his dissertation [20], which was stimulated by a question of

2.5 Spherical and Elliptic Geometry

211

Fig. 2.3. J. Steiner's Roman Surface.

/t'>.

^&m

^:Mm^^C. ' • • V > C^^ ••:::: V - ' . . - -

;?f ./.

Fig. 2.4. Boy's Surface.

D. Hilbert, cf. also W. Boy [21]. The edges visible in this figure are caused by the numerical approximation; in reality, the surface is smooth. D

2.5.2 D i s t a n c e a n d A n g l e The most important invariant of the orthogonal group Gn for its linear action on the Euclidean vector space and of all transformation groups derived from it is the scalar product (p, t)) determining the polarity F. Using this, the spherical distance of two points is defined as p, t) G S"{r)

I—> d{^,t)) := r arccos((p, t))/r^).

(8)

212

2 Cayley-Klein Geometries

i.e., the length of the shortest arc on a great circle through the points p, t). If these points are antipodal, t) = —p, then the distance between t h e m is independent of the chosen great circle and equal to m. Obviously, the distance d has the foUowing properties: 0 < d(?, t)) < rTT, d(j;, t)) = 0 - ^ ^ p = t), <j;,t)) = d(t),j;) = < - j ; , - t ) ) .

(9)

T h e triangle inequality, which any distance has to satisfy, will be derived below as a consequence of the law of cosines in spherical trigonometry. T h e diameter of the sphere <S'"(r), i.e. the supremum of all point distances d(f, t)), is attained for any two antipodal points: d{T,-T)

= r7v

(?G5"(r)).

Exercise 4 . Prove that two pairs (p^,pg), (tli, 1)2) SGn-congruent (Gn-congruent) if and only if

e

S"{r)

x S"{r)

are

In order to define the distance in elliptic geometry, consider again the covering (4). For each g G GL{n + 1, R ) we have g[f] = [g^] by the definition of projectivities, i.e., n is an equivariant m a p . Moreover, Relation (4) implies t h a t f> is an equivariant m a p for the action of G^. Extending this action in the obvious way to sets of A;-tuples and to power sets, g{xi,

...,Xk)

••= {gxi, ...,gxk),

(10)

g{A) := {gx\x e A},

(11)

we again obtain actions on these extensions, for which the n a t u r a l maps are equivariant. A m a p / : P " x . . . x P " -^ M , M any non-empty set, is a Gn -invariant if it satisfies f{gx-i,...,gxk)

= fix-i,...,Xk)

(12)

for all g G Gn, x^ G P " , n = I,... ,k. For other transformation groups we will use a similar terminology. Immediately from the definitions we obtain L e m m a 1. / / / : P " x . . . x P " ^ M is a Gn-invariant k-tuples of projective points, then the pull-back of the map, P*f{n. again is a Gn-invariant; conditions

on the space of

• • - , ? & ) : = / ( p ( ? i ) , • • • ,p(?fc)), moreover,

the map F '•= p*f

F(±Pi,...,±Pfc) = F(Pi,...,Pfc).

satisfies the

(13) symmetry (14)

//, conversely, F is a Gn-invariant on the space of k-tuples of points in the n-sphere S'^''(r) satisfying the symmetry condition (14), then

2.5 Spherical and Elliptic Geometry f{x-i, ...,Xk):=

- F ( j ; i , . . . , ?fc) with p(p^) = x^, K = 1,...

is a correctly defined Gn-invariant, sentatives pf, G p^^{xi^).

i.e., it is independent

213

,k,

of the chosen

(15) repreD

Obviously, the spherical distance (8) does not satisfy symmetry condition (14). Replacing, however, in (8) the scalar product by its absolute value, we obtain a function defined on S"{r) x S"{r) satisfying (14). By Lemma 1 we can thus define the distance of two points in elliptic space as follows: e{x,y)

: = r a r c c o s ( | ( p , t ) ) | ) with a; = [ y ] , y = [t)], y, t) G S'"(l)).

(16)

From (9) we immediately obtain corresponding properties for the metric 0 <e{x,y)


e{x,y)

= 0 ^=> x = y,

e{x,y)

= e{y,x);

(17)

the triangle inequality will be proved below. We equip the elliptic space with the metric defined by (16) and denote it by P'^{r). Obviously, the m a p p is no isometry^: In fact, the diameter of the elliptic space is half the diameter of the sphere of radius r: Exercise 5. a) Prove that in the elliptic space P"{r) the diameter, i.e. the supremum of the distance function, satisfies sup{e(a:, y)\ x,y e P"{r)} b) The projection (4) p : S"{r) -^ P"{r) e{p{T),Pm

= rTT/2.

satisfies:

= dij, t)) ^ ^ dij, t)) < r ^ / 2

(p, t)) e 5"(r).

c) In any space E with a metric d, let B{z,p) denote the open ball of radius p with center z: B(z,p) := {x e E\ d(x, z) 0). (18) Prove that for each open ball B{z,p) C S"{r) whose radius satisfies p < rii/A the map p : B{z,p) -^ p{B{z,p)) is an isometry, i.e., e(p(p),p(t))) = d(p,t)) for a n p , t ) e B ( z , p ) , and that this does not hold for p > rii/A. In general, a surjective m a p p : E ^ P between two metric spaces is called a local isometry if for each point z ^ E there is an open ball B = B{z, e) such t h a t the restriction q := p\B : B -^ p{B) is an isometry. It is easy to prove t h a t an isometry q is always bijective, and t h a t the inverse m a p (/^^ is also an isometry. Taking the open balls of a metric space as neighborhoods of their centers generates the metric topology, in which notions like convergence, open and closed sets etc. can be defined just as is famihar from real analysis, cf. ^ As usual, a surjective map q : Ei ^ E2 between two metric spaces is called an isometry if it preserves the distance, i.e. d2{q{x),q{y)) = di(x,y) for all x,y E Ei.

214

2 Cayley-Klein Geometries

e.g. W. Rinow [94], p. 28 ff. A very illuminating introduction into the real projective geometry of the plane and of three-dimensional space highlighting the metric aspects can be found in the book by H. Busemann and P. J. Kelly [25] In a space E with a metric d let S{z,p) denote the metric hypersphere of radius p with center z: S{z,p):={xeE\d{x,z)=p}

ip>0).

Note t h a t the Euchdean radius of a subsphere has to be distinguished from its spherical radius, which is determined by the metric d on the n-sphere: Exercise 6. Prove: a) As in Exercise 3, let S'' C S"(r) be a fc-subsphere with c > 0. Then it is contained in the metric sphere with center tTlo = tnr/|m| and radius p = r arccos(c/r):

17'= = {y e S"{r) n

H''+'\

d{j,m„) = p}.

(19)

b) If a (fc + l)-plane H^^^ satisfies the equation c = r, then H^^^ is tangent to S"(r), and the intersection S"(r) n H^^^ is the point of tangency. - c) If the subsphere is a great sphere (c = 0), then (19) holds for each point tTlo of the great sphere polar to it, and, moreover, p = r-K/2. - d) Each fc-subsphere S^ is the orbit of a subgroup conjugate to the subgroup 0(fc + l) in 0 ( n + l ) . - e) Metric hyperspheres are (n — l)-subspheres, and vice versa. Since a Euclidean vector space has no isotropic subspaces, we obtain from Proposition 4.2 P r o p o s i t i o n 2. Two finite point sequences {xi,..., Xk), {Vi,..., y^) in the elliptic space P^(r) are Gn-congruent if and only if the corresponding distances are equal: e{xi,Xj)

= e{yi,yj),

i,j =

l,...,k.

D To prove this it suffices to note t h a t the points can be represented by vectors in such a way t h a t corresponding scalar products coincide. From the properties of Gram's determinant, cf. CoroUary 1.6.3.1 or (2.33), we conclude t h a t the dimensions of corresponding subspaces spanned by the points satisfy Assumption a) from Proposition 4.1, so t h a t we may apply it. Exercise 7. Prove that a result similar to Proposition 2 holds for point sequences in the n-sphere. Hint. First prove that the hnear spans £ ( j ; i , . . . , p/j), ^ ( t ) ! , . . . , t);j), h = l,...,fc, have equal dimension, then choose suitably adapted orthonormal frames to represent corresponding vectors by equal coordinates. Using the polarity F : A^^ A we define the notion dual to the distance of two points: T h e angle between hyperplanes X, Y in the elliptic space P'^{r) is taken to be the normalized distance of their poles

2.5 Spherical and Elliptic Geometry aX,Y):=e{X^,Y^)/r.

215 (20)

T h e normalization spoils the duality somewhat, nevertheless, for historical and practical reasons, it is quite common. Since the polarity is a Gn-equivariant involution, the properties (17) of the metric immediately imply 0 < Z ( X , Y) < 7r/2, Z ( X , Y ) = 0 - ^ ^ X = Y , Z ( X , Y) = l(Y, X).

(21)

Applying the polarity we obtain from Proposition 2 C o r o l l a r y 3. Two finite sequences of hyperplanes (Xi,...,Xk) and (Yi,..., Yk) in the elliptic space P^(r) are Gn-congruent if and only if corresponding angles are equal: l{Xi,Xj)

= L{Yi,Yj),

t,j =

l,...,k.

a Exercise 5, a) implies s u p { Z ( X , Y)\X,Y

C P " ( r ) hyperplanes} = n/2.

(22)

Exercise 8. Define the angle between two great hyperspheres to be the angle between the hyperplanes in elliptic space corresponding to them (cf. (20)). Prove the statement analogous to Corollary 3 for sequences of great hyperspheres of S"{r). In contrast to elliptic geometry, in spherical geometry the notion of orientation can be defined and applied in an unrestricted way. We call a sphere S^{r) oriented if the Euclidean vector space V containing it is oriented; a spherical n-simplex (pg, • • •, ?„) is positively oriented if the volume of the cuboid formed by its vertices is positive: [Po, •••,?„] = d e t ( j ; o , . . . , ? „ ) > 0; here the determinant is to be taken for the coordinates with respect to the standard basis, which is assumed to be positive. A A;-simplex ( p g , . . . , pj,) with vertices in general position determines a great A;-sphere and, at the same time, its orientation, if we consider it as positively oriented; if ( t ) o , . . . , t)fc) determines the same great A;-sphere, then the simphces (pg, • • •, Pfc), (t)oj • • • J tJfc) are equally oriented if the determinant of the basis transformation is positive:

J2^i

det(aij) > 0 .

(23)

Obviously, this is an equivalence relation with two equivalence classes, each representing one of the two possible orientations of the great A;-sphere. If the n-sphere S"'{r) is oriented, then the polarity F defines an involution between oriented great spheres: In the image F{S''{r)) = S''^-''-\r), a generating

216

2 Cayley-Klein Geometries

simplex (t)fc+i, • • •, t)„) is to be considered as positively oriented if it spans a positively oriented n-simplex (pg,..., pj,, t)i.^i,..., t)„) in S'^''(r) together with a positively oriented simplex (pg, • • •, Pfc) in S''{r). It is easy to see that this definition does not depend on the choice of the simplices representing the orientations. The oriented great 0-spheres and, by means of the polarity F the oriented great hyperspheres as well, bijectively correspond to the points of S^{r). The pole of a great hypersphere spanned by (pg,..., p„_^) with respect to the polarity F is easily computed using the vector product to be t) = Pg X . . . X p„_ir/|pg X . . . X p „ _ i | .

(24)

Note that the vectors of a generating (n — l)-simplex are linearly independent, and compare Proposition 1.6.3.2. The angle between two oriented great hyperspheres is defined as the normalized distance of their poles: 0 < l{S"-\r), Sr\r))

:= d{F{S), F{S\))/r < n.

(25)

Example 2. A great 0-sphere consists of two antipodal points; it is oriented by prescribing their order S^ = {—f,f), i.e. by choosing p. An oriented great circle on the unit sphere S"^ in the Euclidean space V is determined by a pair (pg,pi) of linearly independent unit vectors; it decomposes S*^ into two hemispheres, one of which contains the pole of the oriented great circle

t) = ?o x?i/l?o x?il; we take this to be the positive hemisphere: IJ+it)):={TeS'\{T,t))>0}, and the opposite one, S- = —S+, as the negative hemisphere. The great circle as the polar of t) is determined by the condition S\t)) =

{TeS'\{T,t))=0};

it is oriented by a pair of its points (pg, p^) satisfying [t),?o,?i] > 0 . The definitions introduced here hold correspondingly for hyperspheres in an n-sphere. • Example 3. We define a biangle as the region in the sphere S*^ bounded by the halves of two great circles intersecting in antipodal points. Two great circles decompose the sphere S*^ into four biangles, among which the two opposite ones are congruent. The four biangles can be determined by choosing one of

2.5 Spherical and Elliptic Geometry

217

the four possible combinations of orientations for the great circles bounding them, or as the intersection of the hemispheres defined by the great circles:

Si^ n S2+, ^1+ ri S2-, ^1- ri S2-, ^1- ri IJ2+As the angle [3 of the hiangle we take the angle [3 = a between the bounding great circles if the biangle is one of the four determined by the great circles, and f3 = 2TI — a otherwise; in other words, /3, 0 < /3 < 27r, is the interior angle of the biangle. It is easy to show t h a t the angle characterizes the biangle up t o G'2-congruence. • 2 . 5 . 3 T h e L a w of C o s i n e s a n d t h e T r i a n g l e I n e q u a h t y In the notations common in elementary geometry let A, B , C be three points in general position on the sphere S'^{r). To fix the concept we will always assume the three points to be arranged into a triple. Spherical trigonometry is concerned with the search for invariants of such point triples and related construction problems. Since each triple {A, B, C) uniquely determines a twodimensional great sphere S'^{r) C 5'"(r), and since two such great spheres are always Gn-congruent, we may concentrate on the case n = 2? T h e point pairs (A, B ) , ( B , C ) , (C, A) then define three great circles, the side circles of the triple subdividing the sphere S"^ into eight triangular regions. Each of these triangular regions is the intersection of three hemispheres determined by the great circles, cf. Figure 2.5. Out of these triangular regions we assign a unique particular one to the triple A, B, C in the following way: Let Sc denote the hemisphere bounded by the great circle S^{A,B) containing C; SB and SA are defined analogously. T h e intersection SA n SB n Sc is called the Euler triangle with vertices A, B, C, cf. H.-G. Bigalke [13]. Note t h a t each of the eight triangular regions occurring in the decomposition of the sphere is an Euler triangle for a corresponding triple formed by points from the triple A, B, C and their antipodes. Exercise 9. Prove that a region bounded by three arcs of great circles in the sphere S\r) is an Euler triangle if and only if these arcs are shorter than TTT, and that this holds if and only if the interior angles of the region at all three vertices are less than TT.

In contrast to the situation in the Euclidean plane, on the sphere there are different ways to associate with a point triple (A, B, C) a triangular region whose boundary consists of arcs of great circles, AB, BC, CA. One of these regions is the complement of the corresponding Euler triangle. Again we start from the side circles of the triple {A,B,C). As the side c we choose one of the arcs AB C S^{A, B) of the great circle through A and B into which it is In the sequel, the parameter r will only be included at important places; if it is omitted, then we refer to the considerations for arbitrarily fixed r.

218

2 Cayley-Klein Geometries

Fig. 2.5. Euler Triangles. divided by these points. Let b = CA, a = BC be the other, correspondingly chosen sides of the triangle. We suppose that the closed curve ahc formed by juxtaposing these sides has no self-intersections. According to the Jordan Curve Theorem^, its complement in S"^ has two components, whose common boundary it is; they are what we called triangular regions before. We consider one of these regions as distinguished and take a spherical triangle to be a triangular region with the vertices A, B, C and the sides a, 6, c defined above. For formal reasons, it may sometimes be useful to consider a hemisphere as a triangle as well: Just choose any three different points on the bounding great circle as its vertices. To introduce orientations on the sides of the triangle and on the great circles they determine we simply take the order of the pairs (A, B), {B,C), {C,A). Furthermore, among the triangles bounded by abc we distinguish the right-hand one from the left-hand one depending on whether, moving along abc on the sphere, the triangular region lies to the right or to the left of this boundary with respect to the orientation of the sphere. The angle a of a triangle at the vertex A is understood to be the angle of the biangle containing the triangular region formed by the adjacent side circles for the sides b and c; the angles f3 at the vertex B and 7 at the vertex C are defined similarly. As usual in elementary geometry, the side lengths of a triangle will be denoted by the same letter as the sides themselves: c = d{A, B), a = d{B, C), b = d{C, A). ^ Compare, e.g., W. Rinow [94], p. 400. The proposition for the plane stated there can be transferred to the sphere S using stereographic projection or one-point compactification.

2.5 Spherical and Elliptic Geometry

219

Recall Example 1; to compute the poles of the oriented sides on S'^{r) we have to identify the points with their position vectors: AxB

,,

BxC

„,

CxA

According to Proposition 1.6.3.2 and Formula 1.(6.3.35) the norms of the vector products satisfy \Ax B\= r^ sin(c/r), \B xC\=

r^ sin(a/r), \C x A\= r^ sin(6/r).

(27)

By Definition (25), the angles at the vertices are equal to the distances of the normalized poles. In order to apply this definition we have to assume that the angles of the triangle are less than or equal to TT; for Euler triangles this was shown in Example 2. From (17) and (14) we obtain c o s 7 = - ( A ' , B ' ) A ^ c o s a = - ( B ' , C " ) / r ^ cosp =-{C,A')/r"^.

(28)

The opposite sign in these equations refers to the fact that in order to define an angle the great circles have to be oriented from the vertex towards both the other points. Now insert equations (26), (27) into (28) and apply the following well-known formula (cf. 1.(6.3.30)) from vector algebra: (Xi X X2,Yi X Y2) = {Xi,Yi){X2,Y2)

-

{XuY2){X2,Yi).

This leads to the law of cosines for the sides in spherical and elliptic geometry: The side lengths a, b, c and the angle 7 of an Euler triangle are related by cos(c/r) = cos(a/r) cos(6/r) + sin(a/r) sin(6/r) cos 7;

(29)

corresponding relations hold for the other angles and sides. From Formula (29) the metric properties of the distance functions d, e are easily derived: Proposition 4. The distance functions defined in (8) and (16) are Gn-invariant metrics on the n-dimensional sphere S'^''(r) and the elliptic space P"{r), respectively. P r o o f . By (9) and (17), respectively, d and e have all the properties required of a distance function except the triangle inequality. Since the Gn-invariance is obvious from the very definitions, it suffices to prove the following lemma for the distance function d: Lemma 5. The distance function defined by (8) and (16), respectively, both satisfy the triangle inequality d{A,B)
A,B,C A,B,C

eS'^ir), e P''{r).

(30) (31)

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2 Cayley-Klein Geometries

In (30) equality holds if and only if C is a point between^ A and B on equality holds in (31) if and only if between A and B on a line segment

A, B, C lie on a common great circle, and an arc of length d{A, B) < rn; similarly, A,B,C lie on a common line, and C lies of length d{A, B) < r-K 12.

P r o o f . Three points in general position on the sphere S"'{r) determine a two-dimensional subsphere S*^ (r), and in this they define an Euler triangle whose side lengths satisfy c = d{A, B), a = d{B, C), b = d{A, C). Because of a,b,c < nv and 0 < 7 < TT, the law of cosines for the sides implies cos(c/r) = c o s ( a / r ) cos(6/r) — s i n ( a / r ) sin(6/r) + s i n ( a / r ) s i n ( 6 / r ) ( l + C0S7), cos(c/r) > cos((a + 6 ) / r ) ) . Since c o s x is monotonously decreasing for 0 < x < TT, we have c < a-\-b, and this is Equation (30) with < instead of < . Because all distances are less t h a n or equal to r7r/2 in the elliptic plane, we find a point triple covering A, B, C in the covering sphere with the same distances. Thus, (31) immediately follows from (30). If equality holds in these inequalities, then the points are not in general position, i.e., they lie on a great circle or a line, respectively. Discussing the obvious cases separately then proves the lemma. • C o r o l l a r y 6. The side lengths a, 6, c of an Euler triangle on the sphere or in the elliptic plane P (r) always satisfy

S'^ir)

0 < a + 6 + c < 2-nr. P r o o f . Consider the complement of the Euler triangle (A, B , C) inside the biangle of the sphere determined by the sides CA, CB. It is itself an Euler triangle, namely {A, B, —C). For the lengths of its sides we have a' = 7rr — a, b' = irr — 6, c' = c. The inequality obtained from Lemma 5, c < a' + 6' = 27rr — (a + 6), is nothing but the assertion. In the case of the elliptic plane, under the covering (4) the complementary Euler triangle in the biangle. A, B, —C, corresponds to the triangle A, B, C bounded by the complementary line segments of the sides a, b in the original triangle together with its side c. T h e side lengths of this triangle are again those in the last formula above. Hence Lemma 5 can be applied. •

We do not introduce the notion of betweenness formally. On curves equipped with with a real parameter t it is always used in the sense of the order transferred from R onto the curve via the parametrization.

2.5 Spherical and Elliptic Geometry

221

P r o p o s i t i o n 7. The angles of an Euler triangle on the sphere or of an arbitrary triangle in the elliptic plane satisfy the inequalities 7 r < a + /3 + 7 < 3 7 r , a + / 3 < 7 + 7r,

a + 7 < / 3 + 7r,

(32) /3 + 7 < a + 7r.

(33)

P r o o f . Choose the orientations on the sides of the triangle so t h a t , moving along the boundary in this direction, the triangular region lies to the left. T h e angles a ' = TT — a, /3' = TT — /3, 7 ' = TT — 7 are called the exterior angles of the triangle A,B,C. Since only the angles are relevant, we may as well suppose t h a t the triangle hes on the unit sphere S"^. The poles a^,b^, c^ of the oriented sides are the vertices of the polar triangle, whose oriented sides are the polars A^, B^, C^ of the vertices in the original triangle. By Exercise 9, the polar triangle is also an Euler triangle. According to (20) the angles between the oriented sides, i.e. the exterior angles a', f3', 7', coincide with the side lengths of the polar triangle. Hence Corollary 6 implies 0 < a ' + /3' + 7 ' = 37r - ( a + /3 + 7) < 27r, and this immediately leads to (33). Now the triangle inequality applied to the polar triangle yields 7 ' < a ' + f3', i.e. the first of the inequalities (32). The remaining ones are proved analogously. Since, by Exercise 8, the angles between great circles are equal to the angles between the lines in the elliptic plane corresponding to t h e m under the covering (4), all the inequalities also hold in this case. • Exercise 10. Prove that, for r ^ 00, the above law of cosines for the sides tends to the law of cosines from Euclidean geometry, c = a -\- b

—2abcos^.

2 . 5 . 4 E x c e s s , C u r v a t u r e , a n d Surface A r e a Equality (32) leads to an essential difference between elliptic and Euclidean geometry: T h e so-called spherical excess of the triangle A, e(Z\) := a + / 3 + 7 - TT,

(34)

is always positive, whereas, as is well-known, in Euclidean geometry the angle sum is equal to n for every triangle. The latter is an immediate consequence of the parallel postulate together with some simple propositions concerning the angles formed by a pair of parallel lines and a line intersecting them, cf. Figure 2.6. P a r a l l e l P o s t u l a t e If H C E is a line in the Euclidean plane, and X £ E \ H is a point not lying on it, then there is a unique line Hi C E not intersecting H.

222

2 Cayley-Klein Geometries

This line is known as the parallel to H through x (cf. CoroUary 1.4.3.1). Obviously, in elliptic geometry as well as in spherical geometry, where great circles are taken to be hnes, there can be no parallels: Two different lines in one plane always intersect. We will encounter a different negation of the parallel postulate in the next section dealing with hyperbolic geometry.

Fig. 2.6. Angles of a Euclidean triangle. The excess can be used to define the elementary surface area in the twodimensional spherical and elliptic geometries: The surface area of an Euler triangle A in the sphere S'^{r) or in the elliptic plane P (r) is defined as the positive number F{A) := e{A)r\ (35) A subset M of a sphere or an elliptic plane will be called elementary if it has a triangulation M = [J-^ Aj, i.e. a representation as the union of finitely many Euler triangles two of which have at most boundary elements (vertices or sides) in common. The area of an elementary set is then defined to be the sum of the areas of these triangles: k

F{M):=J2F{AJ).

(36)

1

In order to justify this definition we have to prove that the sum in (36) does not depend on the chosen triangulation. First we will show that this is true for triangles. / Aj of a triangle A in the sph Lemma 8. For each triangulation A = [j--^ ere S"^ or in the elliptic plane

2.5 Spherical and Elliptic Geometry

223

/

e{A)=J2e{A,).

(37)

j=i

P r o o f . Let ej„t denote the number of vertices of the triangulation belonging to the interior of A, let Cbd be the number of vertices lying on the boundary of A (including the vertices of A itself), let kint denote the number of edges of the triangulation contained in the interior of A, and, lastly, let kbd be the number of edges on the boundary of A. Then the numbers e of all vertices and k of all edges of the triangulation satisfy

According to Euler's Polyhedral Formula (cf. Exercise 13), f-k

+ e = l.

(38)

Since each interior edge belongs to precisely two triangles of the triangulation and each triangle Aj has three edges, we obtain 3 / = 2kint + krd-

(39)

Denoting the angles of A by a, f3,7 yields / y ^ e( A') = a + /3 + 7 + eint2n + (e^d - 3)7r - fn; j=i

in fact, at the vertices of A the angles of the adjacent triangles Aj of the triangulation add up to a, [3 and 7, respectively. On the other hand, at the Crd — 3 remaining vertices on the boundary of A their sum is TT, and at each interior vertex it is 27r. Summarizing we arrive at / ^'^{^j)

= a + 13 + 1 --^{f

-icint

- e r d + 3).

j=i

Now Cint = e — Crd implies / - 2ej„t - e^d + 3 = / - 2e + e^d + 3, and because of e^d = krd a straightforward calculation using (38) leads to / - 2ej„t - e^d + 3 = 3 / - 2A; + krd + 1Taking into account k = kint + Kd equation (39) yields J ~ 2ej„( — Crd + J ^ 'Jj ~ ikint

~ krd + 1 - ^ 1 - !

this is nothing but the assertion. This lemma suffices as a justification for Definition (36):

D

224

2 Cayley-Klein Geometries

Corollary 9. The area of an elementary set M defined by (36) does not depend on the triangulation. The area is additive in the following sense: If M = Ml U M2 is the union of two elementary sets having at most boundary points in common, then F{M) = F{Mi) + F(M2). P r o o f . Let M = [][ Aj = [][ Ak be two triangulations of an elementary set M. Then we find a common refinement of these triangulations, i.e. a triangulation M = U . J, Aj^k of M such that the triangles Aj, Ak are triangulated by the triangles of the refinement:

k

j

Summing over the triangles of the respective triangulations we obtain by Lemma 8:

/

/

j=l

j=l

/ k

k=l

/ j

k=l

Multiplying by r^ proves the first assertion. Additivity immediately follows from the definition. D R e m a r k . The excess a + /3 + 7 — TT quantitatively characterizes, how much the geometry of the triangle differs from that of a Euclidean triangle with the same angles; for this reason it is also caUed the curvature of the triangle. It is the homogeneity of the sphere that permits to make it the basis for the definition of the surface area: The curvature function or Gau£ curvature of the sphere is constant, and the curvature of a region is proportional to its area. Frequently, even in elementary textbooks, cf. e.g. M. Berger [9] or H. G. Bigalke [13], the notion of surface area is based on its differential geometric definition. A synthetic definition of the surface area, based solely on the inner geometry of the surface determined by the metric can be found in Chapter X of the book [2] by A. D. Alexandrow. This book deals with convex surfaces which may even have singularities like vertices and edges; spheres are the simplest and, because of their perfect symmetry also most interesting examples for this type of surface. Discussing non-homogeneous surfaces the excess thus has to be interpreted as a measure for the curvature of a triangle instead of its area. For instance, in the case of a convex surface the excess of a triangle (bounded by shortest lines in the surface) is always non-negative. In greater detail the general relation to differential geometry is presented in M. Berger's book [9]. Using constructions developed in measure theory it is possible to start from elementary sets and introduce measures for area as well as curvature for general sets. If the corresponding surfaces are smooth, then the results coincide with those obtained by surface and curvature integrals in differential geometry. For the sphere S'^{r) and the elliptic plane P (r) the Gaufi curvature K = 1/r^ is the density of the curvature measure. It is obvious that the curvature of a sphere is greater the smaller its radius. Assume

2.5 Spherical and Elliptic Geometry

225

that the surface area of a triangle A in S*^ (r) occurring in Formula (35) is constant and let its radius r tend to infinity, then its excess e{A) tends to zero: With increasing radius the local geometry of the sphere S'^(r) approaches the Euclidean geometry. Hence it is possible to draw to scale plane maps of not too large regions of the earth's surface with acceptable precision. We want to derive a formula for the area of a convex spherical polygon. A spherical polygon P is understood to be a simply connected domain of the sphere S'^{r) bounded by finitely many arcs of great circles, its sides having at most one of their endpoints in common. These endpoints are called the vertices of the polygon. Let n denote the number of vertices of the polygon. Introducing formal vertices on one of its boundary arcs, any hemisphere or biangle can always be considered as a triangle. Hence we may suppose n > 3. To each vertex Ai of the polygon we assign the angle aj to be that of the biangle determined by the sides meeting at Ai containing interior points of the polygon arbitrarily close to Ai. The angle at a formal vertex is thus always equal to TT. The polygon P„ is called convex if for every two points A, B G P„ a shortest arc AB of a great circle belongs to P„. The following lemma contains an expression for the surface area of a convex spherical polygon:

Fig. 2.7. Proof of Lemma 10.

Lemma 10. Let P„ be a convex polygon in the sphere S'^{r) with angles ai at the vertices Ai, i = 1 , . . . , n. Then its area is FiP„)=

iy^ai-in-2)71

]r\

(40)

226

2 Cayley-Klein Geometries

P r o o f . Without loss of generality we may suppose r = 1. If the polygon is a triangle, then (40) coincides with (35). Assume t h a t (40) holds for all integers i i 3 < j
n+l n = ^ a i - (n - 1)^ = ^ a i - (n - 2 ) ^ = F ( P „ ) .

2. Assume t h a t the point An+i does not he on the great circle S. Let Sdenote the closed hemisphere bounded by S t h a t does not contain A „ + i . T h e convexity of P „ + i imphes t h a t all the other vertices of P „ + i he in S^, and, moreover, t h a t , as the intersection of P „ + i with the hemisphere S-, the polygon P„ with vertices (Ai,..., An) is again convex. Let a[, a'^ denote the angle of P „ at the vertex Ai,An, respectively. Assuming again r = 1, we conclude from (40) together with the Definitions (35) and (36) for the polygon P„, F{Pn+l)

= F{Pn) n-1

= ^

+

F{A{AuAn,An+l))

ai + a[ +a'^-

(n-

2)7r + a + /3 + a „ + i - n

i=2

n+l = ^ a i - (n + 1 - 2)7r, since the angles of P„ and A at the common vertices A\,An angle of P n + i , cf. Figure 2.7, we obtain:

add up to the

o.\ = a[ + a, a„ = a'^ + [3. On the other hand, the angle 7 of Z\ is nothing but the angle a „ + i of P „ + i . As the result we obtain Formula (40) for P „ + i . Since the proof works for every triangulation of P „ , and as the result does not depend on the triangulation but only on the invariants of the polygons in question, the proof of the lemma is complete. • R e m a r k . To compute the volume for polyhedra of higher dimension is much more involved. It seems t h a t there does not even exist an elementary formula expressing the volume of a general n-simplex, n > 3, in a space of constant curvature. See J. Bohm, H. Hertel [16]. Exercise 1 1 . Prove that the surface area of a biangle with angle a in S'^(r) is equal to 2r^a. The area of the whole sphere S^{r) is 4r^7r. Exercise 12. Define the notions of convex polygon, its angles, and its surface area for the elliptic plane P^{r), and prove that Formula (40) continues to hold in this case. The area of the whole eUiptic plane P^{r) is equal to 2r^7r.

2.5 Spherical and Elliptic Geometry

227

Exercise 1 3 . Euler's Polyhedral Formula is one of the roots of combinatorial topology. It does not only hold for triangulations in the strict sense above, but for arbitrary, even curvilinear decompositions of planar regions bounded by a simply closed curve into similar regions, where the edges are taken to be the arcs between two vertices. They are required to have at most vertices in common. Prove Euler's Polyhedral Formula (38). Hint. Taking away an edge separating two regions of the triangulation does not change the number (38). Removing a non-separating edge together with one vertex exclusively belonging to it, then the same holds. A proof can be found in H. G. Bigalke, [13], p. 310. (Note that in Proposition 10.1 of [13] the decomposition is one of the whole sphere. So the complement of A has to be taken into account to compute / , hence / —fc+ e = 2 for the decomposition of S'^. This formula holds for the numbers of faces, edges, and vertices of an arbitrary polyhedron homeomorphic to the sphere S .) 2.5.5 S p h e r i c a l T r i g o n o m e t r y In Section 3 we already proved the law of cosines for the sides and formulated the fundamental problem of spherical or elliptic trigonometry, to find invariants of triangles. Below we state several related results from spherical trigonometry as exercises. We confine the discussion to Euler triangles in the unit sphere S"^ and continue to use the notation familiar from elementary geometry. Exercise 14. Prove the law of cosines for the angles: cos 7 = — cos a cos /3 + sin a sin /3 cos c. Hint. Apply the law of cosines for the sides to the polar triangle. Exercise 1 5 . Prove the law of sines in spherical trigonometry: sin a sin o.

sin b sin j3

sin c sin 7

Hint. Prove first det(A, B, C) = sin6sincsina.

(41)

To this end, represent A, B, C in a suitably adapted orthonormal three-frame. Proving (41) also shows the first of the congruence theorems contained in the following exercise. Exercise 16. Prove the following congruence theorems : a) Two Euler triangles are congruent if and only if two corresponding sides and the angle between them coincide. b) Two Euler triangles are congruent if and only if the corresponding sides are equal. c) Two Euler triangles are congruent if and only if corresponding angles are equal. While the first two congruence theorems hold literally also in Euclidean geometry, this is not true of assertion c): In spherical geometry similar triangles

228

2 Cayley-Klein Geometries

are always congruent. We recommend to look at further formulas and propositions from spherical trigonometry in a suitable collection, e.g. I. N. Bronstein, K. A. Semendjajew [23], Section 2.6.4; try and prove them using the tools at hand; see also M. Berger [9], Section 18.6.

Fig. 2.8. Stereographic projection of P^(R)

E x a m p l e 4. The covering map (4) provided us with a picture of the real projective plane as the sphere with antipodal points identified, cf. Exercise 1 and Example 1.1.1. We obtain a representation for this plane that is well suited for detailed investigations by stereographically projecting the upper hemisphere S\ := S'^(e2)US'^(e2) including the equator S^{e2) from the South Pole s = —e2 onto the equatorial plane E : x'^ =0, cf. (6), (7). This projection is (cf. Figure 2.8) st : a; G 5*+ — I y = st{x) := (sV

x)nE.

(42)

It has the following properties: 1 The image of the projective plane is the closed unit disk; antipodal points of the boundary, i.e. the equator, are to be identified. The equator is the image of a projective line, in fact, of the polar for the North Pole n = [62]. Since the projection p is injective on the open hemisphere 5*^(62) and all lines are projectively equivalent, the complement of a line in the real projective plane is homeomorphic to an open disk. The lines through the North Pole n correspond to the great circles through 62; they are mapped to diameters of the equator by st.

2.5 Spherical and Elliptic Geometry

229

T h e lines not containing the North Pole n are m a p p e d by st to arcs intersecting the equator in antipodal points. If the projective plane is the elliptic plane, then the m a p st preserves angles: T h e angle between two lines in the elliptic plane is equal to the angles between their images in the Euchdean plane^ E.

Fig. 2.9. Euler triangles in P ^ ( R )

To prove this statement it suffices to perform the calculations in Euclidean space E^, which we leave to the reader®. For two points in the eUiptic plane (or in the elliptic space P " ) there are two shortest paths connecting t h e m if and only if their distance is equal to the diameter of the space, for the elliptic plane with r = 1, if and only if their distance is 7r/2, cf. Exercise 5. And these shortest paths again are the two hue segments into which the points The angle between two curves at a point of intersection is defined as the angle between their tangents at this point. '^ The n-dimensional stereographic projection will be discussed in greater detail in the next section, cf. Lemma 6.3, Exercise 5.4.

230

2 Cayley-Klein Geometries

divide the line joining them. In general, a shortest path is a connecting curve of least length. The length of a curve is defined in differential geometry; the definition together with the triangle inequality immediately imply that, in elliptic geometry, shortest paths have to be line segments. The cut locus of a point X is the set of all points in a metric space for which there are at least two shortest paths joining them to x. So, in the sphere S*" the cut locus of p is the opposite point —p, whereas in eUiptic space the cut locus of x is the polar F{x). In the stereographic picture 2.8 the polar of the North Pole is the equator with antipodal points identified. The two line segments leading to a point [p] = [—p] G F{n) are represented by hue segments from the center 0 e E oi the diameter determined by p to p and —p, respectively. Obviously, the stereographic projection is no isometry, the distances between pre-images and images are in general different.

^r^:/:?;?^;ii&:. ••/•-,•

"•-/••;:'-'-:-X.

^^^•5$5^ J

#

•^y^'/^/'.

^\:'^^

Wmw Fig. 2.10. A Mobius strip. Now consider three points A, B, C of the elhptic plane in general position. Their connecting lines decompose the plane into four triangles / , / / , / / / , IV, cf. Figure 2.9. Identifying, to begin with, the equator only in triangle / / produces a Mobius strip, see Figure 2.10; this is a surface frequently called onesided, i.e. non-orientable, whose boundary is a closed curve homeomorphic to a circle composed of the two arcs XY of the equator not yet identified. Performing, in addition, this identification inevitably results in self-intersections of the surface, and leads to a surface similar to the cross-cap. Figure 2.2. Applying a suitable motion we may always arrange the North Pole to represent the point C, p{t2) = C; the side lines Ay C,B\/ C then turn out to be diameters in E intersecting at C under the angle 7. The points A, B divide the side line A\/ B into two line segments with lengths c = e{A, B) < 7r/2 and c' = TT — c > 7r/2, respectively; analogously for the other side lines. Since for line segments with lengths smaller than or equal to 7r/2 the arcs of great

2.5 Spherical and Elliptic Geometry

231

circles projected onto them from 5*+ have the same length, — in fact, in this case the spherical distance coincides with the elliptic one — we always find an Euler triangle lying in S^ whose vertices represent A, B,C = n, and whose side lengths are just the distances of the vertices. If the side lengths a, 6, c are all different, or if at most one of them is equal to 7r/2, then this triangle is even uniquely determined. In Figure 2.9 / is the image of such a triangle under the stereographic projection st. If, e.g., all the distances are equal to 7r/2, then the four triangles are all congruent, and their angles are 7r/2 as weU; they correspond to the intersections of 5*+ with the four upper octants of the chosen orthonormal coordinate systems for E . Their stereographic projection leads to the simple Figure 2.11. This decomposition implies that the area of the elliptic plane is equal to 27r, hence for P (r) the area is 2r^7r. D

Fig. 2.11. Decomposition of P^(R) into four congruent triangles.

232

2 Cayley-Klein Geometries

2.5.6 T h e M e t r i c G e o m e t r y of E l l i p t i c S p a c e In this section we formulate some results belonging to the metric geometry of the n-sphere and the n-dimensional elliptic space in the form of exercises. Obviously, the set Iso{M)

:= {/ : M ^ M\f

isometry}

of isometrics of a metric space [M, p] forms a group with the composition o as operation, called the isometry group of the metric space. Exercise 17. Prove that the isometry group of the n-sphere S"{r) (the elliptic space P"{r)) is isomorphic to the orthogonal group 0(n + l) (the projective orthogonal group PO(n+ 1)). (Hint. In order to show that each isometry / belongs to the corresponding group first prove that it preserves angles. Then consider a frame of n + 1 mutually orthogonal vectors; the associated orthogonal n-simplex is transformed by / into another orthogonal n-simplex. On the other hand, there is an orthogonal transformation g e 0 ( n + 1) coinciding with / on the vertices of the simplex. Since the coordinates of an arbitrary point are determined by the distances to the vertices of the simplex, which are preserved by / as well as g, f = g follows.) Exercise 18. A metric space [Af, p] is called two-point homogeneous if it has the following property: For two point pairs (a,6), (a',b') E M x M with equal distances p(a,b) = pia!,b') there is always an isometry g e Iso{M) such that g(a) = a' and gib) = b'. Obviously, every two-point homogeneous space is homogeneous. Prove that the Eucbdean spaces E", the elbptic spaces P"{r), and the n-spheres S"{r) all are two-point homogeneous. Let [M, p] be a metric space, and let B C M be a non-empty set. For each point x G M the distance from x to B is taken to be the infimum of all distances from x to the points of 6 G B, pix,B)

•.= mf{p{x,b)\be

B}.

Exercise 19. Let now 5™ C S" be a great m-spherein S", r = 1, 0 < m < n. Prove: a) The spherical distance d from p e 5 " to 5™ C S" satisfies d(j;,5r) = arccos(|pi(j;)|),

(43)

where pi is the orthogonal projection onto the (m, + l)-plane H spanned by 5™, i.e., 0 < d{f, B) < -K/2. - b) For each point ^ e S" \ {Si U S^) there is a unique point 3 e Si with d{j,l) = d(j,Si). The great circle determined by j;,3 is orthogonal to Si and intersects Si in the 0-sphere {j,—j}. - c) Formulate and prove statements analogous to a) and b) for the elliptic space. T h e result of the last exercise, c), is C o r o l l a r y 1 1 . If B is an m-plane in the elliptic space, and x G P^{r) \B is a point not belonging to the polar of B, then there is a unique point z ^ B such that e{x,B) = e{x,z). D

2.5 Spherical and Elliptic Geometry Since in this case e{x, z) < r7r/2, the hne segment xz uniquely determined; it is called the perpendicular from x the foot of the perpendicular. Obviously, for a; G B we have Let again [M,p\ be a metric space, and let B C M be The distance set 5{B, e) of radius e is defined as 5{B,e):={xeM\p{x,B)

233

C x \/ z is also onto B, and z is x = z. a non-empty set.

= e}.

(44)

Exercise 20. In the notations of Exercise 19, let the great m-sphere 5™ lie in the (m + l)-plane H spanned by the vectors eo,.. . , Em from the orthonormal basis {tj),j = 0,... ,n. Prove: a) For e < 7r/2 the distance set S{Si, e) is the intersection of S" with the hypercone defined by m

n

( ^ ( x ^ ) ' ) cos'e - ( ^

(a;")')sin'e = 0.

(45)

7^ = 771 + 1

^ = 0

Deduce from this that the distance set is a generalized torus; in fact, as a hypersurface in 5 " it can be represented by the map h: (p,t)) eST{a)

xSi{^/l-a'^)

i—> p + t) e 5 " w i t h a = cose.

This map is a homeomorphism. Here 5™ (a) denotes the hypersphere of radius a in H, and Si{^/1 — a^) is the hypersphere of radius V l — a ' in H^. For n = 3 and m = 1 we obtain the torus as the product manifold of two circles. - b) Together with p also —p satisfies equation (45); hence the distance set is invariant under this transformation. The covering p : S" ^ P" maps it to the distance set of the m-plane p(S'5") with respect to the eUiptic metric, and the restriction of p to S{Si, e) again is a double covering. Exercise 2 1 . Prove that each of the distance sets (5(5™, e) considered in Exercise 20 can also be interpreted as the distance set of an (n — m — l)-dimensional great sphere. Which one is it, and what is the radius? Exercise 22. Consider the distance set D := (5(5™, e) C S", and determine its isotropy group Go '•= {g e Gn\gD = D}. Prove that D can be generated in two different ways by a generalized rotation of certain subspheres S: D = U^^p^j hS, where H is the subgroup of all the elements h e SGn leaving each vector in a certain subspace of the Eucbdean vector space W C \^"+^ fixed: h\W = id^y. The isotropy group Go acts transitively on D; determine the isotropy group for this action. Let [M, p] be a metric space, let B C M be a non-empty subset, and take e > 0. T h e set U,{B) := {x e M\p{x, B) < e} (46) is called a neighborhood

of radius e for B .

234

2 Cayley-Klein Geometries

Exercise 2 3 . Prove that the neighborhood U := Uc{A) of radius e, 0 < e < 7r/2, of a line A in the elliptic plane P ^ , r = 1, is a Mobius strip, cf. Figure 2.10. The complement P'^\U is homeomorphic to a closed disk. Hence, as a topological space the real projective plane is a closed surface obtained by gluing a disk to the boundary of a Mobius strip, which, in fact, is homeomorphic to a circle. Exercise 24. Consider the distance set D := (5(5™, e), e < 7r/2, of a great m-sphere Si C S". Take k := niin{m, n — m— 1}. Prove: a) Through each point p G -D there is a great fc-sphere completely lying in D. - b) For all/ > fc there is no great /-sphere lying in D. - c) For fc = 0 the 0-sphere {p, —p} is the only great sphere through p lying in _D. - d) If n = 3 and m = 1, then through each point ^ E D there are precisely two great circles in D. - e) In the cases different from c) and d) there are always infinitely many great fc-spheres through p e -D in _D. (Hint. Recall Exercise 2.4 and Equation (2.10).) In case d) of the preceding exercise let 5*2 C -D be one of the great circles through p G -D. Since 5*2 lies in the distance set D of radius e for 5*1, the distance d{t), Si) is the same for all t) G 5*2, a property characterizing the lines parallel to a given one in Euclidean geometry. Hence 5*2 is called a Clifford parallel to Si. Note t h a t this does not define an equivalence relation, because transitivity is violated: If 5*2, 5*2 are the two Clifford paraUels through p in D, then because of (i(p, 5*2 ) = 0 and 5*2 7^ 5*2, the great circle 6*2 is no Clifford parallel for 6*2. A detailed discussion of Clifford paraUels and references to the literature, in particular also to generalizations, can be found in M. Berger [9], Section 18.8. Exercise 2 5 . Let S2 be a Clifford parallel for the great circle Si. Prove that Si is a Clifford parallel for S2. (Note that the definition is not symmetric!) R e m a r k . T h e notions from metric geometry discussed in this section have far-reaching generalizations. A particularly successful development of metric geometry originated in the work on surface theory by A. D. Alexandrow [2] and his numerous collaborators. A systematic t r e a t m e n t is contained in the monography by W. Rinow [93]. Quite general results concerning the cut locus on convex surfaces can be found in the thesis by J. Kunze [71]. It is known t h a t the cut loci of points in convex surfaces do not contain closed curves. For the eUiptic plane the contrary holds: Although the local geometry is spherical, hence certainly convex, the cut locus of each point is a closed curve, in fact, as shown above, its polar. 2 . 5 . 7 A n g l e b e t w e e n S u b s p a c e s a n d D i s t a n c e of G r e a t S p h e r e s In this section we will deal with the task to find complete systems of invariants for pairs (S'f,S'™) of great spheres in the n-sphere, where great spheres of different dimension are permitted. For all A;, m with 0 < A ; < T O < n w e consider the set of all such pairs with the action of Gn induced by its action on the n-sphere, as described in (10), (11). Since the bijective relations between

2.5 Spherical and Elliptic Geometry

235

great spheres and vector subspaces as well as between these and projective subspaces are G^-isomorphisms of transformation groups, it suffices to find complete systems of invariants for the action of the orthogonal group 0{n) on the corresponding products G„,fc x Gn^m of G r a £ m a n n manifolds. To solve this problem we consider the angle between subspaces, cf. H. Reichardt, [91], § V.3. The general result for unitary spaces was already formulated in Exercise 1.6.4.5. First we discuss this case purely algebraicaUy and wiU then derive the corresponding geometric consequences. For the sake of simplicity and in accordance with the notations used in 1.6.4 we now suppose: Assumption A. Let V" be an n-dimensional real or complex unitary vector space with a linear action of the orthogonal group for K = H, and of the unitary group G := U{n) for K = C, preserving the Euclidean or the positive definite Hermitean scalar product, respectively (cf. Section 1.6.1). This action induces, as described more generally in (10), (11), an action of G over the products G„,fc X Gn^m of Grafimann manifolds: g{ljk^ VF™) = (£,t/^ gVF™), geG,

l
(47)

Let now V " = W(BW be the direct orthogonal decomposition of the vector space V " defined by the subspace W, let p r : V ^^ W be the orthogonal projection onto the first component, and let q := pr\U : U ^ W he \is restriction to the subspace t / ; moreover, lei q* -.W ^^U denote the adjoint operator (Definition 1.6.4.1) corresponding to the dual m a p . P r o p o s i t i o n 12. Suppose that Assumption A holds. For each pair of subspaces (U^W"') there is a unique self-adjoint operator a := q*oq G E n d ( t / ). Its k eigenvalues AQ, are real and, suitably labelled, they satisfy the inequalities 1 > Ai > A2 > . . . > Afc > 0. The sequence of these eigenvalues action of G on G„,fc x G„,m-

is a complete system

(48) of invariants

for the

P r o o f . Corollary 1.6.4.1 implies a* = (q* oq)* = q* o(q*)* = q* oq = a; hence a is self-adjoint. By Proposition 1.6.4.2 a is diagonahzable, and all eigenvalues are real; moreover, there exists an orthonormal basis (&«),« = l,...,k, for U consisting of eigenvectors of a. If b, |b| = 1, is an eigenvector, then 1 > A = (ab, b) = {qb,qb) > 0. The first inequality foUows from the fact t h a t , as the orthogonal projection of b, the vector qb cannot have a larger norm t h a n b. Hence, suitably labelling the eigenvalues we obtain the inequalities (48). A simultaneous transformation (47) of the subspaces by a unitary or orthogonal transformation leads t o the following relations; f>ri,qi,ai are the maps associated with the pair

(gU^gW^:

236

2 Cayley-Klein Geometries priog

= gopr,

qi = g o q o g^^\Ui,

ai = g\U o a o

g^'^\Ui.

Since g\U : U ^ Ui is unitary, the eigenvectors of a are transformed into the eigenvectors of ai with the same eigenvalues. Hence the sequence (48) of eigenvalues consists of G-invariants. We now show t h a t this system of invariants is complete, i.e., pairs of subspaces with coinciding sequences (48) of eigenvalues can be transformed into one another by a m a p g (^ G. To this end we adapt an orthonormal basis (Oj) for V " to the pair (U ^W™) so t h a t , with respect to this basis, the position of the subspaces is expressed by the eigenvalues. Then, for two pairs of subspaces with coinciding eigenvalues the adapted bases determine a unique unitary transformation mapping them, and hence the pairs of subspaces as weU, into one another. Since the basis was chosen so t h a t W = £ ( o i , . . . , 0™) holds, W is the linear span of the remaining basis vectors. Consider, as above, a fixed orthonormal basis for U'' consisting of eigenvectors of a, (bfj), K = I,... ,k,. Decompose the vectors of this basis into their components with respect to W and W : b, = qb,+q^b,.

(49)

T h e orthonormality of this basis (bfj) imphes {qbi_„qb„) = (ab^, bj.) = A^(5^j., ij,,v = I,.. .,k,

(50)

{q^bf,, q^b„) = {bf,, b„) - {qbf,, qb„) = (1 - Xf,)Sf,„.

(51)

If A^ = 1, then by (50) {qb^,qb^) = (b^, b^) = 1, and hence qb^ = b^ G U nW. Obviously, U DW is the eigensubspace for the eigenvalue 1. We set Oa := ba for a = 1 , . . . , d := dim U nW.

(52)

Let now bi3,f3 = d+1,..., s, be the vectors of the eigenbasis whose components in (49) b o t h are non-zero; hence A; — s is the multiplicity of the eigensubspace for the eigenvalue A = 0, which is the kernel of a as well as t h a t of q. Therefore, according to (50) and (51), X^ = 0,b.y = q^b.y eUn

W^

-^^

7 = s + 1 , . . . , A;.

(53)

Note t h a t the cases d = 0 or s = k b o t h are possible. For p = d + 1,... ,s b o t h components in (49) are different from zero; we normalize t h e m and set Op •.= qbp/\qbp\ e W,

am+p-d := q^bp/\q^bp\

By (53) we find additional basis vectors of W Om-d+a •= b<j G W^

e W^,p

= d+l,...,s.

(54)

by

for cr = s + 1 , . . . , A;.

(55)

T h e vectors defined in (52), (54), and (55) are orthonormal by (50) and (51). For s < m OT k — d < n — m we complete O i , . . . , Os to an orthonormal basis for W™ and O m + i , . . . , ttm+k-d to an orthonormal basis for W . This concludes the adaptation process for the basis for V^. Summarizing we have:

2.5 Spherical and Elliptic Geometry

237

1. T h e vectors O i , . . . , 0™ span the subspace W. 2. T h e vectors bi,... ,bk with the basis representations ba = fla for a = I,...

,d,

bp = ttp cosipp + ttm+p-dsinipp ioT p = d + I,...,

s,

ba = Om-d+a for 0" = S + 1, . . . , fc,

span the subspace U. Moreover, taking into account (51), the angles cpn, 0 < (/CK < vr/2, between U and W are uniquely determined by the equations {qbn, qbf,) = AK = cos^ 95K, K = 1 , . . . , A;. (56) Obviously, d, s and the angles (f^ are uniquely determined by the sequence (48) of eigenvalues, and they again determine the form of the adapted basis. Because of the usuaUy necessary basis completion as well as the multiplicities of the eigenvalues, this itself is not uniquely determined. For another pair of subspaces, {TJ ,W ), with the same eigenvalues (48) we find an orthonormal basis (Oj) for V " adapted to it, satisfying the properties 1., 2. for [U ,W ), correspondingly. It is straightforward t h a t the unitary transformation uniquely determined by ga.i = Oj, « = 1 , . . . , n, maps the pair ( t / , W) into(C/',VF™).

D

E x a m p l e 5. Now let again (S'f, S*™), 0 < A ; < m < n , b e a pair of great subspheres of the n-sphere S^,r = 1. Since the dimensions of the corresponding vector spaces exceed those of the spheres by one, we again start the labeUing of indices by zero. According to (19), (43), the quadratic form Q2{T):=cos\d{T,S^^))

= {qT,qT)

is the cosine square of the distance from the point p to the great m-sphere 5*2. T h e restriction of Q2 to the great A;-sphere 6*1 is just the quadratic form whose eigenvalues AQ > Ai > . . . > A^ form the system of invariants determined in Proposition 12. From the extremal properties of the eigenvalues, see e.g. Exercise 1.6.5.4, we conclude t h a t cpo is the minimum of the distances from the points p G 5*1 to the great m-sphere 6*2. It is equal to zero if and only if AQ = 1, i.e. if 5*1 n 5*2 ^ 0. The angles fn are the stationary values of the distance function d(f, S™) under the condition p G 6*1, and cpk is the maximal distance of points p G 5*1 to 5*2. Definition (16) for the distance in eUiptic space implies t h a t the considerations in this example can immediately be transferred to distances of points in a A;-plane to an m-plane in elliptic space. • Exercise 26. Consider two hyperspheres Si, 52 C S". Prove that there is at most one eigenvalue different from one, actually A„_i, and the corresponding angle (p„-i coincides with that between the hyperspheres as defined in (20).

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2 Cayley-Klein Geometries

Exercise 27. Consider all position possibilities for great subspheres in the dimensions n = 2 and n = 3. Discuss the occurring angles as well as distances. Moreover, investigate how the complete systems of invariants for the pairs (81,82), (8^,82), (81, 8^-), (8^, 8^-) are mutually related. Exercise 28. Consider the action of the Euclidean group E(n) on the set of pairs (H''\ M ™ ) consisting of a k- and an m-plane in the Euclidean point space E". Find a complete system invariants for this action. 2.5.8 Quadrics In Section 1.9.6 the classification of quadrics in the real projective space P " was reduced to the classification of real polar maps and hence of symmetric bilinear forms, cf. Example 1.9.2. In elliptic geometry two such forms are equivalent if and only if they can be transformed into one another by an orthogonal transformation of the associated Euclidean vector space; as is wellknown this is decided by the Eigendecomposition Theorem, cf. e.g. Section 1.6.4. The classification of quadrics in eUiptic and spherical geometry essentiaUy proceeds along the same line as in the Euclidean case. Now, however, the facts that the coordinates are homogeneous and the symmetric bilinear form is determined only up to a non-zero factor have to be taken into account. Since the covering map (4) is equivariant, we can deduce the elliptic classification from the spherical: For each elliptic quadric the spherical one that is determined by the same bilinear form is a double cover. Hence elliptic quadrics are equivalent with respect to the action of the orthogonal group 0(n + 1) if and only if their coverings are, and this again is the case if and only if the defining bilinear forms are transformed into one another by an orthogonal transformation and a multiplication by K 7^ 0. In this section we consider quadrics in S*". To do so we will work in model (2) for the spherical geometry with radius r = 1; as described before, this way the classification of quadrics in elliptic geometry is accomplished as well. Let the quadric Q be given by the symmetric bilinear form 6(j;, t)) = Sl'j^ofSijx'x^

mit Aj = fSjf,

here we suppose 6 7^ 0. The x*,« = 0 , . . . , n, are always the coordinates of the vectors representing the points in an orthonormal coordinate system for the Euchdean vector space V"+^, thus homogeneous coordinates for the points as well. The spherical quadric Q = Qb corresponding to the bilinear form b is the intersection of the cone determined by 6(j;, p) = 0 with the unit hypersphere, cf. Figures 2.12, 2.13: g = { p e F " + i | 6 f e y ) = 0,|y| = l }

(57)

In elliptic geometry we may interpret this equation as the definition for the set of representatives, uniquely determined up to multiplication by ± 1 ,

2.5 Spherical and Elliptic Geometry

239

; ;-.:-v'4

t^-

^'.i.'•.. 7-

Fig. 2.12. A spherical ellipse.

for the points of the quadric; hence together with the spherical the elliptic classification of quadrics is accomphshed as weU. As always in this transition, we just have to identify the antipodal points of the spherical quadric to pass to the corresponding elliptic quadric. Definition (57) together with the Eigendeconiposition Theorem (cf. Proposition 1.6.5.1) immediately lead to

Fig. 2.13. Intersection of the sphere with an elliptic cone.

240

2 Cayley-Klein Geometries

P r o p o s i t i o n 13. The spherical quadrics are invariant under reflections in the center of the hypersphere S*" containing them; hence Q = —Q. For each symmetric bilinear form b there is an orthonormal basis (Oj) of the Euclidean vector space V " + such that its matrix satisfies 6(aj, Oj) = I3ij = XiSij, ij

= 0 .. .,n.

(58)

The numbers Aj are the eigenvalues of the self-adjoint operator corresponding to b. By r we denote the rank and by I the index of the bilinear form b. Multiplying, if necessary, the form by —I, we may suppose 0 < I < r/2. Labelling the non-zero eigenvalues Aj in a monotonously increasing way the bilinear form defining the quadric Q satisfies (58) as well as Ao < Ai < . . . < A,_i < 0 < A, < . . . < A^_i ^ 0, A, = . . . = A„ = 0, 0
(59)

Xr-i = 1 yield a uniquely the numbers AQ, . . . , Xr-2 maps corresponding to b, geometry as well. •

In the exercises and examples to follow we want to apply Proposition 13 to the geometry of quadrics. For this we may always suppose t h a t the quadric is given by a bilinear form in the normal form described in Proposition 13. Exercise 29. Prove: a) For I = 0 the quadric is an (n — r)-dimensionaI great sphere; for r = n + 1 it is empty. Different positive values oi \p,p = 0,. .. ,r — 1, always lead to the same quadric; the example shows that non-equivalent symmetric bilinear forms may determine the same quadric. - b) If r < n + 1, i.e. if the quadric Q is degenerate, then it contains the great (n — r)-sphere S determined by the kernel of the corresponding polar map. - c) The intersection Q n S^ of the quadric Q with the great sphere polar to i7 is a non-degenerate quadric in S . E x a m p l e 6. In this example we consider a degenerate quadric Q C P " of rank r < n + 1 in elhptic space. Denote by B the vertex space of Q, i.e. the kernel of the polar m a p F corresponding to 6: B = ker F = [Or,..., a„] = Or V . . . V a „ . In other words, B is an (n — r)-dimensional projective subspace spanned by the last n — r + 1 points in a polar simplex corresponding to a normal form for Q. Denote hy A = B its polar. By Exercise 29 the intersection Q := Q r\ A is a non-degenerate quadric. If it is not empty, then by Exercise 1.9.7

g= y xyBxeQ

2.5 Spherical and Elliptic Geometry

241

in fact, B, A are complementary subspaces in P " . This describes the structure of degenerate quadrics: Degenerate quadrics are cones whose generators are the joins of the vertex space with the points of a non-degenerate quadric in the polar of the vertex space, or, in the case I = 0, the vertex space itself. • Exercise 30. Prove the result corresponding to the preceding example in spherical geometry and discuss, in particular, all degenerate quadrics in dimensions n = 2,3.

Fig. 2.14. Stereographic projection of a spherical ellipsoid.

E x a m p l e 7. Now we consider spherical non-empty, non-degenerate quadrics; so suppose / > 1 and r = n + 1. The points different from the vertex of the cone in the Euclidean vector space V determined by the bilinear form b are the solutions of the equation between two positive quantities -Ao(x°)2 - . . . - A , _ i ( x ' - i ) 2 = XiXx'f

+ ... + A „ _ i ( x ' - i ) 2 + (x")2.

(60)

Since the eigensubspaces are orthogonal, the vector space V " + ^ splits into the sum of a pair of in orthogonal subspaces: yn+i

^ ^i_

^ wi+^-',

W

= [ o o , . . . , o , - i ] , W+ = [ai,...,

o„].

This implies: Each non-degenerate, non-empty quadric in the sphere S" or in the elliptic space P " defines a pair of pole and polar assigned to the quadric Q in an invariant way, which is determined by the eigensubspaces of negative and positive eigenvalues, respectively. Since the equation (60) is homogeneous, it

242

2 Cayley-Klein Geometries

suffices to solve the equation for a common value, i.e., the equation decouples into the two following equations for separate variables: -Ao(x°)2 - . . . - A , _ i ( x ' - i ) 2 = 1, A,(x')2 + . . . + A „ _ i ( x ' - i ) 2 + (x")2 = 1. Each of these equations determines a hypereUipsoid Qin the associated subspace, and the map

V •• (?o,?i) € g - X g+ ^

C W^,QJ^

(?o,?i)/Vl?oP + l?iP e g

C WJ^

(ei)

is a bijective representation of the quadric Q C S" as the direct product of two hyperellipsoids with dimensions l — l,n — l. Moreover, it is a double cover of the corresponding quadric in elliptic space. The classification we just obtained refines the projective classification of quadrics described in Example 1.9.2; the new invariants occurring now are the eigenvalues, which, just as in Euclidean geometry, determine the metric properties of the quadric. Figure 2.12 shows the example of a spherical ellipse; this is the only type of a nonempty, non-degenerate quadric in S"^. Already S'^ eludes any direct Euclidean presentation, hence we use stereographic projection to depict examples for b o t h the types of non-empty, non-degenerate quadrics in the three-sphere in Figures 2.14, 2.15. D

Fig. 2.15. Stereographic projection of a spherical hyperboloid.

Exercise 3 1 . Prove the bijectivity of the map tp defined in (61). Complete the classification of non-degenerate quadrics for the dimensions n = 2, 3. Find geometric interpretations for the eigenvalues in all the resulting cases.

2.6 Hyperbolic Geometry

243

2.6 Hyperbolic Geometry In this section we denote by P " the real n-dimensional projective space and assume t h a t the associated real vector space V is equipped with a nondegenerate, symmetric bilinear form (,) of index one. In general, a real vector space of dimension N with such a bilinear form as scalar product is called an N-dimensional Minkowski space; as the space-time world, four-dimensional Minkowski space forms the basis for the special theory of relativity^ describing electrodynamics, cf. H. Minkowski [77]^. According to 1.9.4, see also Example 2.1.3 and Table 2.1, the scalar product has the normal form n

A basis (ej), i = 0 , . . . , n, for which the scalar product has the normal form (1), is pseudo-orthonormal; i.e., with respect to it we have (ej, efc) = e-iSik with eo = - 1 and tj = 1 for j = 1 , . . . , n.

(2)

T h e corresponding homogeneous or inhomogeneous point or vector coordinates are called pseudo-orthogonal. The isotropy group of the scalar product is the pseudo-orthogonal group 0 ( 1 , n); it transforms the pseudo-orthonormal bases simply transitively among themselves. Orthogonality with respect to (,) defines the polarity F := -Fn+i,i of the projective space; its isotropy group is the projective pseudo-orthogonal group Gn := PO(l,n) generated by the pseudo-orthogonal group, cf. (1.24). The set of isotropic vectors in V " + ^ forms the isotropic cone; the generators of the cone represent the quadric Q determined by F; it is a hyperellipsoid: x = [T]eQ^^{hT)=0,

TeV,T7^o.

(3)

Since all hyperellipsoids are projectively equivalent, we may as well use the t e r m hypersphere Q = S*"^^ C J-*"; in inhomogeneous coordinates the representation of Q is

E (x^r = 1; 1

in fact, for each solution of (3) the inequality x° ^ 0 has to hold, so t h a t we may set x° = 1. T h e isotropic cone (3) splits the vector space V into two regions: one containing the space-like vectors, V+:={TeV\{T,T)>0},

(4)

^ A very clear presentation of the mathematical foundations for the special theory of relativity is contained in P. K. Raschewski [90], Chapter IV. A reprint of this paper with a commenting supplement can be found in J. Bohm, H. Reichardt [17].

244

2 Cayley-Klein Geometries

and another one, consisting of two connected components, with the time-like vectors, cf. Figure 2.16: ^-:={?e^l(?,?)<0},

(5)

Obviously, these regions are invariant under the action of 0 ( 1 , n). Projectively, this division of V \ {o} corresponds to the decomposition of projective space into P" = A{Q)UQUI{Q). (6) Here A{Q) := TV{V^) is the outer and I{Q) := TV{V^) the inner region of the quadric Q, cf. Example 3.1. According to Corollary 3.1 the g roup (j^n acts transitively on each of the sets Q, A{Q), I{Q).

Fig. 2.16. The isotropic cone.

Definition 1. The n-dimensional hyperbolic space H" := I{Q) = 7r(V^_) is the inner region of Q; hyperbolic geometry studies the geometric properties of objects in H"" invariant under the action of the projective pseudo-orthogonal group Gn= PO{l,n). D The action of Gn on the hypersphere Q defines the Mobius geometry on the hypersphere to be discussed below, whereas, as we wiU see in Example 1, the geometry determined by the action of Gn on the outer region A(Q) may be interpreted as the geometry of hyperplanes in hyperbolic space. This section deals with hyperbolic geometry, which is frequently called non-Euclidean geometry. Historically, hyperbolic geometry originated in the

2.6 Hyperbolic Geometry

245

century-long unavailing effort to derive the parallel postulate discussed in the previous section from the other axioms of Euclidean geometry. As we will explain below, through each point in the hyperbolic plane not on a given line there are infinitely many parallels to this line. Hyperbolic geometry was invented at about the same time by C. F . Gau£, J. Bolyai, and N. I. Lobatschewski. T h e history of this discovery including some of the original papers is described in H. Reichardt [92]. The t e r m "non-Euchdean" is to a t t r i b u t e t o the strong contrast between this geometry and the Euclidean, and perhaps also to the fact t h a t no other not Euclidean geometry had been discovered earlier. The geometry on the sphere, discussed in the previous section and known long before, is also not Euclidean; in contrast to the hyperbolic, however, it is realized as the inner geometry of a surface in Euclidean space, and can thus in a way be studied within the Euclidean framework. It is a deep result from differential geometry, proved by D. Hilbert [52] in 1901, t h a t there can be no global, singularity-free realization of the hyperbolic plane as a surface in three-dimensional Euclidean space. The invention of hyperbolic geometry contradicted philosophical prejudices raising Euclidean geometry and all of mathematics based upon it to an a priori given form of our visual perception (I. K a n t ) . T h e "foundations of geometry", which can only be indicated here, were for the first time systematically, and in a way finally, presented in the book by D. Hilbert [53]. An interesting description of these connections can also be found in the contribution by W. Klingenberg [66]. As there are a lot of non-Euclidean geometries by now, it is perhaps better to follow the suggestion by F . Klein [60] (see also [92]) and use the term hyperbohc geometry. 2 . 6 . 1 M o d e l s of H y p e r b o l i c S p a c e Now we consider the hyperbohc spaces i J " , n > 0; in case n = 2 it is also called the hyperbolic plane H , and for n = 1 the hyperbolic line. For each point X G i ? " there are precisely two normalized representatives ±j:GFwith(j:,j:) = - l ,

(7)

belonging to the upper and lower sheet H C V of the n-hyperboloid defined by (j;,j;) = — 1 , respectively; they are determined by (7) and x° = -(eo,j;) > 0 or x° = -(eo,j;) < 0. Let H^ denote the upper sheet of the hyperboloid H, determined by x° > 0, cf. Fi gure 2.17, and denote by E^ '.— i i ( c i , . . . , Cn) the linear span of the basis vectors orthogonal to eo. Since for each vector p^ G E" there exists a unique vector q{fo) G H^ lying above it. q:^„eE"<—>9(?o)

: = e o V l + (?o,?o)+?oei:f+,

(8)

the m a p q is bijective; this is a parameter representation of H^ relating it homeomorphically to E". Since, on the other hand, each point x in the hyperbolic space i J " has precisely one normalized representative in H^, the map

246

2 Cayley-Klein Geometries

* =M

Fig. 2.17. The pseudo-Euclidean model.

p i y G F + ^ H ? ) :=[?]€ If"

(9)

is also bijective. Using it to transfer the topology defined by the parameter representation (8) onto H" implies: L e m m a 1. The hyperbolic space i ? " is homeomorphic to the Euclidean space E"^. The map h := p o q : E"^ -^ H"" defined by (8), (9) is a bijective parametrization of i ? " describing this homeomorphism. • T h e m a p f> defined by (9) corresponds to the equally denoted m a p (5.4), which is only a local homeomorphism. Like there, one can introduce a parameter analogous to the radius r by defining the n-hyperboloid H(r) and its upper sheet: H{r) := {y G F|(y,y) = - r ^ } , H+{r) := {y G H{r)\ - (eo,?) > 0 } .

(10)

Nothing essential will change this way. As the spheres S^{r) are orbits of the group 0{n + 1), the n-hyperboloids H{r) C V " + ^ t u r n out to be orbits for the action of the pseudo-orthogonal group 0 ( l , n ) . It is easy to see t h a t a transformation g G 0 ( l , n ) maps H^{r) to itself if and only if in the matrix of g with respect to an orthonormal basis 700 := -{zo.gzt})

> 0.

(11)

2.6 Hyperbolic Geometry

247

Since one of the transformations g, —g G 0 ( 1 , n) always has this property, and as, according to (1-24), the projective pseudo-orthogonal group is the quotient group G„:=0(l,n)/{±/„+i}, Gn is isomorphic to the isotropy group 0 ( l , n ) + of H^ in 0 ( l , n ) ; we will henceforth identify this isotropy group with Gn. In the theory of relativity the invariance of H^ means t h a t the time orientation is preserved, i.e. past and future cannot be interchanged; what appears to correspond to reality. Obviously, any transformation satisfying gco = —Co does not leave H^ invariant. T h e sign of the determinant det(gr) G Gn decides, whether g preserves the orientation of the hypersurface H^, and hence also t h a t of the hyperbolic space H", or not^. Exercise 1. a) Prove by computation: If a,6 e 0 ( l , n ) , c = aob, and -(eo,aeo) > 0,-(eo,6eo) > 0, then —(eo,ceo) > 0. - b) The relation (p, t)) < 0 is an equivalence relation on the n-hyperboloid H defined by (7); the equivalence classes are H+ and H-. Exercise 2 . Prove that the isotropy group of a point x e H" in hyperbohc space is isomorphic to the orthogonal group 0(n). (Hint. Representing the transformation g e Gn by a pseudo-orthogonal matrix with respect to the standard basis (d), i = 0,...,n, chosen above, Gn appears as the subgroup 0 ( l , n ) + of matrices (ttij) e 0 ( l , n ) satisfying aoo > 0 (Exercise 1); then the isotropy group of eo under the action of Gn is the subgroup of 0 ( l , n ) + for which aoo = 1.) Because of the transitivity of Gn on H" (Corollary 3.1) the hyperbolic space is the quotient space H" ?i^O{l,n) + /0{n). Using the bijectivity of the m a p (9) we identify H^ with H" and choose the vectorial or the projective representation, as appears convenient. T h e structures defined on H^ by the scalar product are invariant, since the scalar product has this property. In analogy to elliptic geometry, these structures may be used to describe a hyperbolic metric and trigonometry, so t h a t H^ proves to be a very suitable model for hyperbolic space; we will call it the pseudo-Euclidean model of hyperbolic space ^. T h e term "model" used here has its origin in the historic development of non-Euclidean geometry. T h e starting point was a purely axiomatic theory One can prove that the group 0 ( l , n ) has four connected components, which are characterized by the property that the transformation g preserves the time orientation and/or the space orientation or reverses them, see e.g. P. K. Raschewski [90], or (4.33) for n = 1. Using differential-geometric methods one can prove that the pseudo-Euclidean scalar product on _ff+ (r) induces a Riemannian metric of constant negative curvature - 1 / r ^ cf. P. K. Raschewski [90], § 118.

248

2 Cayley-Klein Geometries

in which ah axioms of Euchdean geometry apart from the parahel postulate were supposed to hold; the paraUel postulate was replaced by the following N e g a t i o n of t h e parallel p o s t u l a t e . Through each point x of the hyperbolic plane H and every line h not containing x there are at least two lines 91,92 C H in this plane not intersecting h, x = 9^ A 92As with all axiomatic theories, the question arises, whether the theory is free of contradictions. This question can be answered by finding a structure within a contradiction-free theory satisfying the required axioms. Such a structure is called a model for the axiomatic theory. In this sense, Euclidean geomet r y as constructed in real coordinates is a model proving t h a t the system of axioms formulated in "Euchd's Elements" is free of contradictions supposing t h a t computing with real numbers is contradiction-free. This assumption, however, is not at all proved within mathematics, more precisely: within the framework of mathematical logic and the foundations of mathematics. Because of the fundamental role the real numbers are playing in mathematics, no m a t h ematician, and all the more no physicist, will have the slightest doubt t h a t the real numbers are free of contradictions. The possibility to test their properties by measurements based on physical theories, which are formulated with their help, provides empirical evidence t h a t they indeed are contradiction-free. In this sense, we will develop the hyperbohc space H" defined as the inner region of a hypersphere or the corresponding vector set H^ as models for hyperbolic geometry, without discussing its axioms in any detail. This model goes back to F . Klein [60]. A detailed description of the axiomatic references can be found in the paper by W. Klingenberg [66]. Exercise 3 . Prove that the pseudo-Euclidean model H+ for the hyperbolic line is the upper branch of a hyperbola in V^ with the asymptotes defined by (p, p) = 0. Defining i e R I—> j;(t) := eo cosh(i) + ei sinh(i) e H+ (12) yields a bijective parameter representation of the hyperbolic line. D e f i n i t i o n 2. A hyperbolic k-plane B , 0 < k < n, is understood to be a projective A;-plane whose intersection with the inner region H" = I{Q) of the quadric Q is not empty. Formally, the empty set will also be considered as a hyperbolic A;-plane of dimension k = —1. As usual we speak of points (A; = 0), hues {k = 1), and hyperplanes (A; = n — 1). If not stated otherwise, the points or subplanes of a hyperbolic A;-plane will always meant to be the elements of the intersection B^ n H""; the points from B^ n Q are called points at infinity of B'', and those from B'' n A{Q) are caUed their outer points. The set of hyperbolic A;-planes will be denoted by Hnkn Orthogonality arguments, the considerations in Section 1.9.8, and the transitivity statements in Example 2.1 immediately lead to:

2.6 Hyperbolic Geometry

249

Lemma 2. A projective k-plane M is a hyperbolic k-plane if and only if the associated vector space W is a {k + 1)-dimensional pseudo-Euclidean subspace of the pseudo-Euclidean vector space V defining the geometry, and this again is the case if and only if the polar F{M ) lies completely in the outer region A{Q) of the quadric. The hyperbolic k-planes are hyperbolic spaces of dimension k. The group Gn acts transitively on the set Hnk of hyperbolic k-planes. • Example 1. The hyperbolic points x G i ? " correspond to the one-dimensional time-like subspaces; hence their polars x^ are projective hyperplanes lying completely in A{Q), since they correspond to Euchdean subspaces Vt^" C V"+ . On the other hand, each point in the outer region x G A{Q) corresponds to a one-dimensional space-like subspace whose orthogonal complement is pseudo-Euclidean and hence determines a hyperbolic hyperplane, and vice versa. If a; = F{X) G A{Q) is the pole of a hyperbohc hyperplane, then for each of its representatives p, a; = [p], we have the inequality (p, p) > 0; there are always two normalized representatives a; = [p] = [—p] satisfying the equation

••

\

' \

• i

\-

.-^

i

: r --./ /-. •• t~.

/

••••../

••••

X ' \ - - '

"... •••

> . - " "

•'•

.;•

•

\

. < > " • • ' • • • " . • • .

• • ' • . • \ - \ . - :

..

.'•••/~>--/N ;•••

•

\ ' ' C . . - - M . • • • : • • '•

>

:

••

..'^

*• •• • ••

fc

. /

/ /"'•:

/

S ^ / \

/

\

•

•:.;/ .::/xyM • • i.y-rvy^ vi

Li^^mM^

m^ 4' \-:

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rlk-

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.

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>'.

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,-

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Fig. 2.18. The hyperboloid {1,1) = 1.

250

2 Cayley-Klein Geometries n

(y,y) = - ( x ° ) 2 + ^ ( x * ) 2 = i

(13)

i=i

describing a one-sheeted n-hyperboloid Q i in V"^^, cf. Figure 2.18. The space A(Q) of all hyperbolic hyperplanes can be obtained from the hypersurface (13) by identifying antipodal vectors p, —p. This identification yields a correspondence between a vector p G Q i and an oriented hyperbolic hyperplane as follows: A point y G i ? " lies in the upper half-space of the hyperplane X = F{x) if its uniquely determined representative t) G H^ satisfies the inequality (t), p) > 0; the lower half-space determined by the oriented hyperplane is analogously defined by (t),p) < 0. Reversing the orientation, i.e. replacing p by —p, upper and lower half-space are interchanged. Algebraically, these orientations can also be expressed by the orientations of the bases in the associated vector spaces. Since the hyperplane X = F{x) is the polar of x, its points satisfy y = M G X ^ ^ (t), p) = 0 ,

( X = F{x),

X = [p]).

As in the Euclidean and the spherical geometries each hyperbolic hyperplane decomposes hyperbolic space into two disjoint, open regions whose common boundary it is. The space of oriented, hyperbolic hyperplanes bijectively corresponds to the n-hyperboloid (13). • Exercise 4 . a) Prove that under the action of Gn the space of oriented hyperbolic hyperplanes is isomorphic to the quotient space Qi '^O{l,n)

+ /0{l,n-l)

+.

(Proceed in a similar way as in Exercise 2). - b) Represent the sets of hyperbolic fc-planes Hn,k as quotient spaces of the group GnA second model for the hyperbohc space i J " , likewise going back to F.Klein, is obtained by normalizing the representatives [p] of its points x differently: Decompose p into its time- and space-like components with respect to the standard basis chosen at the beginning, p = tox° + pi with (pi, eo) = 0. Then by Definition 1 and (5), because of (p,p) = - ( x ° ) 2 + ( p , , p , ) < 0 and (pi,Pi) > 0 we always have x° 7^ 0, and hence h:x=

[eox° + pi] G i ? " I—> h{x) := ^Jx°

G D"

(14)

is a correct definition, i.e. independent of the chosen representative; here -D" denotes the n-dimensional open ball of radius 1 in the Euclidean vector space £;" = £ ( e i , . . . , e „ ) .

2.6 Hyperbolic Geometry I?":={t)Gi?"|(t),t))
251 (15)

Obviously, the map h is bijective (even a homeomorphism with respect to the usual topologies); its inverse is t)eD'

eo

t)]eH-

(16)

This completes the construction of the second model for hyperbohc space, which we want to call Klein's model, sometimes the name projective model is also used. It has the advantage to display the subspace structure of hyperbohc geometry particularly clearly: Think of D" as embedded into the hyperplane x° = 1, as is suggested by the representation in (16). The hyperbolic A;-planes are thus simply the intersections of D" with the {k + l)-dimensional vector subspaces defining them. Looked at from within the Euchdean space E" the hyperbolic A;-planes appear as the intersections of Euclidean A;-planes with the open baU _D". Figure 2.19 shows the relation between both models for the hyperbolic plane.

Fig. 2.19. The pseudo-Euclidean and Klein's model for H^

Example 2. In Klein's model for the hyperbolic plane H the lines appear as chords of the open unit disk. The line B in H with the parameter representation (12) is mapped by h to the diameter h{[eo cosh(t) + ei sinh(t)]) = ei tanh(t) e E"^, -OO < t < OO.

(17)

For t tending to ±oo the points move towards the unit circle ±ei. If a = [o],0 G -D^, is a point not lying on B, then the two chords through

252

2 Cayley-Klein Geometries

0 and the two points ±ei of the boundary circle do not meet the hne B; they are called the boundary parallels to B through a. They decompose the pencil of lines into two angle regions, one of them consisting of hnes intersecting B, the other formed by the lines parallel to B, cf. Figure 2.20. In the pseudoEuchdean model for H^ the line B corresponds to the upper branch of the hyperbola H^, obtained by central projection of eo + h{B) from the origin 0 G V^ onto i7+. Obviously, in a hyperbohc space i J " with n > 2 there are, moreover, skew lines, i.e. two lines not lying in a plane. In dimensions n > 2, the notions parallel and boundary parallel only apply to lines that are not skew. Hence, like before through each point a G i ? " \ B there are precisely two boundary parallels to the line B and infinitely many parallels; in addition, however, there are infinitely many lines skew to B. D

Fig. 2.20. Parallels in Klein's model for H^. In the sequel we will repeatedly need the general stereographic projection, which, from the systematic point of view, belongs to Euclidean geometry; for n = 2 we already used it in Example 5.4. To define it consider the unit hypersphere S" C £!"+^ in the Euchdean (n + l)-dimensional vector space

2.6 Hyperbolic Geometry

253

with the origin o as its center; the Euchdean scalar product will be denoted by (p, t)). Choose an orthonormal standard basis (ej), i.e. one for which (,£», Ej

"ij J * J.?

0,

The point s := — eo will be called the South Pole, and the hyperplane E" := £ ( e i , . . . , e„) defined by x'-' = 0 the equatorial hyperplane. The stereographic projection st : S" \ {s} -^ E" is defined as the map (cf. (5.42)) st:xeS"\{s}<—>y

= st{x) := (s V a;) n £;" G £;", (n > 1)

(18)

Fig. 2.21. Stereographic projection. Hence the image y of a; is the intersection of the line connecting x to the south pole with the equatorial hyperplane, schematically represented in Figure 2.21. It is obvious that the image point y moves towards infinity if X tends to the south pole s. Viewing the Euclidean space as compactified by a point denoted oo, whose neighborhoods are defined as the complements

254

2 Cayley-Klein Geometries

of closed sets in E"^, and setting st(s) = oo the stereographic projection is extended to a homeomorphism from S*" onto this one-point compactification of £;". We have L e m m a 3. The stereographic projection (18) is a homeomorphism. It maps k-suhspheres A'^ C S*" \ {s} to k-spheres of E"^; k-suhspheres containing the south pole s are mapped to k-planes in E"^. Moreover, the map is conformal. Recall t h a t a m a p is called conformal if it preserves angles. A classical result from differential geometry proved by C. F. Gau£, the "Theorema Egregium", implies t h a t there is no isometric m a p , i.e. one preserving distances, between spheres and planes; hence conformity is the best one can have. Therefore, for n = 2 stereographic projection is applied in cartography and complex function theory; in the latter it establishes the relation between the Riemann sphere and the extended Gau£ number plane. To prove Lemma 3 we recommend t h a t the reader proceeds as explained in the following exercise. Exercise 5. Let Xi,i = 0,... ,n, he the coordinates of x, and let yj,j = 1 , . . . ,n, be the coordinates oi y = st(x) with respect to the standard basis and the origin 0. a) Expressing (18) in coordinates yields St:

yj = - - ^ ,

j = l,...,n.

(19)

1 +a;o b) To obtain formulas for the inverse x = st^^{y) l + (t),t))'

compute

l + (t),t))'

"- '

Here t) denotes the position vector oi y. - c) Prove the assertion in Lemma 3 concerning the images of fc-subspheres. (Hint. First consider the case k = n—1 and use the fact that each hypersphere can be represented as the intersection of S" with a hyperplane. Let the hyperplane be described by its Hessian normal form (n, p) = p, cf. 1.(6.3.37), where n e 5 " is a unit normal vector and p,0 < p < 1, its distance from the origin. Note, moreover, that each fc-subsphere can be represented as the intersection of finitely many hyperspheres.) - d) Prove that st is a conformal map. (Hint. First consider the case n=2. At the arbitrary point x & S^ let 5i,52 be the unit tangent vectors to the meridian and the circle of latitude through this point, respectively. Compute the images of these vectors using the parametrizations of these curves by arc length s and differentiation with respect to s. Show that they are orthogonal and both are multiplied by the same factor, which only depends on the point x. For n = 2 this is nothing but conformity. For n > 2 take an arbitrary orthonormal basis Sj,j = 1,... ,n, of the tangent space TxS". Since x and two of the basis vectors Si,Sj,i / j , determine a great 2-sphere S^ C S", the conformity in this general case is a direct consequence of that for n = 2.) E x a m p l e 3. We want to describe three more models for hyperbohc space. To do so we start from Klein's model. Consider the open unit baU _D" as embedded into the hyperplane E" = £ ( e i , . . . , e„) C £!"+^ with normal vector eo and

2.6 Hyperbolic Geometry

255

take only the upper part S*", XQ > 0, of the unit hypersphere, in whose equatorial hyperplane D" lies. Then for each point x G -D" there is a unique point a;+ € S" lying above it, which is orthogonally projected to x. The hyperbohc hues correspond to semicircles orthogonally intersecting the boundary S*"^^ of D". Each hyperbolic A;-plane is generated by a Euclidean A;-plane M^ d E" intersecting the boundary S*"^^ in a A; — 1-sphere; then the only non-zero angle formed by its lift to S*" with E" is again 7r/2. The upper half S*" of the hypersphere with the structure in which A;-dimensional hemispheres are the hyperbolic A;-planes is again a model for the hyperbolic geometry; we will call it the spherical model. Applying stereographic projection to the upper half of the hypersphere S*" we obtain a further model for hyperbolic space within the open ball D", in which the hyperbolic lines are represented by semicircles or diameters orthogonaUy intersecting the boundary S*"^^. Analogously for the hyperbolic A;-planes: they are mapped to A;-dimensional hemispheres orthogonal to S*"^^. We wiU call this model, also first described by F. Klein, the conformal disk model for hyperbolic space. A fifth model goes back to H. Poincare: First we think of D" as lying in the hyperplane of the meridian determined by xi = 0, then we consider the hemisphere defined by xi > 0 with the hyperbolic structure just described, and, finally, we apply the stereographic projection st. Altogether we obtain the Poincare model: the open half-space of E" determined by xi > 0. Because of conformity and circle invariance, the hyperbolic lines appear as semicircles or half-lines orthogonal to the boundary, i.e. the hyperplane xi = 0 in E"^, and the hyperbolic A;-planes become the A;-dimensional hemispheres or A;-dimensional half-spaces orthogonal to it in the described sense. Figure 2.22 is a transformation of Figure 2.20 to the Poincare model for the hyperbolic plane.

Fig. 2.22. Parallels in the Poincare model for H^. Figure 2.23 shows a family of circle tangents in the conformal model for the hyperbolic plane. It is remarkable that each of the tangents belongs to a quadrangle of tangents in the family whose vertices belong to Q, i.e. they are points at infinity. In each of these quadrangles any two adjacent sides are boundary parallel, whereas opposite sides are parallel. To every line the family contains several paraUels. This picture clearly shows the considerable differ-

256

2 Cayley-Klein Geometries

ences to the Euclidean as well as the eUiptic geometry particularly clearly: In Euclidean geometry, for each tangent to a circle there is a unique second parallel tangent, whereas in elliptic geometry there are no parallels at all. •

Fig. 2.23. Circle tangents in the conformal disk model for H^

2.6.2 Distance and Angle Now we return to the original definition of hyperbolic space as the inner region of the quadric Q; we describe the point x G H^ by its unique normalized representative x = [j;],j; G iJ", (j;,j;) = —1. The fundamental invariant for the hyperbolic geometry of radius r is the distance of two points which, in analogy to the eUiptic distance (5.16), is defined by h{x,y)

:=rarcosh(|(j;,t))|), x= [j;],y= [t)], ? , t ) G F ^ ,

(21)

see also Exercise 6. It is easy to prove that h has the following properties of a metric:

2.6 Hyperbolic Geometry h{x, y) = h{y, x), h{x, y) > 0, h{x, y) = 0 •^=^ x = y.

257 (22)

Below the triangle inequality for h will be derived as a consequence of the hyperbolic law of cosines. For simplicity, we will from now on suppose that the radius is equal to one, r = 1. Exercise 6. a) Prove 1 < \{f,t))\ for p, t) e -ff+, p / t). Hint. Apply (2.33), (2.34) to the pseudo-Euclidean subspace [p, t)] C F"+^. - b) Prove (22). Just like Proposition 5.2 in eUiptic geometry this leads to Proposition 4. Two finite point sequences (xi,..., Xk), ( y ^ , . . . , yi^) in the n-dimensional hyperbolic space i J " are Gn-congruent if and only if the distances of corresponding point pairs coincide: h{xi,Xj)

= h{yi,yj)

for ij =

l,...,k.

P r o o f . The equality of distances imphes the equality for the scalar products of the representatives. Next we have to verify the assumptions of Proposition 4.2. Since in a pseudo-Euclidean vector space of index 1 any subspace containing a time-hke vector is not isotropic. Assumption c) of Proposition 4.2 holds. For the same reason we may apply Formula (2.33) for Gram's determinant, implying Assumption a) (of Proposition 4.1). Finally, because of K = 1, Assumption d) is trivial, and this proves the assertion. D Example 4. Let us now consider two hyperbolic hyperplanes X, Y, Xr\I{Q) ^ 0, Yr\I{Q) ^ 0; if there is no danger of confusion, the last two conditions will not be mentioned explicitly, i.e., we will just speak of hyperbolic hyperplanes X,..., or, more generally, hyperbolic A;-planes, whenever these conditions are satisfied. The polars of these hyperplanes x = X ,y = Y are always points in the outer region, whose normalized representatives ±p, ±t) are determined up to sign and lie on the one-sheeted hyperboloid Qi, cf. (13), Example 1. If the hyperbohc hyperplanes X,Y are oriented, then their normalized representatives are uniquely determined, and so we have a uniquely determined invariant: I{X, Y) := (p, t)), [p] = X ^ , [t)] = Y ^ , p, t) G Qi. For non-oriented hyperplanes X,Y the invariant

(23)

we have to do without the sign and obtain

| / | ( X , Y ) : = | ( p , t ) ) | , [p] = X ^ , [t)] = Y ^ , p , t ) G g i .

(24)

Using representing vectors which are not normalized we have the following general formulas

/(X, Y) = ^ ^^'^^ Now suppose X ^Y

, |/|(X, Y) = J^^'^^'

V(?,?)(t),t)) V(?,?)(t),t)) and distinguish three cases:

.

(25)

258

2 Cayley-Klein Geometries

1. | / | ( X , Y ) < 1. This holds if and only if p, t) span a Euclidean subspace, and this again is satisfied if and only if X A Y = (a; V y ) ^ is a pseudo-Euclidean subspace, i.e., the hyperplanes intersect in a hyperbolic (n — 2)-plane X HY. T h e angle a between the intersecting hyperplanes is then uniquely determined by cos a = I{X,

Y),

cos a = \I\{X,Y),

0 < a < TT, ( X , Y oriented), 0
7r/2, {X,Y

not oriented).

2. | / | ( X , Y) = 1. This holds if and only if p, t) span an isotropic subspace, and this again holds if and only if X A Y = (a; V y ) ^ is an isotropic subspace, i.e., the hyperplanes intersect Q in a tangent (n — 2)-plane X DY; X,Y are said to be boundary parallel. 3. | / | ( X , Y ) > 1. This holds if and only if p, t) span a pseudo-Euchdean subspace, and this again holds if and only if X A Y = (a; V y ) ^ is a Euclidean subspace, i.e. X A Y is an (n — 2)-plane lying completely in the outer region; X , Y are said to be parallel hyperplanes. Obviously, for two pairs ( X Q , , Y Q , ) , X Q , y^ Ya,a = 1,2, of different hyperplanes to be Gn-congruent equality of the invariants | / | is necessary. To prove its sufficiency one can proceed as follows. Starting from the equality of the scalar products for the normahzed representatives of the poles one finds an isomorphism between the two-dimensional subspaces spanned by them. Then one applies E. W i t t ' s Theorem or an elementary construction for adapted pseudo-orthonormal bases to show t h a t equality of the invariants | / | ( X i , Y i ) = | / | ( X 2 , Y2) implies Gn-congruence of the pairs. In particular, two boundary parallel pairs are always G„-congruent, whereas, just like in Euclidean geometry, two parallel hyperplanes have an invariant. • Exercise 7. a) Find for each of the cases in Example 4 and every n > 0 pairs of hyperbolic hyperplanes satisfying the corresponding conditions. - b) Explain the elementary construction mentioned in the final paragraph of Example 4 in detail. - c) Show by adapting the frames that two pairs of boundary parallel hyperplanes (case 2 of Example 4) are always Gn-congruent. E x a m p l e 5. Let X , Y be two hyperbolic half-lines starting from their point of intersection a G H"". In accordance with Equation (12) let these half-lines in the model i 7 " be described by b(t) = ocosh(t) + b i s i n h ( t ) , t > 0, c(s) = ocosh(s) + Ci sinh(s), s > 0, where 0, b(t), c(s) G HI, - ( 0 , 0) = (bi, bi) = (ci, ci) = 1, (0, bi) = (0, ci) = 0. Since tii,Ci are orthogonal to the time-like unit vector 0, they span a Euclidean subspace; moreover, because of the bijectivity of Equation (8) they

2.6 Hyperbolic Geometry

259

are uniquely determined space-like unit vectors. As the angle between the halflines with intersection point a we take the number uniquely defined by cos a = (tii,Ci),0 < a < n. We will show that this definition is compatible with that of Example 4, Case 1. To see this, consider the hyperbolic plane spanned by a and the two halfhnes. The corresponding pseudo-Euchdean subspace W^ is that spanned by 0, bi, Ci. We take it to be oriented, so that a vector product is defined. The orthogonality relations imply that oxbijOXCi are normalized space-like unit vectors representing the poles of the lines X, Y, which are oriented by the parametrizations above. From Formula (2.37), in which we have to set a = det((ej,ej)) = - 1 , we obtain I{X, Y) = (o X bi, 0 X ci) = - ( o , o)(bi, ci) = (bi, ci). In fact, by construction we have (o, bi) = (o, Ci) = 0.

D

Example 6. Next we want to prove that the spherical, and hence Klein's and Poincare's models for hyperbolic space as well, are conformal. Since a proof relying solely on linear algebra would, on the one hand, be too complicated, and, on the other hand, just produce particular results, this time we will apply differential-geometric methods. Consider the upper sheet H^ of the hyperboloid representing H" defined by (7) and x° > 0, and take the coordinates X*, i = 1 , . . . , n, as parameters. Then the parameter representation of this hypersurface is p : (x*) G I?" ^

y(x*) := ( \

l+

J2ix')\i;\...,x-')€H^,

where we inserted the following consequence of (7)

l + J2{a

x° \

The metric induced by means of the differential df from the pseudo-Euclidean scalar product in the tangent spaces of H^ is positive definite because of (?,?) = - 1 , (?,t^?) = 0 . The Riemannian metric it defines is described by the Gn-invariant arc element _. {dl,di)

n

= ^-^((x°)2^(dx^-)2 -

n {Y^xUx^f).

260

2 Cayley-Klein Geometries

Using db • , , , -— = 0 smii t + b 1 cosii t at and the corresponding equation for the derivative of c(s) we obtain the following formula for the angle between the lines in Example 5 at their point of intersection, o, i.e. for s = t = 0, expressed in this arc element (§(0),^(0)) = (bi,ci)=cosa. at as This is what we already computed before. Now consider the spherical model S*" for i 7 " described in Example 3; performing the computations according to the construction, the image of j;(x*) in this parameter representation is described by x^ The arc element of the hypersphere S" C £!"+^ is to be computed starting from the Euclidean scalar product (,). Differentiating the last equation and taking the scalar product yields 1 (dps,dps) = ^-g-2-(dy,dy). This shows the conformity of the m a p p G H" i—> pg G S*". According to Lemma 3 the stereographic projection is also conformal, hence the composition of these conformal maps immediately leads to: The conformal disk model and Poincare's model for the hyperbolic spaces are angle-preserving images of hyperbolic geometry. • E x a m p l e 7. Consider the polar coordinates u, v on the hyperbolic plane; they are the parameters in the representation P(M, V) := Co cosh M + (ei sin v + e2 cos w) sinh u of the pseudo-Euchdean model H'^. Figure 2.24 shows the images of the orthogonal coordinate lines in Poincare's model. In the Klein's model the cartesian coordinates x, y determine a net consisting of hyperbolic lines, which are not orthogonal in the hyperbolic sense. Figures 2.25 and 2.26 show this net in the conformal disk and Poincare's model, respectively. •

2 . 6 . 3 D i s t a n c e a n d A n g l e s as C r o s s R a t i o s In the previous section we proved t h a t the hyperbolic space is two-point homogeneous, i.e. for any two point pairs in H"" there is a transformation g G Gn mapping one into the other if and only if they have the same distance; it is the distance alone t h a t determines t h e m up to congruence. On the other hand.

2.6 Hyperbolic Geometry

261

Fig. 2.24. Polar coordinates in Poincare's model for H^.

Fig. 2.25. Cartesian coordinate lines x,y in the conformal disk model for H^

we can assign a projective invariant to any pair of different points in hyperbolic space in a natural way: In fact, if XQ, J/Q are the points of intersection of the connecting line xV y with the quadric Q, we can form the cross ratio (CR) {x,y;xQ,yQ), which stiU depends on the order of the points. Since it is a projective invariant, it appears natural to ask, whether there is a relation between this C R and the distance of the points. Already in 1859 A. Cayley

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2 Cayley-Klein Geometries

Fig. 2.26. Cartesian coordinate lines x,y in Poincare's model for H

[29] studied this problem quite generally, and later F. Klein [60] resumed the discussion. P r o p o s i t i o n 5. Let x ^ y he two points in the hyperbolic space H^(r), and let XQ, DQ he the points of intersection of the connecting line xV y with the ahsolute Q. Then the distance of the points is related to the cross ration hy h{x,y)

=

-\\n{x,y;xQ,yQ)\.

(26)

P r o o f . First we note t h a t the right-hand side of (26) does not change, if x and y or XQ and J/Q are interchanged, since by (1.4.7) the C R is replaced by its inverse. For the proof of the proposition we pass to the complex extensions of b o t h the vector space V " + ^ and the scalar product (,). Now we prove a more general lemma allowing for more applications. L e m m a 6. Let P^ he the complex orthogonal space with the non-degenerate quadric Qc (Example 3.2j. Let the connecting line x V y of the two points x,y G P c \ Qc he a secant with the points of intersection {XQTVQ} = (aJ V y) n Q c (Example 1.9.8J. Then hy (4.17) the invariant Sq{x, y) G C is defined; the solution set of the equation Sq{x, y) = coi? LP, ^ G C,

(27)

is n = {±c^o + kT^l k G Z } , where cpo (^ C is the solution that is, up to sign, uniquely determined hy (27) and the condition 0 < Re(c^o) < TT. The numher

k{x,y)

:= |Im(^)|

does not depend on the choice of cp (^ il, and, Hx,y)

=

(28) moreover,

-\ln\{x,y]XQ,yQ)\\.

(29)

2.6 Hyperbolic Geometry

263

The real part of f and the argument a of the CR satisfy a{x, y) := arg(a;, y; XQ, yg) = 2 Re(^) mod 27r.

(30)

P r o o f . That i? is the solution set and the statement concerning (28) follow from the properties of the complex function cosz. To prove (29) we first compute the parameter on the line x V y for the points of intersection XQ,yQ. Since x,y do not belong to Q c we find normalized representatives p, t), a; = [p], y = [t)], (p, p) = (t), t)) = 1. The ansatz }{t) := pt + t) G Q c leads to the quadratic equation {i{t),i{t)) = 0 with the solutions tl,2 = - ( p , t ) ) ± V ( ? , t ) ) 2 - l .

(31)

Note that ti 7^ t2- In fact, if this were not the case we had (p, t)) = ± 1 ; the line xV y would then correspond to an isotropic vector space and hence be a tangent to Q c with the point of contact determined by ti = t2From Formula (1.4.16) we obtain for the CR {x,y;z{ti),z{t2))=t2/ti.

(32)

Choosing an arbitrary complex solution cp for (p, t)) = cosc^, inserting it into (31) and the result into (32) a simple computation yields {x,y;xQ,yQ)=e^''^.

(33)

When XQ and yq are interchanged, then according to (1-4.7) the CR is replaced by its inverse, i.e. cp by —cp. Hence cp is only determined up to sign by x,y. We now write the CR in trigonometric form as p e ' " and use the decomposition

^ =

^ix,y)+ivix,y)

of cp into real and imaginary part. Inserting this into (33) leads to the formulas (29) and (30) of the lemma. D We conclude the proof of Proposition 5. Since the connecting line a; V y C H" is a secant of the real quadric Q, the solutions ti,t2 from (31) are real and different; hence we have cos'^i^) = Sq{x,y)

= (p,t))2 > 1 .

Thus cpo is purely imaginary, cpo = i'lp, and by (33) the CR is positive. Taking into account (21), (28), (29) with h{x,y) = rk{x,y) we conclude from (33) the assertion of Proposition 5. • We return now to elliptic geometry and again consider the extension of the elliptic space to the complex orthogonal projective space P^ with the bilinear extension of the scalar product. Because oi x ^ y the normahzed representatives satisfy (p, t))^ < 1, and hence the solutions (31) are complex conjugate. Thus in equation (33) for the CR ip is real, and Definition (5.16) (with r = l ) for the distance in elliptic geometry implies

264

2 Cayley-Klein Geometries

Corollary 7. For the distance of two points x, y in the elliptic space with parameter r the following equation holds e{x, y) = 2 I ln(a;, y; XQ, yq)! = r|^|.

(34)

Here XQ, I/Q denote the points of intersection of the complex extension {x\/y)c for the connecting line with the absolute Qc of the complex extension for the elliptic space. The logarithm is meant to be the principal value of the complex logarithm. • By Definition (5.20) the angle between two elliptic hyperplanes is reduced to the distance of its poles; hence there is a corresponding formula for this as well. The duality of points and hyperplanes, lines and hyperplane pencils with an (n — 2)-plane as support, auto-polar points XQ G Q C and tangent hyperplanes leads to Corollary 8. The angle of two hyperplanes X, Y in the elliptic space satisfies

Z(X, Y) = i | ln(X, Y; X Q ,

YQ)\

= \^\.

(35)

Here XQ, YQ denote the tangent hyperplanes for the complex extension of the hyperplane pencil with support {X A l^)c to the absolute Qc for the complex extension of elliptic space. The logarithm is meant to be the principal value of the complex logarithm. • E x a m p l e 8. Corollary 8 correspondingly also holds for the angle of two intersecting hyperplanes X, Y in hyperbolic space; in fact, according to Example 4, Case 1, the poles of these planes span a line in the outer region of the quadric, whose complex extension again intersects the complex quadric Q c in two complex conjugate points, the polars of which are the (complex) tangent hyperplanes to Q c . If the hyperplanes are parallel but not boundary parallel (Case 3 of Example 4), then their intersection X AY lies in the outer region A{Q) of the quadric; hence its polar ( X A Y)^ = xV y, x = X ,y = Y ,is a secant of the quadric, where now x, y are outer points. Since the CR is real again, the angle cp = i'ip,'ip (^ H, in (33) has to be purely imaginary. Thus the CR is positive and determines via IVI = i | l n ( X , Y ; X Q , Y Q ) | = arcosh | / | ( X , Y)

(36)

a kind of distance for the parallel hyperbolic hyperplanes. This notation is generalized and justified in Proposition 25. • E x a m p l e 9. In the pseudo-orthonormal standard basis we consider the vectors n(t) := eocos(t) + eisin(t). The corresponding points n(t) = [n(t)] oscillate back and forth on the line segment between the inner point [eo] and the outer point [ei] of the quadric Q. The polar hyperplanes N{t) = n{t)^ satisfy:

2.6 Hyperbolic Geometry

265

1. For 0 < t < 7r/4 the vector n(t) is time-like, hence the polar is an outer hyperplane; the scalar product restricted to its vector space is positive definite and determines an elliptic geometry in the corresponding projective hyperplane. 2. For 7r/4 < t < 7r/2 the vector n(t) is space-hke, hence the polar is a hyperbolic hyperplane; the scalar product restricted to its vector space is pseudo-orthogonal of index 1 and determines a hyperbolic geometry in the corresponding projective hyperplane. 3. T h e point p := n(7r/4) hes on the quadric Q, and its polar T := N{-K/A) is the tangent hyperplane to Q at this point. The polar is spanned by T = [oi, 0 2 , . . . , 0„] with oi := eo + ei, Oj = ej for « = 2 , . . . , n . T h e bilinear form induced on the associated n-dimensional vector space W^ is positive semidefinite of rank n — 1 and obviously symmetric. W i t h respect to the basis {cij),j = 1 . . . , n, it has the coordinate representation (j;,t)) = x 2 y 2 + . . . + a;"y". The next exercise contains some properties of the geometry induced on a tangent hyperplane. • Exercise 8. Prove under the assumptions and using the notation of Example 9, 3: a) The isotropy group Gj- of T under the action of G„ coincides with the subgroup of the projective group preserving the polar map determined by the scalar product (, )| j - x T - ^ ^^ '^^^ point of contact {p} = Q DT is fixed for the action of Gj-, and Grp acts transitively on the complement T\{p}. - c) The group Gj- acts transitively on the bundle Tfc(p):={B'=cT|peB"} as well as on the set

Mk{p):={B''cT\p<^B''} of fc-planes not containing p. - d) Let a: / j / be two points of T different from p. Then by (4.17) their invariant Sq{x,y) is well-defined; two such points are Gy-congruent if and only if these invariants coincide. The points x,y lie on a line through p if and only if Sq{x,y) = 1. — The geometry discussed in this exercise is the elementary example of an isotropic geometry. Isotropic geometry was studied and applied by Karl Strubecker in various ways. A complete list of references together with a recognition of his merits can be found in the obituary by K. Leichtweifi [74]. E x a m p l e 10. We still want to show how the Euclidean geometry fits into the scheme described here. To this end, we start from the elliptic geometry, i.e., we consider a real vector space V " + ^ with the positive definite scalar product (,), the orthonormal standard basis {Cj),j = 0 , . . . , n, and the associated elliptic space P " . Let W" := [ e i , . . . , e„] denote the subspace spanned by the last n basis vectors, and let p^ G W" denote the orthogonal projection of

266

2 Cayley-Klein Geometries

p = eoa;° + Po G V"+^ onto this subspace. We now define the scalar product over V depending on the parameter t G R: (?, t))t := -{x°y° cost + (p„, t)„) sint).

(37)

For 0 < t < 7r/2 the corresponding quadric Qt : (?,?)* = 0 is imaginary; hence the geometry is eUiptic, and for —7r/2 < t < 0 it is a real elhpsoid, whose inner region is a model for hyperbolic geometry. For t = 0 the quadric degenerates into the "double counted hyperplane" (x°)^ = 0 with the vector space W". We choose this as the (improper) hyperplane at infinity of an afhne space A " C P " . Since for t ^ 0, t > 0, the factor sint cancels for the points in this hyperplane the invariant

V{To,To)t{^o,^o)t

actually does not depend on t, and defines the angle cp between the vectors in the usual way. The improper hyperplane is thus equipped with an elliptic geometry in a natural way. Its group is the orthogonal group 0(n) acting on W", and this preserves the distance defined for the proper points as usual: Consider their representatives p, t) G V^ normalized by x" = y" = 1, then p — t) G W", and 0{n) as weU as the group of translations indeed preserve p{x,y)

:= V ( ? - t ) , ? - t ) ) .

We thus have the analytic model for Euclidean geometry, which in this context, because of its bordering position between hyperbolic and elliptic geometry, is also called parabolic geometry. According to O. Giering [44] the absolute of Euchdean space consists of the degenerate quadric (x°)^ = 0 together with a given empty quadric Q c in its vertex space (its "cusp"), which thus lies in the complex projective space belonging to the complex extension W^. Considering, at the same time, real projective spaces and their complex extensions he widely generalizes this scheme in the notion Cayley-Klein geometries. The classification he accomplishes in low dimensions is based on the classification of real and complex polarities. • 2.6.4 The Hyperbolic Law of Cosines and Hyperbolic Metric Let A,B,C G i ? " be three hyperbolic points in general position. For the purposes of trigonometry we may again suppose n = 2. Represent the sides of the triangle starting at A as in Example 5, where the pseudo-orthonormal three-frame is suitably adapted: Set ^ = [o] = [eo], B = [b] = [eocoshc+ei sinhc],

(38) (39)

C = [c] = [eo cosh 6 +1) 1 sinh b], f 1 = ei cosa + e2 sina.

(40) (41)

2.6 Hyperbolic Geometry

267

Equation (41) is a consequence of the fact t h a t tii is orthogonal to eo; according to Example 5 the angle a. of the triangle at the vertex A, 0 < a < TT, is uniquely determined by cos a = (ei,t)i), and t^ is uniquely determined by (41); more generally, the three-frame is uniquely determined by the triangle via (38)-(41). Here, by (21) the numbers 6, c are the lengths of the sides in the triangle, which, as usual, are denoted correspondingly: h{A, B) = c, h{A, C) = b, h{B, C) = a. Note t h a t the scalar product of two vectors from i 7 " is always negative, more precisely, we have (o, b) < - 1 for all 0, b G F ^ , a^

b.

(42)

In fact, we may always choose the orthonormal vectors eo, ei so t h a t (38) and (39) hold for A= [a], B = [b], which implies (o, b) = - coshc < - 1 . Now the side length a is easily computed from (21), (40), and (41): (b, c) = — c o s h a = — c o s h c c o s h 6 + (ei,t)i) s i n h c s i n h 6 . Moreover, (41) implies the hyperbolic law of cosines P r o p o s i t i o n 9. Consider the notations for a triangle common in elementary geometry. Then, in the hyperbolic geometry of radius r = 1 the length of the side a opposite to the vertex A satisfies cosh a = cosh b cosh c — sinh b sinh c cos a.

(43) D

Exercise 9. Prove that formula (43) also holds for degenerate triangles (a = 0,7r). C o r o l l a r y 10. The hyperbolic distance satisfies the triangle inequality for any three points A,B,Ce H^: h{B,C)
+ h{A,C)]

equality holds in (44) if and only if A lies on the hyperbolic line B\/C B and C. Hence by (22) h is a Gn-invariant metric on i J " . The group of this metric is the group Gn.

(44) between^ isometry

P r o o f . Because of —cosa < 1, Equation (43) is an immediate consequence of the addition formulas for cosh c o s h a < cosh(6 + c); c o s h a = cosh(6 + c) •<=^ a = n. We do not introduce the notion of betweenness formally. On curves equipped with with a real parameter t it is always used in the sense of the order transferred from R onto the curve via the parametrization.

268

2 Cayley-Klein Geometries

The last condition, however, just expresses the position of the points asserted in the corollary. We now prove the last assertion. If gr is a projective transformation preserving the metric, then any linear map generating it has to be conformally pseudo-orthogonal; by the definition of Gn we may suppose that this linear transformation belongs to Gn- To prove that every isometry of i J " is generated by a map g G Gn we introduce an orthogonal coordinate system as foUows: Set Oo := eo, Oj := eo %/2 + ti for i = 1, Obviously, the edges of the simplex starting at ao = [oo] are orthogonal, the Oj are normalized representatives of its vertices Oj = [Oj] G H"", and by (21) the distances of the vertices can be computed from (oo, Oj) = - V 2 for i > 0, (Oj, Oj) = - 2 for i,j > 0,iy^ j . If, conversely, a o , . . . , a„ are n + 1 points whose normalized representatives Oj satisfy these equations, then eo := ao; Ci := a^ - OoA/2for i = 1,. uniquely determines a pseudo-orthonormal frame (ej), i = 0 , . . . ,n. We call such point sequences orthogonal coordinate simpUces and conclude: For any two orthogonal coordinate simplices (oj), (6j) there is a unique transformation g G Gn mapping them into one another, i.e. g{ai) = bi. g is determined by an appropriate assignment between the pseudo-orthonormal bases corresponding to the point sequences. Now let a; = [p] G i ? " be an arbitrary point; by ^* we denote the coordinates of its normalized representatives with respect to the basis (Oj). Then the distances hi := h{x,ai) satisfy: n

(y, Oo) = -cosh/io = - e ° -

V2J2C, n

(y,Oj) = -cosh/i,- = - V 2 e ° - e - 2

Yl

^'-

The determinant of the matrix for this system of equations is equal to —1. Hence the coordinates (^*) can be uniquely expressed in terms of the distances (hi), and vice versa. Let now / : H" -^ H" be an arbitrary isometry. Then (bi) := (/(oj)) is an orthogonal coordinate simplex as well. The corresponding map g G Gn is obviously an isometry. Hence the distances satisfy h{g{x),bi) = h{x,ai) =

h{f{x),bi).

So all the distances from f{x) and g{x), respectively, to the vertices 6j of the orthogonal coordinate simplex coincide; hence this has to hold for the coordinates of these points as weU, implying f = g & Gn•

2.6 Hyperbolic Geometry

269

For every two hyperbolic points A, B G i ? " , A y^ B, there is a unique hyperbolic connecting hue Ay B D I{Q) and on it (with the notations (38), (39)) the uniquely determined connecting line segment with the parameter representation j;(t) = eo cosht + ei sinht, 0 < t < c = h{A, B).

(45)

For t -^ ±oo the corresponding point on the line tends to its point of intersection with the quadric Q, which thus in hyperbolic geometry has to be considered as the "infinite" of the hyperbohc space. The triangle inequality again implies that the connecting line segment is the shortest curve between two points, which, moreover, is uniquely determined here. Hence in hyperbolic geometry the cut locus is empty for each point. In the next proposition we will determine the distance from a point to a A;-plane, and this will lead to the notion of the perpendicular familiar from Euclidean geometry. Proposition 11. Consider a k-plane A C i ? " and a point a G i ? " in hyperbolic space. Then there is a unique point b £ A such that h{a, b) = min{h{a, x)\ x e A''}. For a ^ A

each line c C A

through the point b is orthogonal to a\/ b.

P r o o f . Let the pseudo-orthonormal basis (ej) of the vector space V " ^ be chosen so that the subspace W^^^ associated with the A;-plane A^ is spanned by the vectors eo,..., e^ (transitivity of Gn on the set of hyperbolic A;-planes). We decompose the representative o G i J " for the point a into its orthogonal components 0 = Oo + ai, Oo G W, oi G W^. Because of (a, o) = —1, there is a unique number c > 0 such that (oo, ao) = — cosh^c; the vector h := Oo/coshc G i J " is the normahzed representative for the point b = [oo] in A . We show that the point b satisfies the assertion of the proposition. In fact, for a; = [p] G A'', p G H", we have by (21) and (42) cosh/i(a, x) = —(o,p) = —(oo,p) = —{b,f) coshc = coshh{b, x) coshc > cosh c. Equality holds if and only if cosh h{b, a;) = 1, i.e. h{b, x) = 0, and hence x = b (cf. (22)). On the other hand, coshh{a,b)

= — (o, b) = — (ao, b) = coshc,

implying the first assertion. To prove the second we may suppose b = [eo] without loss of generality. To compute the angle we determine the vector f i uniquely defined by

270

2 Cayley-Klein Geometries 0 = eocoshc + d i s i n h c , a = [o], ( e o , t ) i ) = 0 , (t)i,t)i) = 1, c > 0.

The case c = h{a, 6) = 0 is only possible for a G A , and then nothing remains t o be proved. An arbitrary line through b and a; G A has the parameter representation 0)

= to cosh(t) + pi sinh(t) mit p = p(/i(6, x)), {to, Ti) = 0, (pi, ?i) = 1-

According to Example 5 the angle then satisfies cos a = (tii, p^). The distance h{a, x{t)) is computed from (21) by

f{t):={a,m) = {to coshc + f 1 sinhc, eo cosh(t) + ^^ sinht) = — cosh c cosh t + (fi, p]^) s i n h c s i n h t = — cosh c cosh t + cos a sinh c sinh t. This function has to be stationary at t = 0; because of f'\t=o

= cos a s i n h c = 0

and c > 0 we immediately conclude a = 7r/2.

D

As usual in metric spaces we call the number h{A , a) := h{b, a) the distance from the point a to the k-plane A , the segment between a and b is called the perpendicular from a ^ A'' onto A'', and b is called the foot of the perpendicular. By means of a suitably adapted basis it is easy to show C o r o l l a r y 12. Two pairs (A ,a), [B ,b) of hyperbolic k-planes A ,B and points a,b ^ H"" are Gn-congruent if and only if the distances satisfy h{A\a) = h{B\b). a Exercise 10. Let A^ be a line in the hyperbolic plane. Prove that the set M{A'^,c) := {x e H'^\ h{A'^,x)

= c},

( c > 0),

is decomposed into two Gn-congruent arcs meeting the line A in its points at infinity A^ n Q. In contrast to Euclidean geometry, these arcs are no lines (cf. Figure 2.27). They are called the equidistants of the line A . Find a parameter representation for these arcs. What do the similarly defined equidistants in spherical and in elliptic geometry look like? By Exercise 5.23 the e-neighborhood of the line A^ in the elliptic plane P^{r): [ / ( A \ e ) := {x e P ^ ( r ) | e(A\a;) < e}, (0 < e < r7r/2), is a Mobius strip. Exercise 1 1 . Let A be a hyperbolic fc-plane, and let a ^ A , 6 e A be two points such that the line a V 6 is orthogonal to every line B^ G A with b & B^. Prove that the line segment between a and b is the perpendicular from a onto A .

2.6 Hyperbolic Geometry

271

Fig. 2.27. Equidistants of a line in the conformal disk model.

Exercise 12. Prove: a) For every two lines A / B in the hyperbolic plane H^ there exists a line C orthogonal to both if and only if they are parallel (and not boundary parallel). (Hint. Apply the angle definition from Example 4, Case 1, and consider the polar of the point of intersection AAB lying in the outer region A{Q).) - b) The common perpendicular C of two parallel lines is uniquely determined. c) li a = A /\ C,b = B /\ C are the points of intersection of the lines with the common perpendicular, then /i(A, B) := h(^a, b) is called the distance of the parallel lines A,B. Prove that this coincides with their metric distance: h{A, B) = mm{h{x,y)\

x e A,y

e B}.

d) Two pairs of parallel lines {A, B), (Ai, Bi) are Gn-congruent if and only if they have equal distances. - e) h{A,B) = r arcosh(|/|(A, B)) (see Example 4). Exercise 1 3 . Prove the following statement generalizing Proposition 11: Let T^"+^ be an arbitrary vector space with scalar product, let P" be the associated projective space, and let F be the auto-correlation determined by the scalar product. If A''" is a non-isotropic subspace, then for each point a ^ A''" U F(A''") there are uniquely determined points bo G A ' ' \ bi e F(A''") such that the line aVbo (aVbi) is orthogonal to A'' (to F{A'')) in the sense described in Proposition 11. (Hint. If W C T^ is the

272

2 Cayley-Klein Geometries

subspaces associated with A , then consider the decomposition of a representative 0 for a into its orthogonal components, 0 = bo + bi, bo e W, bi G W^.) Exercise 14. Conclude from Exercise 13: For a E Q and a hyperbohc fc-plane A''" C H" in the hyperbolic geometry there is a unique point b e A''" such that the line a V 6 is orthogonal to A in the sense described in Proposition 11. Exercise 1 5 . Consider two lines A / B in the hyperbolic plane H^. Prove that there is a unique symmetry axis S, i.e. a hyperbolic line for which there is an isometry g & G2 satisfying the following conditions: g{A) = B,g{s)

= s for sil s E S,g

= idTT2 .

2.6.5 H y p e r b o l i c T r i g o n o m e t r y In this section we again consider triangles in the hyperbolic plane and use the traditional notation. Recall the proof of the law of cosines (Proposition 9). There we adapted a basis {Ci),i = 0 , 1 , 2 , to a given triangle, cf. (38)-(41). Using this it is easy to prove the hyperbolic law of sines: P r o p o s i t i o n 13. The side lengths a,b,c and the angles a,[3,^j degenerate triangle A, B,C ' in in the the hyperbolic hyperbolic plane plam H satisfy sin a

sin (3

sin 7

sinh a

sinh b

sinh c

of a nan

(46)

P r o o f . From equations (38)-(41) one computes the volume of the cuboid spanned by the representatives 0, b, c v{a, b, c) = sin a sinh 6 sinh c = sin/3 sinh a sinh c = sin 7 sinh a sinh 6, where the last two equations are obtained by cyclic permutations of the vertices A, B, C. Division by sinh a sinh 6 sinh c yields the assertion. D Exercise 16. Let {A,B,C) angle 7 = 7r/2. - a) Prove: sma =

be a right triangle in the hyperbolic plane with the

sinh a , coshc tanh6 —, cosh a = -—, cos a = —. smh c cosh 6 tanh c

,,_, (47)

- b) If, for fixed points A and C, the point B moves towards infinity on the line C \/ B, then the side lengths tend to infinity, and the angle /3 tends to zero. - c) In the situation of b) the angle a. tends to a limit UJ satisfying the equation cos uj = tanh b. (Hint. In addition to (47) apply the hyperbolic law of cosines (43).)

2.6 Hyperbolic Geometry

273

T h e last equation implies t h a t the angle to only depends on the length h of the perpendicular from A onto the line B V C; it is called the parallel angle, since it is nothing but the angle of the boundary parallel through A for the hue Cy B. The function uj := n{h) = arccos(tanh(6)) is often called the Lobatschewski function of hyperbolic geometry. T h e definition above implies t h a t the parallel angle is a monotonously decreasing function of the distance 6, and for 6 > 0 it is always less t h a n 7r/2. Moreover, for b tending to infinity it approaches zero. Note t h a t the same definition for the parallel angle in Euclidean geometry would yield the constant value 7r/2. T h e paraUel angle was introduced by N. I. Lobatschewski in his invention of hyperbolic geometry, cf. A. P. Norden[79] and H. Reichardt [92]. T h e hyperbolic law of cosines for angles dual to the law of cosines cannot be proved as easily as in elliptic geometry by just relying on the duality. In fact, in hyperbolic geometry the projective duality is violated, since the polar of a hyperbolic point is an outer line, i.e. something not hyperbolic; nevertheless, the hyperbolic object corresponding to an outer line is the pencil of lines through its pole (n = 2). Here we quote a proof taken from the interesting and clearly written book [79] by A. P. Norden^. P r o p o s i t i o n 14. In the traditional notations, a non-degenerate triangle in the hyperbolic plane (parameter r = 1) satisfies the law of cosines for angles; cos a=

— cos (3 cos 7 + sin (3 sin 7 cosh a.

(48)

P r o o f . By the hyperbolic law of cosines (43) we have cosh c = cosh a cosh h — sinh a sinh h cos 7. Replacing in (43) cosh c by this expression yields cosh a = cosh h cosh a — cosh h sinh h sinh a cos 7 — sinh h sinh c cos a. We apply the hyperbolic law of sines to the last t e r m in this equation, sinh h sinh c cos a = sinh b sinh a sin 7 cot a, then insert the result, and, after a straightforward computation, obtain '^ For many years A. P. Norden held the geometry chair at the University of Kasan, the very same University where N. I. Lobatschewski had worked, who was the first (in the Kasan Messenger, 1829) to publish his discovery of non-Euclidean geometry. The paper of the Hungarian mathematician J. Bolyai only appeared in the year 1832. C. F. Gaufi mentioned his considerations concerning non-Euclidean geometry exclusively in letters. It is generally thought that the foundations of non-Euclidean geometry were laid by these three geometers independently.

274

2 Cayley-Klein Geometries cosh a sinh b = sinh a sinh 6(cosh b cos 7 + cot a sin 7 ) .

Dividing by sinh a sinh b leads to coth a sinh 6 = cosh b cos 7 + cot a sin 7. To the left-hand side of this equation we again apply the hyperbolic law of sines: coth a sinh b = cosh a sin f3/ sin a. Inserting and multiplying by sin a yields cosh a sin f3 = cos a sin 7 + sin a cos 7 cosh b. Similarly, cosh b sin a = cos /3 sin 7 + sin /3 cos 7 cosh a. Inserting this equation into the last but one, we obtain cosh a sin (3 = cos a sin 7 + cos 7 cos (3 sin 7 + sin (3(1 — sin 7) cosh a. A simple calculation then results in (48).

•

Exercise 17. Prove that two triangles in the hyperbolic space H", for which corresponding sides have equal lengths or corresponding angles are equal, are Gn-congruent. Formulate and prove further congruence theorems. Now we t u r n to the essential characteristics of hyperbolic geometry distinguishing it from the elliptic as well as the Euclidean one. In differential geometry the following property is shown to be characteristic of Riemannian manifolds with negative curvature: P r o p o s i t i o n 15. Let A,B,C be three non-colUnear points in the hyperbolic space i J " . Then the sum of the angles in the triangle they span satisfies

a + /3 + 7 < TT. P r o o f . Obviously, we may suppose n = 2. First consider a right triangle with the angle 7 = 7r/2. By the law of cosines for angles, (48), we have cos/3 = sin a cosh 6 > 0, hence f3 < 7r/2. This implies cos(a + /3) = cos a cos f3 — sin a sin /3, = sin a(cos a cosh b — sin /3), = sina(cosacosh6 — V 1 — sin^ acosh 6). Now

2.6 Hyperbolic Geometry

275

V 1 — sin^ a cosh b < v 1 — sin^ a = cos a, and hence cos(a + /3) > sin a cos a (cosh 6 — 1) > 0, i.e. a + f3 < 7r/2. Thus the assertion holds for right triangles. Next consider an arbitrary triangle A, B, C. Then the base point D of the perpendicular from C onto the line A\/ B either lies between A and B, or to the left of A, or to the right of B. In the first case we decompose the triangle A, B, C into two right ones, A, D, C and D, B, C. Then its angle 7 is the sum of the angles 71,72 of b o t h right triangles meeting at the vertex C. Hence a + /3 + 7 = a + 7 i + / 3 + 72 <7i". Suppose now, e.g., t h a t D lies to the left of A. T h e right triangle D, A, C has the angles 7r/2,7r — a, 71, and n — a < 7r/2 implies a > 7r/2. Then it is easy to see — e.g. assigning to each line in the pencil through A a parameter on this line via the points of intersection with the side B y C — t h a t the base point of the perpendicular from A onto the side B y C has to lie between B and C. Again, this triangle is decomposed into two right ones, and then the assertion is proved as above. The case t h a t D hes to the right of B is treated analogously. • Exercise 18. Already in Example 3 we showed that in the conformal disk model hyperbolic lines intersect the quadric Q orthogonally. From this one may conclude that the angle between two boundary parallel lines is always zero. Let x{t) = [eocoshi + eisinhi] e A^ be a parameter representation of the line A^, and let 6 ^ A^ be a point not lying on it. Compute the angle a{t) between the lines A^ and bW x{t) and show that a{t) tends to zero for t ^ 00. In the hyperbolic plane polygons and convex polygons are defined just as in spherical or elliptic geometry, cf. Exercise 5.12. This time, however, we allow the vertices to lie lie at infinity, i.e. on the bounding quadric Q. A polygon is called simply, doubly, etc. or, in general, k-fold asymptotic if one, two, etc., or k of its vertices belong to Q. Since all the sides adjacent to an asymptotic vertex a G Q of the polygon are boundary parallel, the angle at this vertex is zero (Exercise 18). Exercise 19. Consider non-degenerate triangles in the hyperbolic plane. Prove: a) Two threefold asymptotic triangles are always G2-congruent. - b) Two doubly asymptotic triangles are G2-congruent if they form the same angle at their nonasymptotic vertex. - c) For simply asymptotic triangles let, e.g., the vertex C e Q be asymptotic. Since the sides a,b are half-hnes, i.e. have infinite length, a simply asymptotic triangle of this kind has as its essential invariants only the angles a, P, and the side length c. Prove that two such simply asymptotic triangles are G2-congruent if and only if they coincide in two mutually corresponding of these invariants. (Hint for c). Prove first: If G = [eo + £2] e Q, and A : x(t) = [eo coshi + ei sinhi]

276

2 Cayley-Klein Geometries

is a hyperbolic line, then for the oriented angle a(t) from A^ to x(t)\/C equation holds

the following

cos a(i) = - t a n h ( i ) , - o o < i < o o , 0 < a < 7 r .

(49)

This statement holds for an arbitrary hyperbolic line A^ and any point C e Q at infinity, C (^ A^.) T h e definition of the area for elementary sets in Section 5 can almost literally be transferred from spherical or elliptic to plane hyperbolic geometry. The only difference is t h a t now the excess a + /3 + 7 — TT is negative for every hyperbolic triangle Z\; therefore, the area of a triangle in the hyperbolic plane with parameter r is defined to be the quantity F(Z\) : = ( ^ - ( a + /3 + 7))r2.

(50)

Similar t o Lemma 5.8 and Corollary 5.9 one proves P r o p o s i t i o n 16. If M is an elementary set, and {Ai),i triangulation of M, then the definition of the area for M

= l,...,k,

is any

k

F{M) •.= Y^F{A,)

(51)

i=i

is independent

of the chosen triangulation.

•

Obviously, the area is an additive, monotonous, and G'2-invariant function on the system of elementary sets. Similar to what is done in the theory of the Lebesgue measure, this area can be extended to a large class of measurable sets in a G'2-invariant way. It is not difficult to see t h a t the hyperbolic plane H^ itself is not an elementary set. The next exercise implies t h a t its measure is infinite: Exercise 20. Prove: a) Formula (50) also makes sense for asymptotic triangles. Each trebly asymptotic triangle has area 7rr^. - b) In the hyperbolic plane there are elementary sets of arbitrarily large area. Exercise 2 1 . Prove: a) In the hyperbolic plane there are no rectangles. - b) There exist G2-congruent, fourfold asymptotic quadrangles. Exercise 22. In analogy to (5.40) prove that the area of a convex hyperbolic polygon P „ , n > 3, with the angles a.i at the vertices Ai, i = 1 , . . . ,n, satisfies n

F(P„) = ( ( n - 2 ) 7 r - ^ a i ) r l

(52)

In particular, each convex n-fold asymptotic n-angle has area (n — 2)7rr^. Exercise 2 3 . In the complement H^ \ M of each elementary set M there are elementary sets of arbitrarily large area. Find examples for this.

2.6 Hyperbolic Geometry

277

2.6.6 Hyperspheres, Equidistants, and Horospheres As a metric space the hyperbolic space i J " contains the metric hyperspheres S^^^{z, r) of radius r with center z = [3]; its points x = [p] satisfy the equation {hi) = - c o s h r , ((p,p) = - 1 , (3,3) = - 1 ) .

(53)

Hence the notions circle (n = 2), sphere (n = 3), and hypersphere are defined just as in elementary Euclidean geometry; and like there we have P r o p o s i t i o n 17. The isotropy group Gz C Gn of the center z for the hypersphere S*"^^ = S{z,r) acts transitively on S*"^^; it is isomorphic to the orthogonal group 0{n). Thus the isotropy group of a point XQ G S*"^^ under the action ofGz is again isomorph to 0{n — 1), and, as homogeneous spaces, the hyperbolic hyperspheres are equivariantly isomorphic to the Euclidean hyperspheres: S*"^^ = 0{n)/0{n — 1). The hyperspheres are projective hyperellipsoids, each of whose tangent hyperplanes orthogonally intersects the line in the pencil through the center z to its point of contact. P r o o f . As Gn acts transitively on the hyperbolic points, we may suppose 3 = eo. T h e isotropy group of [eo] in (?„ is the orthogonal group of the Euclidean vector space [ e i , . . . , e „ ] , which acts transitively on the unit sphere S*"^^ of this vector space in the usual linear way. Since it preserves the distance, and as the points x G S*"^^ can be represented bijectively and equivariantly in the form ei G S*""^ I—> X = [eocoshr + e i s i n h r ] G 5*""^

(54)

the first four statements of the proposition are immediate. Passing to the homogeneous coordinates of x in equation (53) and using 3 = eo we obtain n

- ( x ° ) 2 ( s i n h r ) 2 + ( ^ ( x * ) 2 ) ( c o s h r ) 2 = 0. i=i

This is the equation of a hyperquadric of index 1. The reflection in a radial line z \/ X leaves each point of this line fixed, hence z as well. Since it is an isometry, the hypersphere S*"^^ is m a p p e d into itself. As x also remains fixed, the tangent hyperplane at x has to be mapped into itself, too. Thus it is orthogonal to the radial line z\/ x. D Thus, just as in Euclidean geometry, the hyperspheres are orbits of the isotropy group of their centers. It is certainly a promising principle to consider the elementary geometry of a given transformation group [G, M] as the study of G-invariant properties of the orbits of suitable subgroups H C G in M. As will soon become clear, in this respect hyperbolic geometry offers more opportunities t h a n the Euclidean or elliptic ones, since the pseudo-orthogonal groups have a richer subgroup structure as compared to the orthogonal one of

278

2 Cayley-Klein Geometries

the same dimension. Consider now a hyperbolic hyperplane B " ^ C i J " . Its pole b = B is an outer point. The group Gn-i C G„ is the corresponding isotropy group in (?„, and at the same time it is the isotropy group of B; again it consists of two components of 0 ( l , n — 1). Choosing b = [e„], the vector space associated with B is W" = [eo,..., e„_i]. The hue by x = [p, e„], a; G B , is orthogonal to each of the lines x\/ y = [p, D] through x lying in B , which we may always consider as spanned by a unit vector D G Vt^ = [tn\^ orthogonal to p. Hence the family of hyperbohc lines {b V x}^^^ consists of the normals to the hyperplane B. They are parallel, and all go through the same outer point b. Any two of these normals have a common perpendicular, the connecting line from their point of intersection to B. Obviously, B is the orbit of 030 = [eo] under the action of Gn-i- We may consider the orbit of the point Xr := [eo cosh r + e„ sinh r] G 6 V a;o as described by the parameter representation Vrix) := (;[eo coshr + e„ sinhr] = [pcoshr + e„ sinhr]

(55)

with p G W, {^,^) = —1. Since Gn-i consists of isometries, the hyperbohc distance h{x, y^(a;)) = h{g{xo), gixr)) = r is constant; hence the hypersurface described by (55) is called the equidistant A{r, B) of the hyperplane B with distance r. Note that negative values are also allowed for r; they define the opposite equidistants.In Exercise 10 we already introduced the equidistants of a line in the hyperbolic plane and learnt already that they are no lines. Obviously, the equidistants of a hyperplane in hyperbolic geometry are no hyperplanes. We now want to show that the tangent hyperplane TyA(r; B) of the equidistant at the point y is orthogonal to the line b\/ y. To see this we insert the parameter representation for the normalized representative of a point in the hyperplane B p(t, ps) = eo cosht + ps sinht with p^ G [ei,..., e„_i], (pg, pg) = 1

(56)

into representation (55): t)^{t, pg) = p(t, pg) coshr+e„ sinhr. Then we compute the differential: iit)^(t, ps) = iip(t, ps) coshr = ((eo sinht + pg cos\it)dt + sinhtdpg) coshr. Since the vectors p^ lie in the unit hypersphere of a Euclidean vector space, we have (p^, d^g) = 0, and a straightforward computation leads to (t)r(t, Ps), (it)r(t,Ps)) = 0, the asserted orthogonality. The arc element for the equidistant ((it)r, (it)r) = (dp, dp) cosh^ r (57) differs from the arc element of the hyperbolic hyperplane only by the constant factor cosh r, so that its inner (metric) geometry itself is hyperbolic. As is

2.6 Hyperbolic Geometry

279

easy to verify, the isotropy group of the point Xr G A(r; B) for the action of G n - i is conjugate to the orthogonal group 0{n — 1). We summarize what we proved concerning the equidistants in the following proposition: P r o p o s i t i o n 18. The normals of a hyperbolic hyperplane B are parallel; all go through the same outer point, the pole b = B of the hyperplane. The orthogonal trajectories of the family of normals are the hyperplane itself and its equidistants. Each equidistant is an orbit for the isotropy group of the hyperplane; as a homogeneous space it is equivariant to the hyperbolic hyperplane:

A(r,B"-i) = G „ _ i / 0 ( n - l ) . Moreover,

its inner geometry

is

hyperbolic.

Fig. 2.28. Equidistants as orthogonal trajectories for the normals of a line in the conformal disk model (cf. Figure 2.27).

Exercise 24. Prove that the equidistants of arbitrary hyperbolic hyperplanes with the same distance parameter r are Gn-congruent.

280

2 Cayley-Klein Geometries

Let now a; G Q be an arbitrary point of the hyperquadric Q; for transitivity reasons we may suppose a; = [eo — e„]. In order to determine the isotropy group of this point, it is useful to introduce a different type of frame, which we wiU also need in later applications. For each pseudo-orthonormal basis (ej), « = 0 , . . . , n, we introduce a new basis by Oo := (eo - e„)/A/2, Oj := e^,« = 1 , . . . , n - 1, o„ := (e„ + eo)/V2;

(58)

it will be called isotropic orthogonal. A basis (Oj) is isotropic orthogonal if and only if the matrix containing its scalar products has the form ((Oj,Ofc)) = B :=

0 I„-i

0

;

(59)

here / « - ! denotes the unit matrix and o the zero column vector of order n— 1, so o' is the corresponding zero row vector. Conversely, setting eo := (ao + o„)/v 2, e^ := Oj,« = 1 , . . . , n - 1, e„ := (o„ - Oo)/v 2,

(60)

we assign a pseudo-orthonormal basis to every isotropic orthogonal one. The elements a in the isotropy group Hx of a point x = [oo] have to satisfy the condition a(oo) = aoA^^, A G R*. Starting with the foUowing ansatz for the form of its block matrix 'A-i o' c 0 A d

0 bV,, and taking into account that, because of a G Gn, it preserves the scalar product, i.e. a'Ba = B, a straightforward computation involving block matrices shows Lemma 19. With respect to an isotropic orthogonal basis (Oj) the isotropy group Hx of the point x = [oo] consists of all transformations in the group Gn which have a matrix of the following form in this basis: a(Aa,A):=

/ A - i X-^a'A 0 A V 0 o'

(o,o)/2A\ a with A e 0{n - I), a eIi"^\X> 0. A / (61)

It is easy to see that, in contrast to the isotropy group of the hyperbolic as well as the outer points, the isotropy group Hx acts transitively on hyperbolic space, Hx[co] = H''. (62)

2.6 Hyperbolic Geometry

281

This can be verified by a computation, or else one could argue as follows: The one-parameter subgroup 0(1,1) = {a{E,o,e*)} acts transitively on the hyperbolic line [eo, ^n\ Cr.• a{E, 0, e*)eo = a{E, o, e*)(oo + a „ ) / v 2 = (oo e * +o„ e* = eo cosh t + e„ sinh t, and the isotropy group Hx acts transitively on the subpencil of all hyperbolic lines through x, so that [eo] can be transformed into any other point y G H". In order to obtain interesting orbits, we consider the subgroup E{n - 1) := {a{A, a,l)\Ae

0{n - 1), 0 G R " - ^ } .

A calculation using block matrices of the form (61) shows that a{A, 0, A)a(B, b, /J) = a{AB, Ab + ajj,, XJJ).

(63)

This implies that E{n — 1) is a normal subgroup in Hx, since it is the kernel of the homomorphism a{A, o, A) G Hx i-^ A G R*,. Comparing this to the representation (1.6.2.42) of the Euchdean groups immediately shows that E{n — 1) is isomorphic to the Euclidean group of the (n — l)-dimensional Euclidean space. Formulas (58), (60), (61) applied to elements z = E{n — l)y, y := [eo] of the orbit yield 2 = a ( A a , l ) y = [eo(l + (a,o)/4) + o / V 2 - e „ ( o , o ) / 4 ] ,

OGR""^

(64)

Hence z = y holds if and only if o = o; thus the isotropy group of y under the action of E{n — 1) is the orthogonal group 0{n — 1). This orbit E{n — \)y is called a horosphere or limit sphere; horocycle or limit circle in the case n = 2. Since Gn is transitive on Q, we may choose an arbitrary point from Q &s x and define the restricted isotropy group Hi^x C Hx isomorphic to E{n—1) as above by A = 1. Since the subpencil of hyperbolic lines through the arbitrary point a; G Q simply covers the hyperbolic space i J " , for every point pair X e Q,y e H" there is a horosphere HixV- From the considerations so far we conclude Proposition 20. All horospheres in hyperbolic space are Gn-congruent. If Hixy is a horosphere, then choosing eo, e„ with y = [eo], x = [eo — e„] Equation (64) becomes a parameter representation for this hypersurface. The horospheres are equivariant to the (n—l)-dimensional Euclidean space E"^ = E{n-l)/0{n-l). a It is remarkable that, as a subgroup in the isometry group of hyperbolic space, the Euclidean group and its orbits appear as Euclidean spaces of codimension 1: This way, Euclidean geometry is embedded into non-Euclidean geometry. This also holds for the (inner) Riemannian metric generated by the

282

2 Cayley-Klein Geometries

Fig. 2.29. Horocycles in the conformal disk model. hyperbolic metric on the orbits. Taking derivatives of the normalized representative 3 for z with respect to the components a* of n-l 0 = ^ ejtt* i=i

(64) imphes the equations dijda'

= (eo - e„)aV2 + ei/V2,

(65)

{di/da\di/da^)=Sij/2. This proves the assertion. Moreover, from (65) we see (3, di/d(i)

= 0 and (eo - e„, di/d(i)

= 0.

This implies (cf. Figure 2.29) C o r o l l a r y 2 1 . The horospheres Hi xV are the orthogonal trajectories for the pencil of boundary parallel hyperbolic lines through the point at infinity X eQ. •

2.6 Hyperbolic Geometry

283

2.6.7 Stationary Angles Let A , B™ C i ? " be an /-plane and an m-plane in the n-dimensional hyperbolic space, respectively. We want to determine a complete system of invariants for pairs like ( A ' , B ™ ) with 0 < I < m < n. The case of points, / = TO = 0, is the topic of Proposition 4, and hyperplanes, / = TO = n — 1, were discussed in Example 4. In order not to have to distinguish too many cases we introduce: Condition A. There is no hyperplane L"^ C H" containinq A U B™. In other words, we suppose that the set A UB™ generates the whole space. Now we consider all pairs of hyperplanes L " " \ M " " ^ C i ? " with A' C L " " \ B™ C Ai""^ and determine the relative extrema of the invariant I{L, M) leading us to the notion of stationary angles. Similar to the Euclidean or elliptic geometries they form the complete system of invariants we are looking for. Let u'^^,W™^^ be the pseudo-Euclidean subspaces associated with A' and B™, respectively. The oriented hyperplanes containing A are determined by the unit vectors in the orthogonal complement U , and analogously for the hyperplanes containing B™:

u e U^, {u,u) = 1, ro eW^,{ro,ro)

= 1.

(66)

Hence, according to (23) we have to find the relative extrema of the scalar product (u, ro) under the constraint (66). Since the arguments u, to vary on unit spheres in Euclidean vector spaces, the existence of minima and maxima is guaranteed. Consider the function /(u,w,A,/i) = ( u , w ) - A ( ( u , u ) - l ) - / i ( ( w , w ) - l ) , ue U^, w e W^

(67)

with the Lagrange multipliers A, jj,. Differentiating with respect to u, ro we obtain the necessary conditions for the relative extrema: {du, tn) - 2X{du, u) = {du, tn - 2uA) = 0, {dvo,u) - 2ij,{dvo,Vo) = {dvo,u - 2tti/i) = 0. Since du e U ,dvoeW only if

can vary freely, these conditions are satisfied if and W - 2uA G t/ and u - 2tti/i G W.

(68)

Let f>rjj,prTj± denote the projections resulting from the direct decomposition

then (68) is equivalent to prjj±VO = 2uA and pr-^^u Define the operator A as follows:

= 2VOIJ,.

(69)

284

2 Cayley-Klein Geometries p:=prjj±\W^, q:=pr^±\U^, A := q o p eEnd{W^).

(70)

Then (69) implies: Ifu, ro is a solution of the extremal problem for the function (67), then TV (^ W is an eigenvector for the operator A: Am = tti4A/i.

(71)

This completes the analytic part of the considerations, which essentiaUy serves to motivate the definition of the operator A; obviously, this operator is assigned to the pair A , B™ in an invariant, more precisely, equivariant way. Since A is an endomorphism of the Euclidean vector space W , its diagonalizability immediately follows from Lemma 22. The operator A defined by (70) is self-adjoint, more precisely, p*=q,

q*=p,

A*=A.

(72)

The eigenvalues of A are non-negative. The eigensubspace of A for the eigenvalue 0 is the kernel of p. P r o o f . The formulas (72) immediately follow from the definition of the adjoint map together with the relation (u,w) = {qn,vo) = {u,pvo) holding for aU u G t / ,V0 ^W . So A is a self-adjoint operator in a Euclidean vector space, hence diagonalizable with real eigenvalues. If Vo is an eigenvector for the eigenvalue a, then by (72)

{vo,Avo) = {vo,vo)a = (pro, pro), immediately implying the last two assertions. • Hence by dim W = n — m there exist n — m not necessarily different eigenvalues aj for A, which we label according to their magnitude: a i > "2 > • • • > "n-m > 0.

(73)

Let rA be the rank of the operator A, i.e. (if ra < n — m) *^rA + l

•••

^n—m

^•

We fix an orthonormal eigenbasis (Oj) for A with Aai = ttiUi. Then (72) implies

(Oj, Aoj) = (pai,paj)aj = SijUj. Hence bi:=pai/y^iior i = l,...,rA

(74)

are rA orthonormal vectors from U , which we complete to an orthonormal basis {bj),j = I,... ,n — I, ior U . Then

2.6 Hyperbolic Geometry (bj, Oj) = Sijy/oj

holds for i = I,...

In fact, because of bj G t /

,n — I, j = I,...

,n — rn.

285 (75)

we have

(bj, Oj) = (bj,pOj) = (JijA/oT for i = I,.. .,rA by the definition of the bj. Since the remaining vectors bp belong to the image of p and are obviously also orthogonal to U, in addition, (bp,Oj) = {bp,paj)

= 0=

Spj^

for yO = r^i + 1 , . . . , n —TO.From (75) and the following rule generally holding for adjoint maps, (Imp) = Keip*, it is easy to deduce Imp = [ b i , . . . , br^], Keip = [o^^+i, • • •, o„-m] = U D W^

(76)

I m ^ = [ o i , . . . , ( V ^ ] , K e r ^ = [ b ^ ^ + i , . . . , b „ _ ; ] = WnU^,

(77)

and (/(bj) = OJA/OI for « = 1 , . . . , n —

TO.

(78)

T h e qualitative properties of the pair A , B™ are reflected in the magnitude of a i . The extremal properties of the eigenvalues, cf. (73) and (75), imply A / O I = max{(u, tti)|u G U^, {u,u) = 1, tn G W^,

( « , « ) = 1}.

(79)

Exercise 2 5 . Prove (81) by a direct estimate. L e m m a 2 3 . / / Condition cases:

A is satisfied,

then a i distinguishes

the

following

1. a i > 1 -^ A DB"^ is a projective subspace lying in the outer region or the nopoint; 2. a i = l -^ A n B™ is a projective subspace tangent to Q; 3. a i < l -^ A n B™ is a hyperbolic projective subspace and different the nopoint.

A{Q)

from

P r o o f . Let a i > 1. Then by (75) the hnear span [oi, b i] is a pseudo-Euchdean subspace oiU +W , and hence U +W is also pseudo-Euclidean. Consequently, the intersection U nW = (U + W ) ^ i s Euclidean or equal to {o}, hence the associated projective subspace A ' n B ™ hes in A{Q), or it only consists of the nopoint. Conversely, if this is the case, then there is a time-hke vector p = u + tti,uGL'" ,V0 eW . Let UQ, Wo be the corresponding normalized vectors. Since the hnear span [u, tn] = [uo,ttio] is pseudo-Euchdean, the determinant of the matrix of scalar products 1 — (UQ, WO)^ has to be negative. T h e fact t h a t a i is maximal implies

286

2 Cayley-Klein Geometries 1 < {uo,rvof < ai

proving Case 1. Let a.i = I. Then by (75) we have (oi, bi) = 1. Thus the vector 3 := Oi — bi G t/ +W is isotropic. Actually, if 3 = o, then Oi = bi G U n Vt^ , in contradiction to Condition A being equivalent to either of the statements

u + w = v, u^r\W^ = {0}.

(80)

Moreover, (75) implies that 3 is orthogonal to each of the generating vectors Oj, bj of t/ + W \ hence this subspace is isotropic. The same then also holds for its orthogonal complement U C\W, and thus A' n B™ is tangent to Q, cf. Corollary 1.9.19. Conversely, if this is the case, then U +W has to be isotropic. Thus we find an isotropic vector t ) = u + t t i G t / +W, u G t / ^ , tn G W^, orthogonal to U^ + W^. In particular, (t), U) = (U, U) + (W, U) = 0, (t), W) = (U, W) + (W, W) = 0, implying

(u,w)^ (u,u)(w,w)

1.

Since a i is maximal, the relation a i > 1 has to hold. By what we proved in Case 1, a i > 1 implies that the space U +W is pseudo-Euclidean. Hence a.i = I. This proves Assertion 2. Since both sides of the equivalences stated in the lemma are complete disjunctions, the third assertion follow from the first two; thus the lemma is proved. • The result of this lemma gives rise to the following definition: Definition 3. Two hyperbolic subspaces A\B"' are called parallel if the section A AB™ of the associated projective subspaces hes in the outer region A{Q) and, moreover, does not only consist of the nopoint; they are called boundary parallel if this section is tangent, and intersecting if it is a hyperbolic subspace. Finally, recall that two projective subspaces A', B™ C P " are skew if their intersection is the nopoint. • Obviously, the eigenvalues ai,i = 1 , . . . , n — m, are invariants of the pair A ,-B™, since the operator A is equivariantly assigned to this pair. In order to prove that this system of invariants is complete, we construct a basis for V such that the following holds: 1. The basis {bj),j = 0 , . . . ,n, is pseudo-orthonormal. 2. (bj), j = 1 , . . . ,n —/, is a basis for t/ . 3. For the eigenvalues aj and eigenvectors Oj of A the relations (73), (74), and (75) hold. 4. The coordinate representation of the eigenvectors in the basis (bj) is canonical in the sense that the coordinates of Oj are uniquely determined by the eigenvalue aj.

2.6 Hyperbolic Geometry

287

Such a basis will be called adapted to the pair A , B"\ Proving the existence of adapted bases suffices to prove the completeness for the system of invariants: If A , B ™ is a pair of hyperbolic subspaces with coinciding invariants (73), and (b') denotes a basis adapted to this pair, then the pseudo-orthogonal transformation of V determined hy g{bi) = b[,i = 1 , . . . , n + 1, generates an equally denoted hyperbohc isometry such t h a t gA^ = A'\ gB™ = B'™. Before we start the construction of an adapted basis, we introduce the following notation: Let Ai"(m), (m, m) > 0, be the hyperbolic hyperplane corresponding to the pole [m] G A{Q). T h e quantities (cf. (75)) V ^ = ( 0 i , b i ) = |/|(M(0i),M(bi)) are called the stationary

invariants

of the pair A , B™. If a i > 1, then we set

y^a7 = cosh/3i

(81)

and call /3i the maximal distance of hyperplanes containing A (36)). Eventually, if aj < 1, then we set Ui = cosl3i, (0 < l3i < 7r/2)

and B

(cf.

(82)

and call /3j the stationary angles of the pair A , B™ (cf. Example 4). We already constructed the vectors b i , . . . , b „ _ i e U of the adapted basis; they satisfy the Conditions 2. and 3. In order to adapt the remaining vectors, i.e. a pseudo-orthonormal basis for t / ' + ^ , {bo,bj),j = n — / + 1 , . . . , n, to the situation, consider the projections of the eigenvectors (recall (74)) prifOi

= Oj - bjA/al, « = 1,. . . , n - m,

onto the space U. Relations (74) and (75) imply {prjjai,prjjaj)

= Sij{l - aj) for «, j = 1,. . . , n - m.

(83)

Hence these projections are orthogonal; we normalize those different from o in accordance with the cases considered in Lemma 23 and complete t h e m orthogonally to form the adapted basis we were looking for. 1. C a s e . If ai > I, then this eigenvalue has multiplicity 1, and all remaining eigenvalues are less than one, a^ < l,i = 2,n — rn. There is an adapted basis bj for V with Oi = b 1 cosh /3i + bo sinh /3i, (84) Oi = bjCos/3j + bn-i+i-i

sin fJi, i = 2,...

,n - m.

(85)

P r o o f . According to (83) the projection prjjai is time-like; normalization gives rise to the basis vector bo and shows the validity of (84). Since V does not contain two orthogonal time-like vectors, 0.2 < I. If now 0.2 = I, then (83) implies t h a t the projection prjja2 had to be isotropic and orthogonal to bo, or else the null vector. Since the subspace orthogonal to bo is Euclidean, this

288

2 Cayley-Klein Geometries

projection then has to be the null vector. But this again means O2 € t/ DW contradicting Condition A, cf. (80). Normahzing the projections prjjai, i > 2, leads to orthonormal vectors bn-i+i-i, i > 2, where the largest index is 2n - {m + I + 1) = n - d, d := dim U

nW.

For d > 0 we complete the set of vectors bj defined up to now by choosing d more vectors, so that all together form a pseudo-orthonormal basis for V satisfying conditions 1-4. 2. Case. / / a i = 1, then this eigenvalue has multiplicity 1. There is an adapted basis bj for V such that Oi = bi +bo +b„_;+i,

(86)

Oi = bicosfJi + bn-i+iSinfJi, i = 2,.. .,n - m.

(87)

P r o o f . By (83), in this case prjj{W ) is isotropic. If, in addition, a2 = 1, then we have two isotropic vectors Oi — bi, 02 — ^2, in this projection. They are linearly dependent, since V is pseudo-Euclidean of index 1, cf. Definition 2.2.3 and the remark following it. This implies c = OiA + 02M = biA + b2M with A/i ^ 0, i.e. 0 7^ c G U^ n W^. But this contradicts (80). According to (83) we determine the vectors bj by normalizing the projections of Oj,« > 2, so that (87) holds. The orthogonal complement of the hnear span [bn-i+2, • • •, b^n-i-m] in U is pseudo-Euclidean. Hence we may represent the isotropic vector Oi — bi G t/ by choosing bo, bn-i+i & U so that (86) holds. 3. Case. If ai < I, then there is an adapted basis bj for V with Oi = bjCos/3j + bn-i+iSinfJi, i = 1,.. .,n - m.

(88)

P r o o f . The proof follows immediately from (83) by normalizing the projections of the vectors Oj onto U. Finally, consider the case that Condition A is not satisfied. Let k + I = d i m t / + W, and let A; < n be the dimension of the hyperbolic subspace A' V B™. Then we have

dfiA^B"")

•.= diinU^nW^

=n-k>0.

We caU d = df{A\ B"") the deficit of the subspaces A', B™. Obviously, the deficit is an invariant of these subspaces; for the pair A , B"\ considered as subspaces of A V B"\ Condition A is satisfied, and we obtain k — m = n — rn — df{A\ B™) eigenvalues as essential invariants of this pair. We remark that all vectors from U O W are eigenvectors of A for the eigenvalue 1, which always occurs for d > 0. Summarizing we formulate:

2.6 Hyperbolic Geometry

289

P r o p o s i t i o n 24. / / Condition A is satisfied, then for each pair of hyperbolic subspaces A , B™ C H", 0 < I < m < n, there is an adapted basis for V with the properties stated in the Cases 1-3. In general, two pairs with the same eigenvalues ai and equal deficit are congruent; these quantities form a complete system of invariants for the action of the isometry group Or^ on the product set Hni x Hnm of these pairs. If Oj is an eigenvector for an eigenvalue ai > 0, then the hyperplanes M ( 0 i ) D B™, M ( p ( 0 i ) ) D A ' realize the stationary invariant \I\{M{ai),M{p{ai)))

= V^i,

(a^ > 0).

Each eigenvector W for the eigenvalue «« = 0 belongs to U; the M{ro) is orthogonal to all hyperplanes M containing A . "^

hyperplane D

As an application of this classification we now want to prove a generalization of Proposition 11: P r o p o s i t i o n 25. Let A', B™ C H", 0 < I < m < n, be two subspaces of hyperbolic space. There exists a common perpendicular XQ V yg, xo e A'- \ B"', yij e B"' \ A ' - , for A \ B"' if and only if the largest invariant satisfies ai > I, i.e. if the subspaces are skew or parallel. In this case, the hyperbolic distance satisfies h{xo, yg) = min{/i(a, 6)| a G A , 6 G B™} = [3i with o.\ = cosh/3i. Here orthogonality is meant as formulated in Proposition 11 for the perpendicular. The points ajg, yg for which the minimal distance is attained are unique. P r o o f . Considering only the subspace A V B™ Condition A is satisfied. If the subspaces intersect, and if ^ G A n B ™ is a hyperbolic point, then, supposing the existence of a common perpendicular, the triple (a;g,yg,2) forms a hyperbolic triangle whose angle sum is larger t h a n TT contradicting Proposition 15. Exercise 12 implies t h a t boundary parallel subspaces ( a i = 1) also have no common perpendicular. Now consider the case a i > 1 of parallel or skew subspaces; for / = 0, A = a, skew just means t h a t a ^ B™, and we are again in the situation of Proposition 11. In order to complete the general case now, we also need a suitable basis for W t h a t is adapted to the situation. This again will be constructed by considering projections of the vectors bi e U ,« = 1 , . . . , n — /. We have pr\Ybi

= bi -pr-^±bi,

i =

l,...,n-l.

Relation (75) implies pr-^±

bi = OJA/OT, a j = 0 for « = TA + 1, . . . , n -

/.

^ Essentially the same classification was originally proved within the framework of Mobius geometry, cf. R. Sulanke [102].

290

2 Cayley-Klein Geometries

This last arrangement takes into account t h a t i = VA + l, • • • ,n — I, ci. (77). Hence pr-^bi

bj

G W

D U

, where

= bi - Oi^/ai ioT i= I,.. .,n-I.

(89)

A straightforward calculation involving the scalar products of these projections leads to (bj - OJA/OI, bj - ajy^)

= Sij{l - ai), i,j = l,...,n-l.

(90)

In particular, the projection of bi is time-like; so we define Oo := - ( b i - o i c o s h / 3 i ) / s i n h / 3 i = bo cosh/3i - bisinh/3i,

(91)

where the last equation foUows from (81) and (84). Taking into account (82) and (85) for « = l , . . . , n —/ — 1, the normalization of the remaining projections leads to a„-m+i •=

,

= bj+isin/3j+i - b„_;+jCos/3j+i.

(92)

If necessary, we again complete this basis to a pseudo-orthonormal basis for W. Now define XQ := [bo] G A ' , yg := [oo] G B"\ Since (oo, bo) = - c o s h / 3 i , the distance of these points is h{xo, j/o) = /3i- We show t h a t the hue XQ V yg is orthogonal to each line through yg in B™. W i t h the initial point yg the line xo V yg has the parameter representation ... 1,^ , f'o - aocosh/3i . p(t) = Oocosht H smht. sinh pi An arbitrary line through yg in B™ has the representation 3(s) = oocoshs + o ^ s i n h s , o^ G W, ( o ^ , o ^ ) = 1, (oo, a^) = 0.

(93)

It is easy to conclude from (91) and (92) the foUowing (bo - ao cosh/3i, o^) = 0. The angle definition for intersecting lines (Example 5) implies orthogonality. Analogously one proves the orthogonality of XQ V yg and each line through XQ in A ' . T O show the asserted distance property consider, in analogy to (93), the parameter representation t)(t) = bo cosht + b ^ sinht, b ^ G U, (b^, b^) = 1, (bo, b^) = 0.

2.6 Hyperbolic Geometry

291

Now we investigate the function describing the distance of the (essentially arbitrary) points [t)(t)] G A\ [^(S)] G B ™ : / ( s , t) := — (t)(i),3(s)) = cosh/3i cosh s cosht — cos7 sinh s sinht = cosh/i([t)(t)],[3(s)]). There we could set C0S7 = (0^, b ^ ) ,

since both these unit vectors are orthogonal to Oo and bo, and consequently span a Euclidean subspace. An elementary argument shows that this function has an extremum only for s = t = 0, which, moreover, is the asserted minimum. Furthermore, the foot points XQ, UQ are uniquely determined. • On the basis of Proposition 25 we define the distance of parallel or skew hyperbolic subspaces as h{A ,B™) := h{xo,yo) = Pi- For intersecting subspaces the distance is obviously zero; as usual, the distance of two sets A, B in a space with metric p is defined as follows: p{A, B) := mf{p{a, b)\aeA,beB}.

(94)

In this sense, for boundary parallel subspaces A , B™ the distance is zero, h{A\ B™) = 0. (The proof is left to the reader as an exercise; it suffices to consider boundary parallel lines in the plane.) 2.6.8 Quadrics A quadric in hyperbolic space H" is defined geometrically as the intersection of a projective quadric B with the hyperbohc space: BH := B D H". Obviously, such a quadric may be empty; already the intersection QH of the hypersphere Q defining the hyperbolic space with H" is, by the definition of hyperbohc space as its inner region, empty; the points of B belonging to Q are called the limit points or the points at infinity of B. To determine the points of BH amounts to finding the hyperbolic points of the projective quadric B, i.e. the time-like solutions of the quadratic equation K?) := K?, ?) = S'^j=obirx'x^ = 0, (y, y) = - 1 , (hj = bji);

(95)

6 is a symmetric, real bilinear form with the matrix (bij). Since the equation is homogeneous, it suffices to consider the equation only for the normalized vectors p, i.e. time-hke unit vectors. Equation (95) suggest to consider a quadric as the intersection of the cone in the vector space V"+^ defined by 6(j;, p) = 0 with the pseudo-Euclidean model H^ for the hyperbohc space, cf. (10) with r = 1. To classify these quadrics with respect to congruence under the action of the group Gn it is not enough to use the Eigendecomposition Theorem as in the Euchdean, the spherical, or the elhptic geometries. This time it is necessary to apply the results concerning the normal forms of symmetric endomorphisms to the endomorphisms associated with a symmetric bilinear form.

292

2 Cayley-Klein Geometries

cf. Exercise 2.17. T h e variety of algebraic possibilities resulting from this as well as from the pseudo-Euclidean structure on t h e vector space correspond to t h e intricate geometric situation. In fact, now we have to investigate the position of an arbitrary quadric with respect to the absolute, which itself is a quadric, namely a hypersphere. It is an interesting, b u t in higher dimensions not quite simple task to establish the relations between t h e geometric and the algebraic properties. E x a m p l e 1 1 . C e n t r a l q u a d r i c s . A non-empty quadric B 7^ 0 is called central if there is a point m G i ? " such t h a t the reflection sm in this point maps the quadric into itself: sm{B) C B; m is caUed a center of t h e quadric B. Note t h a t in this definition all points of B are taken into account, i.e. those possibly lying in t h e outer region of Q as well. To describe sm algebraically we choose the pseudo-orthonormal basis (ej) so t h a t m = [eo] is represented by t h e time-like unit vector eo; then Sm •• x = [^] = [tox° + pi] I—> sm{x) Now we may identify [eo]^ = space; then t h e isotropy group group 0{n), which acts simply Let 61 : Pi G £;" ^

= [eox° - p j with p^ G [CQ]^.

(96)

E^ with t h e n-dimensional Euclidean vector of eo under t h e action of Gn is the orthogonal transitively on t h e orthonormal frames of E^. ^ 6i(Pi) = S^^^^^^hijx'x^

be the restriction of 6 on t h e Euclidean space E^. equation

G R It is obvious t h a t the

6(p) := 6oo(x°)2 + 6i(p,) = 0 for p = eox° + p^, (eo,Pi) = 0

(97)

determines a central quadric in H"", if there are solutions for (97) at aU. In a moment we will prove t h a t every central quadric can be represented in the form (97), which wiU easily lead to a classification of central quadrics. • L e m m a 26. A point z G i ? " is a center of the non-empty quadric B if and only if B can he represented by an equation of the form (97) with z = [eo]. P r o o f . Example 11 shows t h a t t h e condition is sufficient. Now we consider a point z G H". Assuming there is no representation for B in the form (97) with the given z we wiU see t h a t the point cannot be the center of B . To this end we take a pseudo-orthonormal coordinate system with z = [eo]. In the notations of Example 11 we then have for t h e equation of the arbitrary quadric B 6(p)=6oo(a:°)2+2x°c^(Pi) + 6i(Pi), where the linear form LU is deflned by u;(pi) •.= S^^^bojxJ

=: (D,Pi).

2.6 Hyperbolic Geometry

293

T h e second equation determines the vector W in the Euchdean vector space E" = [co]^ corresponding to the form LV. Choosing a suitable orthonormal basis for the Euclidean vector space E" we may suppose t h a t the symmetric bilinear form 61 is diagonal (Eigendecomposition Theorem, cf. Proposition 1.6.4.2). Next we pass to Klein's model for hyperbolic space; to do so we normalize the representatives p of the points by requiring x^ = 1 (cf. (16)). T h e quadric B then consists of all points a; = [eo + Pi] satisfying 6oo + 2 ( D , j : i ) + i : ; = i A j ( x J ' ) 2 = 0 , Pi G N ^ -

(98)

Such a representation exists for each quadric; we call it a representation of the quadric B related to z ^ H"". Since we assume t h a t there is no representation of the form (97), we conclude D 7^ 0. Now we decompose the space E'^ into three mutually orthogonal subspaces, where we do not exclude trivial subspaces:

u+ := [(ei)A.>o], t/_ := [(ei)A. 0. We decompose the vector D = Di + Do components Di G U^ © t / _ , Do G UQ and show: / / Do 7^ 0, for B. To see this note t h a t there is 0 G t / + © t / _ satisfying Since Do 7^ 0, we find a vector b e UQ such t h a t p^ = 0 + point of B, hence a + 2(D,

PI)

= 600 + &i(a) +

2(DI,

0) +

2(DO,

assertion is trivial. into its orthogonal then z is no center a := boo+bi{a) ^ 0 b corresponds to a

b) = 0.

Because of a 7^ 0 we also have (D,pi) ^ 0, and since replacing p^ by —p^ the value of a does not change, the corresponding point does not belong to B; z is no center for B. Now we consider the case t h a t Do = 0, equation (98) for p^ only depends on the first TO- variables x " , a = 1 , . . . , TO. Rewriting it slightly it takes the form A + i:™=iA„(x° + a ° / A „ ) 2 = 0 , where (a") are the coordinates of the vector D, and we have set A:=6oo-i:™=i(a°)VAa. We suppose A>0

and order the equation according to the sign of its terms:

294

2 Cayley-Klein Geometries A + i:LiAa(a:" + a°/A„)2 = S^^^^^ - A„(x° + a°/A„)2.

(99)

In case A < 0, we subtract this number and obtain in each case an equation containing only non-negative summands on both sides. For k = m the righthand side is equal to zero. Since B is not empty, in this case A has to be zero, and the only solution is the vector p^ G t/+ with coordinates x" = —a"'/Xc( fiir a = 1,. .., m. Since, by assumption, D 7^ o, the vector —p^ is no solution, and hence z is no center for B. In the case k < rn we find a solution for any value of the left-hand side; in particular, this holds for the value

which occurs for x" = 0, a = 1 , . . . , A;. If Ai = 0 , then a" = 0, a = 1,... The only solution is then

,k.

since D 7^ 0. In this case, however, —p^ is no solution, hence z is no center. Let now Ai > 0. Then the equation to determine the remaining components of ?i, Ai = -i:™^,+iA„(x°+a°/A„)2, is the equation for a hyperellipsoid in the Euclidean vector space t/_. If not all a" are zero, a = A; + 1 , . . . ,TO,then we certainly find a solution ?i = ^™=fc+ieax° mit (D,yi) = S^+.a'^x'^ ^ 0. But then again, —f^ cannot be a solution and z no center. All that remains is to consider the case a" = 0, a = A; + 1 , . . . ,TO.Then we write (99) in the form i : L i A a ( x ° + a°/A„)2 = -i:™^,+ iA„(x°)2

-A=:A2

and fix the x", a = A; + 1 , . . . , TO, SO that A2 > 0. This time the solution is a hypereUipsoid in the Euclidean vector space t / + . But since D 7^ 0, not all a", a = 1 , . . . , A;, can be equal to zero as well, we again find a solution ^i G U^ for which —f^ does not solve the equation. Rewriting the equation as described the case A < 0 is treated along the same lines. We leave this to the reader. D As already carried out in Euclidean geometry we want to describe the geometric properties of a quadric, e.g. to be central, in terms of its invariants. As we know, rank and index of a symmetric real bilinear form are invariant for all linear transformations, hence for the pseudo-orthogonal g e Gn as well. A symmetric bilinear form and the corresponding quadric are called degenerate if the rank is less than n + 1. The quadric does not change, when its equation is multiplied by a number K G R * ; so we may always suppose

2.6 Hyperbolic Geometry index(6) < rk(6)/2.

295 (100)

Furthermore, there stiU is the possibihty to suitably normalize b by multiplying it with a positive number. However, we do not want to generally require this normalization; instead we will vary it according to the specific situation. In Lemma 2.20 we associated a linear endomorphism with each biform in a vector space with scalar product; moreover, this map is a G^-isomorphism, and in our case hnear as weU (cf. Exercise 2.17). Hence the Gn-invariants of the associated endomorphism are, at the same time, invariants of the symmetric bilinear form, and, having agreed upon its normalization, also invariants of the corresponding quadric. Equation (2.49) for the matrix {a\) of the associated linear map and the form (2) of Gram's matrix immediately imply that, for a pseudo-orthonormal basis, it is simply obtained by multiplying the first row of the matrix (6jj) for the symmetric bilinear form by —1: a"' \

f-1

o'\ r

/ - I Ho ^ n n

/

\

0

In J

fb° I \ \ ^

D' A D D„„ J

f-boo

/ - I \

^

-D' D

/•

-^nn

/

(101)

Here A„„ = B„„ is a symmetric square matrix of order n, /„ is the unit matrix, and the vectors denoted by ' are the corresponding row vectors. From this formula it is obvious that (6jj) is diagonal with respect to a pseudoorthonormal basis if and only if this basis consists of eigenvectors for the associated operator; in fact, both matrices are simultaneously diagonal or not, respectively. Note that, in general, the matrix of the symmetric operator a is not symmetric; this only holds for D = o. Lemma 26 and the form of the associated operator imply Corollary 27. A non-empty quadric B C P " ( R ) determined by a symmetric bilinear form b is central if and only if the corresponding associated endomorphism a has a time-like eigenvector, and this again holds if and only if a has a pseudo-orthonormal basis of eigenvectors. P r o o f . Lemma 26 and Formula (101) immediately imply that eo is an eigenvector for a, and vice versa. If eo is a time-like eigenvector, then the subspace [eo] is invariant for a. By Exercise 2.17, a is symmetric, and so according to Lemma 2.21 the Euchdean subspace E := [eo]^ is invariant for a. By (101) a\E is a self-adjoint endomorphism in a Euclidean vector space. According to the Eigendecomposition Theorem, Proposition 1.6.4.2, a has only real eigenvalues, and, moreover, there is an orthonormal basis of eigenvectors for a\E; hence, completed by eo it is the pseudo-orthonormal basis of eigenvectors for a we were looking for. The converse is trivial. • Proposition 28. Let B C P"(R) be a non-empty, non-degenerate central quadric in hyperbolic projective space. Then there is a unique symmetric bilinear form b of rank n + I such that B is the solution set of the corresponding quadratic equation

296

2 Cayley-Klein Geometries 6(y) = -(x°)2 + Ai(xi)2 + . . . + A„(x")2 = 0.

(102)

The index k + 1 ofb satisfies the inequality k <{n— l ) / 2 , (x*) are the coordinates off in a pseudo-orthonormal basis of eigenvectors for the endomorphism a associated with b, and -1 < Ai <

< A,

(103)

Two of these quadrics are hyperholically congruent if and only if the sequences of eigenvalues for the associated endomorphisms coincide, provided that the fixed normalizations are taken into account. Such a quadric contains a hyperbolic point X G BH = B n H"" if and only if X„ > 1. It only consists of hyperbolic points if and only if Xi > 1.

Fig. 2.30. Ellipse in the conformal disk and in Poincare's model, Ai = 1.1, A2 = 4. P r o o f . By (100) we may suppose that the index k + 1 of 6 satisfies the inequahty k < {n — l)/2, since b is non-degenerate. The form b cannot be definite, since then the quadric B would be empty. According to Corollary 27 the form b has a pseudo-orthonormal eigenbasis. We order the eigenvalues by their magnitude and divide the form by |Ao|; then relations (102) and (103) hold. Note that all eigenvalues are different from zero, since b is nondegenerate. By Lemma 2.20 the map b 1-^ a, assigning the associated linear endomorphism to each symmetric bilinear form, is a Gn-map, and by Exercise 2.17, d) a linear isomorphism as well. Hence the eigenvalues are G^-invariants of the bilinear form 6, and, because of the uniqueness we accomplished by the normalization, of the quadric B as well. If two quadrics have the same system of eigenvalues satisfying (105), then they are G„-congruent; the transformation g G G„ transforming the corresponding pseudo-orthonormal bases into one another realizes the congruence (assignment by equal coordinates). The quadric B contains hyperbolic points if and only if B n H^ ^ 0; i7+ again

2.6 Hyperbolic Geometry

297

denotes the pseudo-Euclidean model of hyperbolic geometry (of. Figure 2.17), described by the equation of time-like unit vectors: -(x°)2 + (xi)2 + . . . + (x")2 = - l .

(104)

Subtracting (102) from (104) yields the equation (l-Ai)(xi)2 + ... + (l-A„)(x")2 = - l , which all points of the intersection B D H^ satisfy. If now A„ < 1, then, by (103), the left-hand side of the last equation contains only non-negative terms, and hence there can be no real solutions. If, on the other hand, A„ > 1, then using (x°)2 = A„/(A„ - 1), xi = . . . = x " - i = 0, (x")2 = 1/(A„ - 1) we obtain two solutions of both equations determined by x'-' > 0 and the roots x" = ±Y^1/(A„ — 1). To show the last claim we look at the quadric B in Klein's model, i.e., in V"+ we consider the intersection of the cone (102) with the hyperplane x'-' = 1. It is described by the equation Ai(xi)2 + . . . + A„(x")2 = l.

(105)

If Ai > 1, then, because of (103), for « = 1 , . . . , n we have Aj > 1, the solution of the equation is a hyperellipsoid which according to

completely lies in the interior of the n-dimensional ball -D" of radius 1. If Ai < 1, then for Ai < 0 it is a hyperboloid, which at any rate contains points outside of -D". For 0 < Ai < 1 it is a hypereUipsoid with largest semi-axis V ^ l / ^ i ) > 1, which thus contains at least two limit points (for Ai = 1) or points lying in the outer region of -D". • This easily leads to the classification of non-degenerate central quadrics in hyperbolic geometry: Proposition 28 immediately implies Corollary 29. Under the assumptions of Proposition 28 the sequence ( A i , . . . , A„) of coefficients occurring in Equation (102) forms a complete system of invariants in hyperbolic geometry for the class of central quadrics considered there. If Xi ^ 1 for i = 1 , . . . ,n, then there is just the single center z = [eo] for the quadric. In the general case, the centers of such a central quadric form a hyperbolic (d— l)-plane, where d denotes the dimension of the eigensubspace for a corresponding to the eigenvalue 1. P r o o f . To prove the second assertion note that by (101) and (102) the zeroth eigenvalue of the endomorphism a associated with h is always aoo = 1 under the assumption of Proposition 28. Thus, because of Aj ^ 1, the corresponding eigensubspace is always orthogonal to eo and hence Euclidean. If

298

2 Cayley-Klein Geometries

the eigensubspace for the eigenvalue 1 of a, which is always pseudo-Euclidean, has dimension d > I, then each time-like unit vector in this space determines a center, and aU these centers together are a hyperbohc {d — l)-plane. • E x a m p l e 12. Already in Proposition 17 we saw that the metric hyperspheres are quadrics. Their defining equation derived there (formula line following (54)) implies that their invariants all coincide; more precisely, Aj = (coth(r))^, « = 1,.. .,n, where r denotes the hyperbolic radius of the hypersphere. Obviously, the center is uniquely determined. Figure 2.30 shows hyperbolic ellipses having a unique center, just like the circles. The equidistants are central quadrics as well. Equations (55) describe the time-like unit vectors representing the points of an equidistant. Hence the corresponding cone in the pseudo-Euclidean vector space has the parameter representation t){x,s) = t)^(a;)s, where by (55) we have to set x = [p] with ^ e W = [e„]^ and (j;,j;) = —1. Eliminating the coordinates of x and s in this parameter representation implies that the equidistant with distance r to the hyperplane x" = 0 is a central quadric, whose normalized equation has the following form:

-iy°f + {y'f + ••• + {y^'-'f + (y")'(cothr)2 = o. The invariants for these equidistants are thus Ai = . . . = A„_i = 1, all points in the hyperplane determining them are centers, cf. Figure 2.27, and the last invariant is a function of the distance parameter. All equidistants with the same distance parameter are Gn-congruent, which was clear anyway (cf. Exercise 24), but is again evident from this representation. FinaUy, Figure 2.31 shows a central quadric not completely belonging to the hyperbolic plane. In the projective plane these are ellipses whose hyperbolic points have two branches orthogonally intersecting infinity, i.e. the boundary. As in the Euclidean situation their tangents become asymptotes for the curve point moving towards infinity. Hence it is certainly justified to call them hyperbolas in nonEuclidean geometry. We recommend that the reader proves the orthogonality; a proof as well as a collection of formulas to work with in the hyperbolic plane and to produce the pictures included here can be found in the Mathematica notebook noeuklid.nb. •

E x a m p l e 13. The horospheres are also quadrics; they are a first example for quadrics whose associated endomorphisms do not have a basis of eigenvectors. From the definition of a horosphere it is obvious that they cannot be central quadrics: In fact, they have a unique limit point, which, for symmetry reasons, is impossible in the case of central quadrics (cf. Figure 2.29). In order to derive a quadratic equation for the horosphere (64) we multiply this equation by the parameter s and thus pass to the equation for the corresponding hypercone.

2.6 Hyperbolic Geometry

299

Fig. 2.31. Non-Euclidean hyperbolas, Ai = 0.5, A2 = 1.8. Then we eliminate the parameters 0, s. The coordinates (z*) of the points in the horosphere satisfy the equation n\2

I ri

0

^r=i(-2*) +2(^") +2z"z

n

0,

(106)

which one can also easily verify directly. Hence the matrix for the associated endomorphism in the pseudo-orthonormal standard basis is

The characteristic equation of this matrix is

xa(t) = ( i - t r + ^ Thus the only eigenvalue is 1, it has multiplicity n + 1. There are, however, only n linearly independent eigenvectors, which are even mutually orthogonal: a{ti) = ej for « = 1 , . . . , n - 1, a(e„ - eo) = e„ - eo. But the last eigenvector is isotropic, it corresponds to the hmit point of the horosphere. This may be seen by dividing the representing vector on the righthand side of (64) by (0, 0) and then letting 0 tend to infinity. In the basis Oj := ej for « = 1 , . . . , n - 1, o„ := eo, a„+i := e„ - eo, which is no longer pseudo-orthonormal, the matrix for a has the Jordan normal form '/„-i 0 0

, - •

300

2 Cayley-Klein Geometries

This describes the invariants for the horospheres, which are aU Gn-congruent, as already mentioned. At the same time this provides an example for Exercise 2.19, which was to show that the eigenvector for a Jordan cell of order k > l always has to be isotropic. If, conversely, the operator associated with any quadric has the same Jordan normal form as that of a horosphere, then this quadric itself is a horosphere; as explained before, it suffices to adapt a pseudoorthonormal basis (ej) to the Jordan basis (Oj); then the equation of the quadric in the orthonormal basis has the form (106), which is the equation of a horosphere. • We called the points in the intersection B n Q of the quadric B with the absolute Q the limit points of the quadric B. The formula 6(o, p) = (a(o),j;) = (a, j;)A, if a(o) = oA,

(107)

immediately leads to Corollary 30. The isotropic eigenvectors for the endomorphism a associated with a quadric B determined by the symmetric bilinear form b are limit points for the quadric. • The example of non-Euclidean hyperbolas, cf. Figure 2.31, shows that the converse of this corollary does not hold. Misusing the terminology we will frequently speak of the Jordan normal form, the Jordan basis, the eigenvectors etc. of a quadric to mean the corresponding quantities of its associated endomorphism. To classify the quadrics it is crucial to decide, which Jordan normal forms may at all occur as normal forms for quadrics. For each of these types of normal forms the eigenvalues appear as parameters, which, taking into account a normalization convention, are invariants of the quadric. Now we will proceed with the classification of non-degenerate, plane quadrics. The variety of possibilities already occurring in this simple case gives an impression of the difficulty to be expected of the classification in arbitrary dimensions. A report on the results of the classification in the n-dimensional hyperbohc space can be found in Section 3.7.3 of the book [95] by B. A. Rosenfeld. There J. L. Coolidge [33] is also quoted, a first systematic presentation of hyperbohc geometry treating the classification of quadrics in the hyperbolic space H as well. Now let a be the endomorphism associated with a plane quadric B. Its characteristic polynomial has degree 3 and real coefficients; hence a has at least one real eigenvalue A. For non-degenerate quadrics all eigenvalues are different from zero, and there are only the following possibilities: 1. There is a time-like eigenvector. Then the quadric is central. 2. All eigenvalues are real, but there is no time-hke eigenvector. 3. There are a real eigenvalue A as well as two non-real, conjugate complex eigenvalues JJ,, JJ,

2.6 Hyperbolic Geometry

301

Obviously, these cases exclude each other; in fact, in Case 3 A can have no time-like eigenvector, since then, according to Corollary 27, all eigenvalues were real. In Case 1, according to the same corollary, the quadrics are classified by Proposition 28. Considering Case 2 we distinguish the following excluding subcases: 2.1. There is a unique J o r d a n cell of dimension 3. 2.2. There is a J o r d a n cell of dimension 2. Three one-dimensional J o r d a n cells cannot occur in case 2, cf. the next Exercise 26. We wiU show t h a t quadrics of the types 2.1, 2.2, and 3 exist and are classified by the corresponding J o r d a n normal forms. Exercise 26. Let a be a symmetric endomorphism of a pseudo-Euclidean vector space V" of index 1. Suppose that a is diagonalizable, i.e. a has n linearly independent eigenvectors. Then a has a time-like eigenvector as well. Hint. First prove that two linearly independent isotropic eigenvectors for a have to belong to the same eigenvalue. E x a m p l e 14. We consider the family of symmetric bilinear forms 6j'3(A) depending on a real parameter A G R , which, in pseudo-orthonormal coordinates, are determined by a matrix of the same form denoted by the same symbol /-A01\ 6j3(A) := 0 A1 . (108)

V1 IV T h u s with the notations z = x^,x

= x^,y = x'^ the equation of this quadric is

A ( x 2 + y 2 - z 2 ) + 2y(x + z) = 0, Xy^O.

(109)

For A = 0 this is just a degenerate quadric. In the same coordinates the associated operator has the matrix

So the J o r d a n normal form of this matrix is of the form we wanted (110)

We call these quadrics J3-curves; Figure 2.32 shows some of these curves in the conformal and in Poincare's model. We may suppose A > 0 to hold in (109); multiplying by —1 and reflecting in the x-axis (x, y, z) i-^ (x, —y, z) we again obtain Equation (109) with positive A and a G'2-congruent curve. Now we only consider orientation preserving transformations. Then each sign of A

302

2 Cayley-Klein Geometries

in (108) corresponds to one class of J3-curves, and the reflection transforms one into the other. Figure 2.32 displays together with each curve its mirror image. Since the eigenvalues of the associated endomorphisms are G-invariants for the bilinear forms, no two of the quadrics determined by (109) (in pseudoorthonormal coordinates) with different values of |A| can be G'2-congruent. To classify the quadrics of Type 2.1 it remains to prove t h a t each quadric whose associated endomorphism has the J o r d a n normal form (110) is G'2-congruent to a J3-curve (109). D

Fig. 2.32. J3-curves, A = ±j/2,j

= 1, 2,

.10.

So let B be a non-degenerate quadric of type 2.1. We want to find a pseudo-orthonormal basis for the pseudo-Euclidean vector space V in which the quadric is described by an equation of the form (109). By assumption, there is a Jordan basis (Oj) for V in which the matrix of the operator a associated with B has the J o r d a n normal form (110); obviously, this basis is not pseudo-orthonormal, since the eigenvector 03 belonging to this basis is isotropic (cf. Exercise 2.19). We will see this again in a moment. T h e symmetry property of a, (a(?), t)) = (?, a(t))) for aU y, t) G F ,

(111)

implies for each J o r d a n basis of a cell with dimension A; > 1 (Oi+i,Oj) = (Oi,Oj+i),

i,j = l,...,k,

(112)

where we have to set Ofc+i = 0. This immediately follows from the form of the matrix (110) together with (111):

(a(oi), Oj) = (Oi, Oj)A + (Oi+i, Oj) = (Oj, a{aj)) = (Oj, Oj)A + (oj, Oj+i). Setting j = k in (112) and using Ofc+i = 0 yields (Oi,Ofc) = 0 for «

,, k]

(113)

2.6 Hyperbolic Geometry

303

in particular, we arrive at the asserted isotropy (Ofc, Ofc) = 0. Moreover, for i = k — 2,j = k — I we have: (Ofc-i,Ofc-i) = (Ofc-2,afc),

A; > 3.

(114)

This has a consequence, which very much restricts the possibilities occurring in the classification of symmetric endomorphisms in Minkowski spaces of arbitrary dimension: C o r o l l a r y 3 1 . If X is a real eigenvalue of a symmetric endomorphism a in an n-dimensional pseudo-Euclidean vector space of index 1, then the cells Z\'^(A) in its Jordan normal form have at most dimension A; = 3. P r o o f . If A; > 3, then (113) and (114) implied (ofc-i,Ofc-i) = 0. But then the subspace [ofc-i,Ofc] were totally isotropic, and this cannot occur in Minkowski space, cf. Exercise 2.4. D In our case, k = 3, (114) only imphes (02, CI2) = (cii, CI3) ^ 0; in fact, for the same reason as in the proof of Corollary 31 02 cannot be isotropic. More precisely, since the index of V is equal to 1, the vector 02 orthogonal to 03 by (113) is space-hke, hence /x := (02, CI2) > 0. As (Oj/,//!) is also a J o r d a n basis, after this normalization we may suppose (02,02) = (01,03) = 1.

(115)

Moreover, for each J o r d a n basis we have (02, 03) = (03, 03) = 0 ,

(fc = 3).

(116)

It is easy to verify t h a t for each z/ G R the transformation Oi : = Oi + 02*^, 02 : = 112 + 03*^, 03 : = 113

again defines a J o r d a n basis for a, which also satisfies (115). Together with (116) this imphes (01,02) = ( 0 1 , 0 2 ) + 2i/,

and choosing z/ = — (oi, 02)/2 we can achieve t h a t the scalar product is zero, (oi, O2) = 0 . Returning to the former notations there now is a J o r d a n basis for which, apart from (115), (116), we also have (ai,O2)=0.

(117)

Lastly, for each /x G R the transformation Oi : = Oi + 03M, &2 : = 02, &3 : = 03

also transforms a J o r d a n basis again into a J o r d a n basis, and Equations (115)(117) are preserved. The equality

304

2 Cayley-Klein Geometries (oi,Oi) = (ai,Oi) + 2 / i

implies that, suitably choosing jj,, passing to the old notations we may arrange that (oi,Oi) = - l . (118) This proves the first part of the following lemma. Lemma 32. For each symmetric endomorphism a in the Minkowski space V whose Jordan normal form consists of a single Jordan cell ^^(A) there is a Jordan basis satisfying (115)-(118). Starting from such a basis the relations to ••= a i , ei := Oi + 03, e2 := 02

define a pseudo-orthonormal basis for V .

(119)

D

The orthogonality relations for the basis (ej) are easily verified using (115)(118). obviously, the assumption implies A G R. The matrix for the basis transformation defined by (119) is (ei) = (Oi)T mit T =

/llO^ 001

Since the operator a has the matrix (110) in the Jordan basis (Oj), its matrix in the basis (ej) is 'A0-1\ 0 A 1 11

A ;

/A00\ = T ^ M 1 A 0 T. VOIA;

Passing to the associated bilinear form we obtain the matrix 6j3(A) defined by (108) for the J3-curve (109). Hence we obtain the result we aimed at: Proposition 33. Each non-degenerate quadric of type 2.1 in the hyperbolic plane is G^-congruent to a unique J^i-curve (109) with A > 0. The J^i-curves with parameters A, —A are transformed into one another by the reflection in the line connecting their limit points. • Now let B be a non-degenerate quadric of Type 2.2. We want to adapt a pseudo-orthonormal basis to B in such a way that the suitably normalized equation determining B has a normal form still to be found depending only on the eigenvalues A, /x of the associated operator a. This aUows to conclude that two quadrics of this type with equal eigenvalues are congruent. For each Jordan basis 0, b, c of the operator a we have a(o) = oA + b, a(b) = bA, a(c) = c/x.

(120)

2.6 Hyperbolic Geometry

305

A Jordan basis cannot be pseudo-orthonormal, since by Exercise 2.19 b is isotropic. If /x 7^ A, then by Proposition 2.24 c is orthogonal to b and hence space-like, since a pseudo-Euclidean vector space of index 1 cannot contain an isotropic vector orthogonal to a time-like or isotropic vector and linearly independent of it. But even in the case X = jj, the vector c is space-like, since the symmetry property of a implies (a(o), c) = (o/i + b, c) = (0, a(c)) = (0, c)/i, and hence (b, c) = 0. Then one argues as before. We normalize c and set c = e2. Since a is symmetric, according to Proposition 2.24 the pseudo-Euclidean subspace [C2]^ is also invariant under a, and (0, 62) = 0 implies (b,e2) = (a(o) - aA, 62) = 0. Hence [{0, b}] is a pseudo-Euclidean vector space, and because of the isotropy of b we have (0, b) ^ 0. The foUowing is easy to verify: If {0, b} is another Jordan basis for a|[o, b], then 0 = OS + bt, b = bs, (s ^ 0).

Since (0, b) = (0, b)s^, the sign of (0, b) is invariant under the transformation. We choose s so that (0, b) = ± 1 . Then we compute the unique t solving the equation (0, 0) = (0, o)s^ + 2(0, b)st = - 1 . Thus we adapted a pseudo-orthonormal basis to the operator a in such a way that 0 = eo, b = eo + ei, c = e2 for (0, b) < 0, (121) and 0 = eo, b = -eo + ei, c = e2 for (0, b) > 0,

(122)

is a Jordan basis for the operator a in each case. For the matrix of the operator a in the pseudo-orthonormal basis {Cj),j = 0,1, 2 this imphes {c4)=

I

1

A - l 0 I if (o,b) < 0 ,

(4)=

/ A - l -1 0\ 1 A+ l 0

(123)

and

\

0

if (o,b) > 0 .

(124)

0 i^J

The matrices of the associated symmetric bilinear forms are (/3|)=(

-A-l 1 0\ 1 A - 1 0 I or (/3|)= ( 0 0 /i

1

A+10|.

(125)

306

2 Cayley-Klein Geometries

We now intersect the cones in the vector space V corresponding to these bihnear forms with the coordinate plane x^ = 1 and rewrite the variables as X = x,x = y. This way we obtain the normal forms for the equation of the quadric in Klein's model we were looking for: - A - 1 + 2x + (A - l)x2 + /iy2 ^ 0 if (o, b) < 0,

(126)

1 - A + 2x + (A + l)x^ + /iy2 = 0 if (o, h) > 0.

(127)

and We call the solutions to these equations J2-curves; more precisely, these are the solutions of (126) with J2^(A, fi) and those of (127) with J2+(A, fi). Since l-i y^ 0, they are very simply expressed algebraically in terms of roots, e.g. in the case of (126) as y

±V^(A + 1 - 2x + (1 - A)x2)//i.

(128)

The normalization for the equations is the requirement /3oi = 1 determining the only mixed coefficient. We ask under which condition two J2-curves are congruent. This holds if and only if their normalized equations can be transformed into one another by a pseudo-orthogonal transformation of the coordinates. Hence no curves in any of the families J2+, J 2 ^ can be congruent, since they differ in the eigenvalues of the associated operators. Substituting in (126) X -^ —X and multiplying the resulting equation by —1 yields Equation (127) for the parameter value —A, —jj,, which leads to the congruence J 2 + ( - A , - / i ) - J2"(A,/i).

(129)

What we discuss here is the algebraic congruence, since the equivalence of equations in question is that under pseudo-orthogonal coordinate transformations. To the existence of real or hyperbolic points, respectively, whose coordinates satisfy these equations, we will return later. Obviously, algebraic equivalence of the equations implies the geometric congruence of their solutions. As already mentioned in Euclidean geometry the converse does not necessarily hold. Summarizing we formulate Proposition 34. For all numbers A, /x G R satisfying Xjj, y^ 0 there is a unique equation for a quadric in the normal form (126) or (127), respectively, whose associated operators have these numbers as their eigenvalues. J2-quadrics are congruent if and only if corresponding eigenvalues of the operators associated with the normal forms (126) or (127) are equal. Quadrics of the families (126), (127) only differ by a reflection in the y-axis, where (129) holds. • The algebraically simple form of the solutions allows to decide, whether the quadric contains hyperbolic points. To this end we need a series of elementary observations: We need to find the positive domains for the radicands of (128),

2.6 Hyperbolic Geometry

307

have to determine the intersection with the interval — 1 < x < 1, and must then investigate, if the image points with coordinates (x, y) belong to the unit circle for the resulting sets. These considerations can be found in the Mathematica notebook noeuklid2D.nb on the homepage of R. Sulanke. This notebook contains many tools for the investigation of the hyperbolic plane; the modules to produce the figures reproduced in this section can be obtained there as well. Here we only present the results of these considerations and, at the end of this section, the corresponding pictures computed in the conformal model. Now we still have to find the quadrics of Type 3 whose associated operators have two conjugate complex eigenvalues X ^ X and a real eigenvalue fi ^ Q. Consider the complex extension Vc of the pseudo-Euclidean vector space V . Bilinearly extending the pseudo-Euclidean scalar product V^ becomes a complex Euclidean vector space, cf. Lemma 1.10.15, Example 1.10.9, and Example 2.2.5. The linear extension of the symmetric linear endomorphism a, uniquely determined by Lemma 1.10.5 (and denoted the same way), is a symmetric linear endomorphism of the complex Euclidean vector space Vc] this is immediate from the linearity of the extensions considered, if the symmetry property is expressed in a basis for the real form V, which, at the same time, is a basis for Vc- Since the eigenvalues of the endomorphism are different, according to Proposition 2.24 the corresponding eigensubspaces are orthogonal, one-dimensional complex subspaces of V^, and, since the scalar product is non-degenerate, they are not isotropic. Dividing by the real eigenvalue yU, normalizes the equation of the quadric and the associated operator so that we may suppose /x = 1. We then find a real eigenvector c G V, which has to be space-like: If it were time-like, then we had Case 1 of a central quadric; moreover, it cannot be isotropic according to what we just mentioned. Next we normalize it and take it to be the last vector t^ in a pseudo-orthonormal basis for V . The complex extension of W := [t^]^ C V is the complex orthogonal complement W"^ = [62]^ C Vc, which is spanned by the eigenvectors 0, b for the eigenvalues A, A; moreover, these are not isotropic. We normalize 0 so that (0, 0) = - 2 . Let A = a + /3i, o = j; + t)i, a , / 3 G R , p, t)G W, be its decompositions into the real components. Passing to the complex conjugate quantities, a(o) = oA imphes that b := p — t)i is an eigenvector for the eigenvalue A. From (0, 0) = (y, y) - (t), t)) + 2 i(j;, t)) = - 2 , (0, b) = (j;,j;) + (t),t)) = 0

we immediately conclude the orthogonality relations -(?,?) = (t),t)) = 1, (?,t)) = 0 .

We set eo := p, ei := t) and represent a in this basis taking into account the relations holding for an eigensystem; then

308

2 Cayley-Klein Geometries

Proposition 35. For each non-degenerate quadric B of Type 3 in the hyperbolic plane there is a pseudo-orthonormal basis {cj) for the associated vector space such that, having normalized the equation to have the real eigenvalue 1, the matrix of the associated operator a has the form (130) where A = a + i/3,/3 7^ 0, denotes a complex eigenvalue of a. Thus the normal form of the equation for such a quadric B in Klein's model is a(x2 - 1 ) - 2 / 3 x + y2 = 0 .

(131)

Two such quadrics are congruent if and only if their normal forms coincide up to the sign of [3. All these quadrics contain hyperbolic points. • It is straightforward to see the solutions from the simple form of Equation (131). Moreover, one notes that replacing [3 by —[3 the corresponding curve is obtained from the first by a reflection in the y-axis; it is also equivalent to exchanging A and A. The last assertion in the proposition is obtained by discussing the domains where the solutions are real as well as their magnitude; this is explained in the Mathematica notebook noeuklid2D.nb already cited. It also contains a module qc[a, [3], which computes the quadric corresponding to the parameters a and /3, more precisely, its hyperbolic points, in the conformal model; for brevity we call them qc-curves. The Figures 2.44, 2.45 were plotted using this module.

2.6 Hyperbolic Geometry

309

2.6.9 Pictures of Hyperbolic Quadrics In this subsection we collect pictures of some quadrics discussed in t h e last subsection.

Fig. 2 . 3 3 . J2"-curves, A = 1, /u = 1.1,1.4,1.7,... , 3.2.

Fig. 2.34. J2"-curves, A = 3, /u = 3.1, 3.8, 4 . 5 , . . . , 8.

310

2 Cayley-Klein Geometries

Fig. 2.35. J2"-curves, A = 1.5, 2.5,.. . , 9.5, /u = 10.

Fig. 2.36. J2"-curves, A = 0.2, /u = .3, 2.3,4.3,

Fig. 2.37. J2"-curves, A :

-1.1,-2.1,

,18.3.

-8.1,/Li:

2.6 Hyperbolic G e o m e t r y

F i g . 2 . 3 8 . J 2 " - c u r v e s , A = 0, - 0 . 2 , - 0 . 4 , - 0 . 6 ,

-2,/Ll= 1.

F i g . 2 . 3 9 . J 2 " - c u r v e s , A = - 1 0 , ^J, = - 1 ,

Fig. 2.40. J2"-curves, A :

-ll,/Ll = A + 1.

311

312

2 Cayley-Klein Geometries

Fig. 2.41. J2 -curves, A = -10,/K :

Fig. 2.42. J2"-curves, A :

Fig. 2.43. J2"-curves, X = /j.

-11,

-19.

-ll,/Ll = A -

-10, for A = /K : -1 the limit circle.

2.6 Hyperbolic Geometry

Fig. 2.44. qc-curves, A = 2^ ± i v 7 , j = 1,

Fig. 2.45. qc-curves, A

^^27Tij/10j

±1,

.10.

,±9.

313

314

2 Cayley-Klein Geometries

2.7 Mobius Geometry As already mentioned in the introduction to Section 2.6, from the projective perspective, Mobius geometry is nothing but the geometry on the hypersphere in projective space determined by the action of the isotropy group for this hypersphere. By CoroUary 3.1, this group, in the context of Mobius geometry caUed the Mobius group, acts transitively on the hypersphere and leaves, as will be shown in a moment, the angle between intersecting circles invariant; using methods from differential geometry it can be proved t h a t this group coincides with the conformal group of the hypersphere for the Riemannian metric induced by the standard embedding of the spheres into Euclidean space. HistoricaUy, the Mobius group was first introduced as the group of globaUy defined conformal maps on the Riemann sphere S within the framework of complex function theory. On the other hand, the Mobius group is the isometry group for the hyperbolic metric defined on the inner region of the hypersphere, cf. CoroUary 6.10. In the context of projective geometry this leads to a close and natural relation with hyperbolic geometry t h a t will be applied in Section 7.1 to transfer the angle invariants from hyperbolic to Mobius geometry.

2.7.1 Spheres in M o b i u s Space In accordance with Section 2.6 we agree to use the foUowing notations: Let V = V"+2 be the (n+2)-dimensional pseudo-Euclidean vector space equipped with the scalar product (,) of index 1, and let P " ^ be the associated projective space. The n-dimensional Mobius space S*" is understood to be the hypersphere Q for the scalar product (,) (cf. (6.3)) in the ( n + l)-dimensional real projective space P " + ^ . Thus, we essentially retain the notations of Section 2.6, the dimension, however, is increased by 1 in order to keep the letter n for the dimension of the geometric space to be studied. In Lemma 6.19 we already described the isotropy group of a point a; G S*" with respect to an isotropic orthogonal basis in terms of block matrices; hence P r o p o s i t i o n 1. The Mobius space S*" is a homogeneous space for the action induced from the projective action of Gn+i = 0 ( 1 , n + 1)+ C 0 ( 1 , n + 1). As a quotient space it is isomorphic to S" = Gn+i/CE{n), where CE{n) denotes the conformal Euclidean group, which is embedded Gn+i via the matrix representation (6.61) (with n — 1 replaced by n).

(1) into D

To justify this notation we first define the group of similarities for the Euclidean space E": A similarity is a product of a Euclidean transformation with a dilation. These transformations form a subgroup of the affine group for the Euclidean space E"^; using methods from differential geometry and, in case n = 2, function theory one can prove t h a t it coincides with the group of

2.7 Mobius Geometry

315

all conformal (i.e. angle-preserving) transformations of .B", n > 2. T h e group CE{n) defined in Proposition 1 is isomorphic to the group of similarities: Exercise 1. Prove that the group described by the matrices (6.61) is isomorphic to the group of similarities for the Euchdean space E". (Hint. Use the following model for the Mobius space S": Consider the unit hypersphere cut out of the isotropic cone (p, p) = 0 by the Euchdean hyperplane

S"+^ -(eo,j;) = -(ao + o„+i,?)/V^=l in the pseudo-orthogonal vector space V. Here and in the sequel let (ei) denote a pseudo-orthonormal basis, and let (Oi), i = 0 , . . . ,n + 1, be the isotropic orthogonal basis associated to it via (6.58). Take the fixed point s := [Oo] = [eo — En+i] e 5 " as the South Pole for the stereographic projection st : S" \ {s} -^ E" from S" onto the equatorial plane E" C E"+^ of S" determined by (e„+i,j;) = 0 (cf (6.18) and Exercise 6.5). Assigning g e CE{n) i—> stogost^^ describes the isomorphy we were looking for. Here, for any element g = a{A, 0, A) the entries A, 0, A correspond to the rotations or rotoinversions, translations, or dilations in the group of similarities of E", respectively.) We may also identify the group of similarities with the conformal Euclidean group. Obviously, each similarity preserves the angles between intersecting lines, and hence it is conformal. T h e next two exercises contain geometric characterizations for the conformal Euclidean group. Exercise 2. Prove that for n > 2 the conformal Euclidean group CE{n) coincides with the group of bijective maps of E" transforming pairs of orthogonally intersecting hues into the same kind of pairs. (Hint. Apply Proposition 1.5.6 and consider the images of lines intersecting orthogonally at the origin with direction vectors Zi + Zj, ti — tj for i / j.) Obviously, this immediately implies that for n > 2 the conformal Euchdean group CE(n) coincides with the group of bijective maps of E" transforming lines into lines and preserving angles between intersecting lines. Exercise 3 . Prove that for n > 2 the conformal Euclidean group CE{n) is the group of bijective maps of E" transforming hues into hues and circles into circles. (Hint. RecaU Thales' Theorem and apply the preceding exercise.) Using isotropic orthogonal coordinates is often advisable to concentrate on the points of S*" as the simplest basic objects. Apart from the points the most important elementary objects of Mobius geometry are the m-spheres, for the investigation of which we will again use pseudo-orthonormal coordinates because of their close relation to the hyperbolic (m + l)-planes. As an m-dimensional subsphere S™ C S*", for short m-sphere, we consider the intersection of a hyperbolic ( m + l ) - p l a n e B™^ with S*". More precisely: liW C V is an (m + 2)-dimensional pseudo-Euclidean subspace, and B™^^{W) is the corresponding projective subspace, then the associated m-sphere be defined as gm ^ gm^^-j ._ B'^+^W) n S*", m = 0 , 1 , . . . , n. (2)

316

2 Cayley-Klein Geometries

So 0-spheres are just arbitrary two-element sets {x, y}, x,y e S*", x ^ y; 1-spheres are circles, and (n — l)-spheres are hyperspheres of S*". As nondegenerate quadrics of index 1 in (m + l)-dimensional projective spaces the m-spheres themselves are m-dimensional Mobius spaces as well. Denote by Sn,m, m = 0 , . . . , n, the set of m-spheres in the Mobius space S*". Obviously, the map F : B™+1

G Hn+l,m+l

^

5™ = B ™ + ^ n 5 " G Sn,m

(3)

is bijective and an equivariant isomorphism from the transformation group -ffn+i.m+i of hyperbolic (m + l)-planes onto S'„,m, i-e. F{gB"'+^) = gF{B"'+^) for aU g G G„+i,B™+i G F„+i,™+i. This correspondingly also holds for the transformation groups of Gn+i defined by extending the action to power sets, product sets, etc. The transitivity properties for the actions of Gn+i in hyperbolic geometry immediately imply: Corollary 2. The Mobius group Gn+i acts transitively on the sets S'„,m of m-spheres in the Mobius space S*", m = 0 , 1 , . . . , n. D The Mobius group was defined by restricting the isometry group Gn+i of hyperbolic space to the bounding quadric (5 = 5*". The following proposition states geometric characterizations of this group. Proposition 3. The Mobius group Gn+i, n > 2, is the group of all bijective maps from S*" onto itself transforming circles into circles. Moreover, it is the group of bijective maps from S*" onto itself transforming hyperspheres into hyperspheres. P r o o f . According to CoroUary 2 any Mobius transformation g G Gn+i transforms circles among themselves. Let now, conversely, F : S" ^ S" he a bijective map transforming circles into circles. Use the model for S" described in Exercise 1 and the stereographic projection st. Since, by Proposition 1, the group Gn+i acts transitively on S*", we find a Mobius transformation g G G„+i such that g{F{s)) = s. If we are able to show that the map Fo = g o F is a Mobius transformation, then F = g^^ o Fo is a Mobius transformation as weU; without loss of generality we may thus suppose that F leaves the South Pole fixed: F{s) = s. Now consider the map / := st o F o st^^, which is a bijection of the Euchdean space E" onto itself. Since F transforms any circle through s into a circle through s, which again are bijectively related to the lines in E" via st, the map / maps the set of hues in E" bijectively onto itself. Hence, by Proposition 1.5.6 it has to be an affine map. According to Lemma 6.3 the circles in Euchdean space are mapped by st^^ to the circles in S" not passing through s. Since this set is also mapped bijectively onto itself by F, the map / transforms the set of circles in E" bijectively onto itself. Hence by Exercise 3 the map / is a similarity. So according to Exercise 1 F belongs to

2.7 Mobius Geometry

317

the conformal Euclidean subgroup CE{n) C G „ + i . The proof of the second assertion proceeds along the same lines; we leave it to the reader. D E x a m p l e 1. Examples of Mobius transformations t h a t do not belong to the conformal Euclidean group are the reflections in hyperspheres also called inversions. Frequently, the term inversive geometry is used instead of Mobius geometry, compare e.g. the very instructive introductory text with this title by J. B. Wilker [111]. In the Euchdean space compactified by the point oo to the n-sphere S*" the reflection s in the hypersphere S"^^ with center m = o + xn and radius r is deflned as foUows: s{x) = s{m + p) := m + - — - for x ^ m , oo; s{m)

:= oo, s(oo) := m.

(4)

\?j ?/

Hence the distance from the image y = s{x) to the center is inversely proportional to the distance from the pre-image to the center: \y — m\\x

— m\ = r^,

and the image point y hes on the radial hue through x ^ m,oo; the last two points are interchanged. It is easy to verify t h a t s is involutive, and t h a t the set of flxed points of s coincides with the hypersphere S. Let us show t h a t s corresponds to a Mobius transformation of S*". Applying a similarity one may always arrange t h a t the hypersphere deflning s is the unit hypersphere of the Euchdean space E" with the origin o as its center. Then p is the position vector of the point x, and the position vector t) of the image point y is determined by t) = p/lfl^. Now consider the reflections Fi in the coordinate hyperplanes x* = 0 of the pseudo-Euchdean vector space V, as usual they are deflned with respect to a pseudo-orthonormal basis by Fi{ti) = -ti,

Fi{tj) = tj fiir j ^ i; «, j = 0 , 1 , . . . , n + 1.

Obviously, each of these reflections is pseudo-orthogonal, hence it generates a Mobius transformation. By means of the stereographic projection already used in Exercise 1 one can show t h a t s = st o Fn+i o st^^. Conversely, each reflection in a pseudo-Euchdean (n + l)-dimensional subspace W"^^ C V obviously generates a reflection in a hypersphere or hyperplane of E^; in the sense of Mobius geometry these reflections do not differ. It is easy to see t h a t they may be even deflned as the Mobius transformations, different from the identity, whose flxed point set is a hypersphere. The next proposition justifles the name inversive geometry. D P r o p o s i t i o n 4. Each Mobius transformation uct of finitely many inversions}

can be represented

J. B. Wilker [111] proves that each Mobius transformation g & G n-\-l represented as the product of at most n + 2 inversions.

as the prod-

C&n GV6I1 DG

318

2 Cayley-Klein Geometries

P r o o f . According to Proposition 2.3.5 each transformation g G G„+i can be represented as the product of at most n + 2 reflections in non-isotropic, ( n + l)-dimensional subspaces. The reflections in pseudo-Euchdean subspaces are just the inversions considered in the previous example. If the subspace of flxed vectors is Euchdean, then we may choose the basis (ej) for V so that this reflection coincides with the map -Fb deflned in Example 1; hence it has the coordinate representation y = —X , y-' = X-' for j = l , . . . , n + l. Since now FQ and —FQ lead to the same Mobius transformation, we may well insert —FQ instead of FQ into the representation for g. Again this map is a product of flnitely many inversions: -Fo

= F-i O

...oFn+l.

a This proposition is frequently chosen to be the starting point for a deflnition of Mobius geometry: Accordingly, Mobius geometry is the totality of geometric notions and properties which are invariant under arbitrary inversions. Note, moreover, that Mobius space can be equipped with an orientation. To each pseudo-orthonormal frame (ej), « = 0 , . . . , n + 1, we assign the South Pole s := [eo — e„+i] as well as the coordinate simplex (oj) := ([ej + eo]), « = l , . . . , n + l . The normalized representatives occurring in the definition lie in the hyperplane XQ = —(eo,?) = 1 and have the positive determinant 2. We may express each oriented n-simplex (xi,..., a;„+i) in this frame by normalized representatives n+l

h = ^o + J2^J^f'^ for the determinants we then obtain D := [eo,?!,...,j;„+i] = d e t ( e | ) . For D > 0 the n-simplex is positively oriented with respect to the basis (ej). Obviously, there are two classes of pseudo-orthonormal frames: Those whose coordinate simplex with respect to a fixed frame is positively oriented, and those for which it is negatively oriented. If one of these classes is distinguished in the vector space V deflning S*", then we speak of an oriented Mobius space; in this situation we can distinguish between positively and negatively oriented n-simplices. As already mentioned in the case of hyperbolic spaces, a map g G G„+i preserves the orientation, if its determinant is equal to one, i.e. if g belongs to the special pseudo-orthogonal group S'0(1, n + 1 ) . Since this group also acts transitively on S*", the oriented Mobius space is described as the quotient space

2.7 Mobius Geometry

S*" =

319

SO{l,n+l)/SCE{n),

where SCE{n) denotes the group of similarities with positive determinant; it is cahed the group of orientation preserving similarities of the Euclidean space. 2.7.2 Pairs of Subspheres The definition of the Mobius space S*" and the equivariance of the action of Gn+i over the hyperbohc (m + l)-planes as well as the m-spheres discussed in the preceding chapter immediately allows to conclude properties of the invariants for pairs of subspheres from those of the invariants for pairs of hyperbolic subspaces described in Proposition 6.24 in a purely formal way: Proposition 5. Let F he the eqivariant isomorphism described in (3). Consider the hyperbolic subspaces A and B corresponding to the subspheres SiU)

e Sn,l,

SiW)

e Sn,m,

1.6.

A'+i = F-\S{U)),

B™+i =

F-\S{W)),

with 0 < I < m < n. Then the stationary invariants and the deficit of these subspaces are Mobius invariants for the pair of spheres as well; this deficit and the eigenvalues (6.73) of the operator A G End(Vt^ ) associated with the pair of subspaces are a complete system of invariants for the action of the Mobius group on S„^i x S'„,m. • Note that for the dimensions of the subspaces we now have d i m t / = / + 2,

dimt/

=n — l,

dimVt^ = m + 2,

dimVt^ = n — m,

so that, obviously, despite the difference in the dimensions of the spaces t/, W, V, the number of invariants nevertheless coincides. Now we want to interpret these invariants geometrically within the framework of Mobius geometry and, if possible, express them in terms of Euclidean invariants for the pairs of spheres; in fact, Mobius invariants are all the more invariants for the subgroup of Euclidean transformations. Hence they have to be functions of the invariants in a complete Euclidean system of invariants for the pairs. Before discussing particular dimensions, we consider the Cases 1-3 described in Lemma 6.23 in general. First we again suppose that Condition A from Section 6 is satisfied. In Mobius geometry this condition amounts to the following: Condition A. For the concerned pair 5*1, 5*2 C S*" of subspheres there is no hypersphere i^ C S*" such that SiU S2 C S. As is common for projective spaces we denote the subsphere generated by a set M C S*" by [M]; this is the intersection of all subspheres containing M. Finally, set

320

2 Cayley-Klein Geometries Si V 5-2 := [Si U 5-2]

As before, the number d satisfying dim S\\/ S2 = n — d is called the deficit of the pair of subspheres, df(Si, 6*2) = d. Example 2. The hyperspheres bijectively correspond to hyperbolic hyperplanes, i.e. to (n+l)-dimensional pseudo-Euchdean subspaces W CV. These again are bijectively determined by their one-dimensional orthogonal complements W = [u], {u,u) > 0. The hypersphere corresponding to u is denoted by S{u). Thus we have

i:(u) :=^([u]^)ns'".

(5)

As for the space of hyperbohc hyperplanes, the one-sheeted hyperboloid {u,u) = 1 of space-like unit vectors is a double cover for the space S'„,„-i of hyperspheres, cf. Example 6.1, Figure 2.18. Introducing oriented hyperspheres by means of the orientations for hyperplanes, the set of oriented hyperplanes is bijectively represented by the space-like unit vectors. Two hyperspheres Si = Sill), IJ2 = ^ ( w ) have a unique invariant:

cf. Example 6.4. For oriented hyperspheres the sign of the scalar product has to be taken into account as well:

T(y

y)-!^hl^ |u||tti|

If \I\{'Ei,'E2) < 1, then the intersection U D W oi the vector spaces determining them is pseudo-Euclidean and hence defines an (n —2)-sphere. By definition the angle [3 between the intersecting hyperspheres is equal to the angle between the hyperbohc hyperplanes describing them: cos/3 := /(Z'i,Z'2), 0 < /3 < TT, for oriented hyperspheres, or cos/3 := j/KZ'i, S2)^ 0 < /3 < 7r/2, for non-oriented ones. If Condition A is satisfied, then |/|(i^i, S2) = 1 if and only if the hyperspheres are tangent to each other; then the intersection U C\W \s isotropic and hence defines a unique point in the Mobius space S*". Condition A is not satisfied if and only if Si = T,2. But even in this case \I\{T,i,T,2) = 1. In contrast to the case of different but meeting hyperspheres the deficit is now equal to 1. In the case |/|(i^i, S2) > 1 the intersection U DW is Euchdean, the hyperspheres do not intersect. Further below we will find a geometric interpretation for the invariant |/|(i^i, S2) also in this case. • We now return to the general situation described in Proposition 5. This proposition refers to Proposition 6.24 and the eigenvalues (6.73); we keep the notations used there. The extremal problem with constraints for the function (6.67) now has the following geometric interpretation: Find the relative

2.7 Mobius Geometry

321

extrema of the invariant I{Ui, U2) for all the pairs of hyperspheres (Z'l, S2) containing the suhspheres, i.e. satisfying Si C Si and 6*2 C S^- We discuss the cases one by one. Case 1. a i > 1. In this case UC\W = {U +W )^ is Euchdean or equal to {0}, the projective subspaces intersect in the outer region A{S^), and the subspheres are disjoint. In addition, the foUowing more precise statement can be proved: Lemma 6. / / the maximal invariant a.i of the subspheres Si = S{U), S™ = S(\V) is larger than I, then these subspheres are disjoint, and there are hyperspheres IJi,IJ2 separating S[, S™, i.e. satisfying S[cUi,

SJp CU2 and UinIJ2

= 9.

(7)

P r o o f . The dimension of the pseudo-Euchdean subspace U + W is not less than 2. Hence we may find a time-like vector ^ = u + VO eU +W such that u eU , vo e W . Consequently, for the intersection of the hyperplanes corresponding to the space-hke vectors u, tn the subspace

[ u ] ^ n [ < = ([u] + N ) ^ is Euclidean; hence the hyperplanes intersect in the outer region of S*", and the corresponding hyperspheres Si, IJ2 satisfy (7). • Case 2. ai = 1. Assuming Condition A the subspace UDW is isotropic, cf. Lemma 6.23; thus the intersection 6*105*2 consists of a single point. This point is the point of contact with the n-sphere for the tangent subspace A ^ PiB™^ . Case 3. a i < 1. In this case the subspace A ' + ^ n S™+i is hyperbolic. Now there are, however, two situations we have to distinguish: Case 3.a. a i < 1, and Dim A ' + ^ n B™+^ = 0. Lemma 7. Let S\, S™ C S*" be two subspheres with the maximal eigenvalue a i < 1 and suppose that l + rn = n — l for the dimensions. Then the subspheres S[,S™ are disjoint and linked, i.e. Si O S2 = 0, and any two hyperspheres Si, S2 with S\ C Si, S™ C S2 always intersect: Si O S2 y^ 0. P r o o f . Applying the dimension formula shows that by Condition A we have Case 3.a. Since in this case the intersection A ' + ^ nB™+^ is a hyperbohc point, the corresponding subspheres are disjoint. The one-dimensional vector space corresponding to the point is time-like. Consequently, the subspace U + W = (t/ n W)^ is Euclidean. Thus, the subspace generated by two unit vectors u e U , vo e W is Euchdean, i.e., its orthogonal complement is an n-dimensional pseudo-Euclidean subspace [u, tn]^ = [u]^ n [tn]^, which determines the intersection of hyperspheres Si{u) D S2{TO). Hence the latter is an (n — 2)-sphere. Since u e U , vo e W are arbitrary unit vectors, this holds for arbitrary hyperspheres with S[ C Si, S*™ C S2. • Case 3.b. 0.1 < 1, and k := DimA'+^ n B™+^ > 0. In this case the subspheres Si, S™ intersect in a {k — \)-sphere.

322

2 Cayley-Klein Geometries In each of the cases considered above the relations ^/ol = cos/3j, if ai < 1,

(8)

determine by definition the stationary values f3i, 0 < f3i < 7r/2, of the angle, under which the hyperspheres IJi,IJ2 with S[ C Z'i,S'™ C IJ2 intersect. Example 3. Already in Example 2 we discussed Condition A and the Cases 1, 2, 3.b for two hyperspheres IJi,IJ2 C S*". The dimension condition in Case 3.a can only hold on the circle, i.e. for n = l ; this case will be the topic of Exercise 6 and the next example. For n > 1 and I(IJi,IJ2) < 1 the hyperspheres intersect; its angle /3 was defined by means of the invariant / in Example 2. To interpret the invariant / geometrically also in the case of non-intersecting hyperspheres, we want to express it in terms of the Euclidean invariants for the hyperspheres. Since the Mobius invariants obviously are Euclidean invariants as weU — the Euclidean group is a subgroup of the Mobius group — this has to be possible. It is easy to see that the radii r, R of the hyperspheres together with the distance d = ^(31,32) of their centers 31,32 form a complete system of Euchdean invariants for the pairs of hyperspheres. As already in (6.2) let (ej), « = 0 , 1 , . . . , n + 1, (recall that the dimension was increased by 1) denote a pseudo-orthonormal basis for the vector space V"+ . With respect to this the equation -(eo,?)=x° = l (9) defines a linear subspace, which we may interpret as the point space E"^ of an ( n + l)-dimensional Euclidean geometry by means of the scalar product on the associated vector space. Then the Mobius space S" is represented as the unit hypersphere with center eo once we normalize the representatives by (9): n+l

? = eo + ?i, n = Yl ^'^* with(j;i, Pi) = 1.

(10)

i=i

We identify S" with this unit hypersphere and project it stereographically from the South Pole eo — e„+i onto its equatorial hyperplane E" C £!"+^ spanned by the vectors e i , . . . , e „ , cf. Exercise 1, Lemma 6.3, and Exercise 6.5. Let S{},r) C E" denote the hypersphere of the Euclidean space E" with center 3 and radius r, and let U{},r) := st^^{S{},r)) be its inverse image under the stereographic projection. By Lemma 6.3 this is a hypersphere in S*". We show: Lemma 8. In the notations introduced above, according to (5) the hypersphere U{},r) := st^^{S{},r)) corresponds to the space-like unit vector u{i,r) = (eo(l - r2 + (3,3)) + 23 + e„+i(l + r^ - (3,3)))/2r,

i.e., S{hr) = S{u{i,r)).

(11)

2.7 Mobius Geometry 323 In order to see this we denote the coordinates of the point t) G S{}, r) by (y*) and the coordinates of the center 3 G E"^' by (z*), « = 1 , . . . , n. They are related by n

{^-h^-l)=Y.^y'-z'f=r\

(12)

i=i

> F r o m Formula (6.20) for the inverse of the stereographic projection, in which now the index n + 1 corresponds to the South Pole — e„+i, we compute using (10) the normalized representative of the point a; = [p] G ^ ' ( j , r) to be ? = eo + ^ + e „ , , i ^ . l + (t),t)) l + (t),t))

(13)

Taking into account (12) the scalar product of the vectors (11) and (13) shows t h a t these vectors are mutually orthogonal, which imphes the assertion. D An elementary calculation together with this lemma shows C o r o l l a r y 9. The Mobius invariant of two oriented hyperspheres 5*2(321^2) in the n-dimensional Euclidean space is equal to IiS,ii,,r,),S,ii,,r,))

= '-^+'-^2rir2

Here d = \}i — h\ denotes the distance of the

.

Si(}i,ri),

(14)

centers.

The expression (14) is frequently called the Coxeter distance of the hyperspheres, even though it, obviously, does not have the usual properties of a metric; therefore, we prefer to call it the Coxeter invariant for the hyperspheres. An elementary geometric consideration shows t h a t , in the case of intersecting hyperspheres, the Coxeter invariant again determines the cosine of the intersection angle. • Exercise 4 . Derive formula (11) for u{]i,r) by taking the vector product of n + 1 isotropic vectors in the pseudo-Euclidean vector space V"^^ which represent points in general position on the hypersphere S"^-^ C S". Exercise 5. Let (b, t)) = p be the equation for a hyperplane H" ^ in the Euchdean space E" ,n > 1. By Lemma 6.3 its inverse image under the stereographic projection is a hypersphere through the South Pole. Prove that this hypersphere is the one defined in (5) as m

with a = ^ + (^°|-|^"+^)^

(15)

Use this to verify that the invariant \I\{st^^{Hi), st^^{H2)) for two intersecting hyperplanes is equal to the cosine of their intersection angle; for parallel hyperplanes it is equal to one. An analogous fact can be proved for a pair consisting of a hyperplane H" ^ and a hypersphere S"^^{z,r); in this case the invariant is |/| = z/r, where z denotes the distance from the center 3 to the hyperplane.

324

2 Cayley-Klein Geometries

Exercise 6. Show by examples tliat tlie Cases 1, 2, 3.a may occur for two 0-spheres in S^; of course, Case 3.b is impossible there. In Case 3.a the invariant for the point pairs is equal to the cosine of the intersection angle between their generating hyperbolic lines. Figure 2.46 shows two orthogonal point pairs on a circle embedded into the Euclidean space E^.

Fig. 2.46. Eigenspheres of orthogonal point pairs in .S'^ C S^

Exercise 7. Find an explicit example for the configuration in Figure 2.46, i.e. two orthogonal 0-spheres in a circle contained in the three-dimensional sphere S^. (Note that the the ambient Euclidean space E^ in the figure now has to be interpreted as the sphere S^.) For this configuration the deficit is equal to 2; the maximal eigenvalue is a i = 1, it has multiplicity 2; the eigenspheres for the eigenvalue 1 form the pencil of spheres through the circle S^ (cf. Figure 2.47), whereas the eigensphere for the eigenvalue 0 intersects all spheres in this pencil orthogonally. (Hint. Eigenvalues refer to the operator A defined in (6.70), which in the case considered here acts in a three-dimensional Euclidean vector space; the eigenspheres are defined by S{u), U an eigenvector of A.) In general, we call the hyperspheres S{TO) for the eigenvectors tn of the operator A defined in Proposition 4 the eigenhyperspheres of the pair of spheres (5*1, 5*2). Each eigenhypersphere IJ2 '•= ^ ( w ) contains 6*2; if the corresponding eigenvalue is a > 0, then the projection u := p{vo) G U defines a hypersphere Si := S{u) D 5*1 for which the stationary invariant i / a = | / | ( i ^ i , S2)

2.7 Mobius Geometry

325

Fig. 2.47. The sphere pencil of a circle

is realized. If, however, a = 0, then S2 is orthogonal to all hyperspheres S containing Si (cf. Proposition 6.24). E x a m p l e 4. Let us again consider the one-dimensional Mobius space, i.e. the circle S*^. Here the hyperspheres are the 0-spheres, i.e. point pairs 5*1 = {s, n } C S*^ with s ^ n. According to Proposition 4.7 the circle is equipped with the structure of a projective line; thus for each throw of points in S*^ the cross ratio C R is defined in an invariant way. Since the Mobius group Gi C PL2(R-) consists of all projectivities of the plane leaving S^ invariant, the C R is also invariant under the action of the Mobius group. As the points from two different 0-spheres determine a throw, it is n a t u r a l to ask, how the C R of this throw is related to the Mobius invariant |/|. To describe this relation we use the notations and constructions from the proof of Proposition 4.7. So we represent the points s,n by isotropic vectors 5,n and adapt the pseudo-orthonormal basis (Cj), « = 0 , 1 , 2, to the point pair so t h a t (5,n) = - l / 2 ,

(16)

Co = 5 + n , ci = 5 - n , C2 = Co X ci.

(17)

Because of (5, C2) = (n,C2) = 0 , the tangents Ts, Tn

intersect in the outer point

326

2 Cayley-Klein Geometries T s n T n = b := [ca]

defined by C2. Consider the projective scale ^ on the tangent Ts defined by the vectors 5, C2 and transfer it by stereographic projection onto S^. By (4.57) we thus obtain a parameter representation for S^ assigning to a number, i.e. scale value, the corresponding point:

/: e e R ^

/(e) = im] ••=[B + C2^+ne].

(is)

Take now a second 0-sphere: 5*2 = {xi, X2}, Xi = [T{^i)],i = 1, 2. Using Formula (1.4.16) we compute the CR {s,b; xi,X2)

= —.

(19)

On the other hand, for the space-like unit vector determining the 0-sphere 6*2 we obtain ^ ^ HXh ^ 2 s + 2neie2 + 0 2 ( 6 + 6 ) .20)

I?ixy2l

6-6

Relation (19) for the invariant |/| and its expression in terms of the CR then lead to the result we were looking for: 1/1(^1,^2) = |(C2,W)| = | ^ | . (21) r] — I Note that the CR depends on the chosen order of the points xi,X2; the other order, however, does not lead to a new value for |/|. Considering oriented 0-spheres, i.e. ordered point pairs, reversing the orientation results in a sign change for / and r/, so that we might as well write I{S\,S2)

= {c2,rv) = ^ . r] — I

(22)

Considering the inverse relation and taking into account (19) we obtain for / = I (Si, 5*2) the same functional dependence -f+1 These formulas will now be applied to prove that the real projective group PLi coincides with the Mobius group. • P r o p o s i t i o n 10. The projective group P L i ( R ) for the projective structure defined on the circle S^ according to Proposition 4.7 is equal to the Mobius group G2-

2.7 Mobius Geometry

327

P r o o f . Let (01,02,03,04) be a harmonic throw in S^; thus the CR r] of this throw is equal to —1. Then by (22) the invariant / of the oriented 0-spheres 6*1 = (oi, 02), 5*2 = (03, 04) is equal to zero. The same also holds for the images of these 0-spheres under an arbitrary Mobius transformation (; G Gi. So by relation (23) the images also form a harmonic throw. According to Staudt's Main Theorem, Proposition 1.4.7, g is a projectivity. Conversely, let g G PLi be a projectivity, and let Xi e S^, i = 0,1, 2, be points in general position, i.e. a projective frame for S^. Let y^ := g{xi) denote their images under the projectivity g. We prove: Lemma 11. / / (xi), ( y j , i = 0,1,2, are two triples of points in general position in the Mobius space S*", then there always is a Mobius transformation h G G„+i such that hxi = y^. In the case n = 1 the transformation h is uniquely determined. P r o o f . Each of the point triples uniquely determines a circle. Since G„+i acts transitively on the set of circles by CoroUary 2, it suffice to consider the case n = 1. Let 5, n, p G V^ be isotropic vectors representing the points of the triple {xi), xo = [5], 031 = [n],a;2 = [?]• Take, moreover, a pseudo-orthonormal basis (Cj) for V such that (16) and (17) hold. Obviously, the representatives of the points and the basis vectors are not uniquely determined by these conditions, yet. It is just the vector C2 that is uniquely determined, if we suppose that the point pair (XQ, a;i) is oriented; in fact, 5 = (co + Ci)/2, n = ( c o - C i ) / 2 , 5 x n = - C 2 / 2 . (24) Now consider the basis representation ? = 5Cl+n^2 + C26. As p is isotropic and the three points are in general position, all three components ^j have to be different from zero. Dividing p by ^3 we obtain the representative for 032 that is uniquely determined by ^3 = 1. Since it is isotropic, we obtain

6 6 = 1. Next we look at the transformations stiU possible without violating the present conditions 5 = BA, n = n/A, (A 7^ 0), with A = ^1. Because of the previous equation, we see that the additional condition j; = 5 + n + C2 = Co + C2 (25) uniquely determines the representatives 5, n, p for the points of the triple, and hence by (24) the basis (Cj) is also uniquely determined by the point triple.

328

2 Cayley-Klein Geometries

Analogously, the point triple ( y j uniquely determines a pseudo-orthonormal basis (bj). T h e transformation h e Gi defined by the pseudo-orthogonal transformation 7 G 0 ( 1 , 2) with 7(Ci) = bi, i = 0 , 1 , 2 , obviously satisfies hxi = y^. Since, moreover, each of these transformations maps the uniquely determined bases into one another, 7 and, hence also h, are uniquely determined. • Now we conclude the proof of Proposition 10. Let h e G2 he the Mobius transformation t h a t is uniquely determined by h(xi) = y^ according to Lemma 11. From what we already proved h, and thus also h^^, is a projectivity of S^. Hence the m a p / := h^^ og is a projectivity as weU. Since this leaves the points Xi fixed, i.e. f{xi) = Xi for « = 0 , 1 , 2, by Proposition 1.3.14 / has to be the identity. It suffices to choose theses points as the base points and the unit point of a projective scale on S^. Hence g = h is a Mobius transformation. D Exercise 8. Let 0:1,0:2,2:3,2:4 G S^ be four different points, a) Prove the following identities for the Mobius invariants of the oriented 0-spheres determined by them: Setting I{{xi,X2), (0:3,0:4)) = A the following equations hold /((o:i,o:2) (0:4,0:3)) = -A, /((o:i,o:3) («^2,o:4)) = /((o:i,o:3)

(0:4,0:2))

=

/((o:i,o:4)

(0:3,0:2))

=

/((o:i,o:4)

(0:2,0:3))

=

| ^ , A + 1' A+ 3 A-1' A+ 3 A'

These imply /((o:i,o:2), (0:3,0:4)) = /((o:3,o:4), (0:1,0:2)) = /((o:2,o:i), (0:4,0:3)) = /((o:4,o:3), (0:2,0:1)). Compare these equations with the identities proved in Section 1.4 for the CR (0:1, 0:2; 0:3, 0:4). - b) Under these conditions none of the right-hand sides of these equations can be equal to one. - c) The throw (0:1,0:2,0:3,0:4) is harmonic if and only if A = 0, i.e., if the point pairs are orthogonal. E x a m p l e 5. Consider now the case n = 2, i.e. the sphere S*^ under the action of the Mobius group G3. Already in Example 1.1.3 we identified the complex line PQ with the Riemann sphere S"^. By Example 1.4.3 the continuous projectivities and anti-projectivities on P ^ induce the automorphism group A u t P c which, in a projective scale, is represented by fractional linear and conjugate fractional linear transformations. At the same time, this group is

2.7 Mobius Geometry

329

the Mobius group Ga of the sphere S"^, as we wih soon show by interpreting the calculus of complex numbers geometrically. Historically, for the first time the Mobius group occurred in this very form. An instructive presentation of this complex form of the Mobius group G3 can be found in Chapter 2 of the introductory textbook [27] by C. Caratheodory. D P r o p o s i t i o n 12. The Mobius group Gs of the Riemann sphere S"^ = PQ as identified with the projective line is the group of continuous automorphisms AutP^. P r o o f . As in Example 1.4.3 we represent the group A u t P j - , by the fractional linear transformations: „, ^ a,z -\- b « , ^ / ( . ) ^ - ^

or

- - / ( ^ - ) - ^ cz + a

, , (26) (27)

with ad — cb y^ 0, where we have to set f{—d/c)

= 00 and / ( o o ) = a/c for c 7^ 0, / ( o o ) = 00 otherwise.

(28)

T h e totality of these transformations forms a group; in the proof and for brevity we will simply denote it by G. So we have to show t h a t G = G3. T h e transformations (26) constitute a subgroup, which we already encountered considering the transformations of projective scales on a hue (cf. Example 1.2.2 and Exercise 1.2.4); as a homomorphic image of the linear group GL{2, C) it is isomorphic to the group PLi(C) of projectivities on the complex projective line: The coefficients a, 6, c, d in (26) are just the entries of the matrix A G GL{2,C), cf. (1.2.10). Hence their determinant, ad— cb, is always different from zero. Multiplying numerator and denominator by the same number k G C* the m a p / defined by (26) or (27), respectively, does not change; if required, we may thus always suppose

ad-cb=

1, i.e. Ae

SL{2,C).

According to Example 1.4.5 or (1.7) this imphes t h a t the set of all transformations f : S"^ ^ S"^ which can be represented in the form (26) is the group of projectivities PLi{C) = SL{2, C ) / { ± / 2 } . Still adding conjugation and taking (27) into account we generate the group A u t P i ( C ) . Obviously, the maps (26), (27) satisfy / ( o o ) = 00 if and only if c = 0; then we may represent t h e m in the form w = az -\- b or w = az -\- b with a 7^ 0. But this is the conformal Euclidean group of the complex plane, which again can be identified with the Euclidean plane, cf. Section 1.2.3. Using this, multiplication by a non-zero number becomes a rotation composed with a dilation.

330

2 Cayley-Klein Geometries

addition corresponds to a translation, and the assignment z ^ z is the reflection in the real axis. Obviously, each similarity g G CE{2) can be represented by one of these transformations / . For c^Q the m a p s : z e z is the inversion in the unit circle with center 0: Exercise 9. a) Verify the last assertion. - b) Prove that each transformation / G G can be represented as the composition of a similarity with the inversion s in the unit circle. (Note that s may occur several times in this composition.) - c) Prove that the map h(z) :=

^ = m + with r > 0 (29) z—m z —m is the inversion in the circle with center m e C and radius r > 0 in the complex number plane. - d) Find an analogous formula for the reflection in the real line J/ = ai + 6, a / 0, i e R, in the complex plane. W i t h these calculations at our disposal we can now complete the proof of the proposition: Statements a) and b) together with Exercise 1 and Example 1 imply the inclusion G C G3. Conversely, according to Proposition 4 each Mobius transformation g €z G3 can be represented as a flnite product of inversions, which are elements of G by Exercise 9, c), d). Since G is a group, we conclude G3 C G. D Exercise 10. Let a:i, i = 1 , . . . , 4, be four different points in S^ = Pj^. Prove that the four points lie on a common line or circle in C if and only if their cross ratio {xi, X2] Xi, X4) is real. C o r o l l a r y 13. The group of fractional linear transformations f defined by (26) consists of all orientation preserving Mobius transformations of the Riemann sphere S"^. This implies the following isomorphy SL{2,C)/{±l2}

=

SO{l,i).

P r o o f . It is not difficult to show t h a t each transformation of the form (26) may be represented as the composition of maps from the following list: 1. Translation: tb '• z 1—> z -\- b. 2. Rotation with subsequent dilation: da : z 1—> za, a G C*. 3. Taking the reciprocal: h : z 1—> l/z. All these maps are orientation preserving. For the translations this is obvious; in the second case, using a = | a | e ' " , z = l-zje"'', we obtain: d„(z) = |a||z|ei('^+°), and this is the rotation through the angle a followed by the dilation with factor |a|. Finally, forming the reciprocal is the product of two reflections: h{z) = l/z is the composition of the inversion in the unit circle, s{z) = l/z,

2.7 Mobius Geometry

331

with subsequent reflection in the real axis. Since each reflection has determinant — 1 (cf. Example 1), this proves our assertion. Hence, cf. Proposition 4, every Mobius transformation of the form (26) can be represented as the product of an even number of reflections.^ Now transformations of the form (27) are obtained from those of (26) by composition with the reflection in the real axis from the right; hence they can always be represented as the composition of an odd number of reflections. Thus they reverse the orientation. Therefore, the maps of the form (26) and (27) each constitute one of the components of the group Ga. According to the Examples 1.4.3 and 1.4.5 the group described by (26) is just the group SL{2,C)/{±l2}; on the other hand, according to Proposition 12 and what was explained at the end of Section 1 it coincides with SO{l, 3). An explicit formula for the homomorphism F : SL{2,C) -^ SO{l,3) in terms of a pseudo-orthogonal matrices whose entries are functions of the complex parameters a, b, c, d can be found in J. B. Wilker [111], Theorem 10. Note t h a t in this paper the last vector in a pseudo-orthonormal frame is always time-like, i.e. its norm square is equal to - 1 . 3 D

2.7.3 Cross Ratios and the R i e m a n n Sphere In the preceding section we identifled the Mobius groups of the circle S^ and the sphere S"^ with the projective groups P L i ( R ) and A u t P ^ , respectively. This implies (cf. Examples 4, 5 and Propositions 10, 12): C o r o l l a r y 14. The cross ratio of four points on invariant. On the Riemann sphere S"^ = PQ, the is preserved by each orientation preserving Mobius transformed into its conjugate by every orientation mation.

the circle S^ is a Mobius cross ratio of four points transformation, and it is reversing Mobius transfor•

For higher dimensions, n > 2, the projective and the Mobius groups differ substantiaUy. nevertheless, even in the n-dimensional case the Mobius groups can be characterized as the invariance groups for suitably deflned cross ratios. This method goes back to J. B. Wflker; in Section 6 of his paper [111] he actually calls this quantity the fundamental invariant of Mobius geometry. It is defined as follows: Let xi,X2,X3,X4 be four different points in the n-dimensional Euclidean space. As the absolute cross ratio we take the expression | . . . 2 ; . 3 , . 4 | : ^ ^ ^ : ^ ^ , 031 - 0 3 4

(30)

032 - 0 3 4

^ In the textbook [27] C. Caratheodory proves that each transformation of the form (26) can be represented as the product of two or four inversions. ^ The reader can find Wilker's matrices together with the necessary background form computer-algebra in the Mathematica notebook riemsph.nb at http://wwwirm.mathematik.hu-berlin.de/~sulanke.

332

2 Cayley-Klein Geometries

where |a; — y | denotes the norm of the position vector x — y, i.e. the Euchdean distance of the points x, y. Taking in Formula (1.4.15) the absolute value we see that Definition (30) coincides with the value of the cross ratio computed there. Obviously, the cross ratio depends continuously on its arguments; letting in (30) one of the points, e.g. X4, tend to 00 we obtain a;i,a;2;a;3,oo := lim ^

^ i ^

034^00 \xi — X4\

4 = F

\X2 — X4\

'r

31

\X2 — Xs\

Hence, taking into account the symmetry in the definition, the absolute CR is also defined if one of the points is equal to 00. Thus Proposition 15. An injective map f of the n-dimensional Euclidean space extended by the point 00 to form the n-sphere S*" is a Mobius transformation if and only if it preserves the absolute CR for quadruples of arbitrary points. P r o o f . We first show the invariance of the absolute cross ratio. By Proposition 10, in the case n = 1 the cross ratio itself remains invariant. Now each Mobius transformation can be represented as the product of finitely many inversions by Proposition 4. As reflections in hyperplanes are even isometrics, it suffice to show that the absolute cross ratio is preserved by any inversion s in a hypersphere. According to Formula (4) the distance of the images of two points X, y is computes as

k(?)-s(t))l = r ^ l ? - t ) l Here p, t) are the position vectors of the points x, y with respect to the center of the hypersphere defining the inversion. A straightforward calculation shows that (30) imphes the invariance of the absolute cross ratio for points different from 00. Then, taking as above the hmit y = X4 ^ 00, shows that the assertion also holds, if one of the points is 00. Conversely, let now / : S*" ^ S*" be an injective map preserving the absolute cross ratio and consider y = f{x). Since the Mobius group acts transitively on S*", we find Mobius transformations s,t e Gn+i such that s(oo) = x, t{y) = 00. Since, by the above, Mobius transformations preserve the absolute cross ratio, the same is also true of the map / i := t o f o s. Moreover, this map has the property /i(oo) = 00 and hence maps the Euclidean space into itself. Consider now an arbitrary triangle 031, 032, 3:3 and its image under / i . Because of the invariance of the absolute cross ratio, by (31) the image points y^ = fi{xi), « = 1, 2, 3 satisfy: I I yi,y2;y3.oo =

Vl-Vs r =

I I 031,032:033,00 =

y2-y3

This implies ll/i -Vsl ^ 1^/2 -Vsl |a3i-o;3| |o;2-o;3|

331-033 r 032-033

2.7 Mobius Geometry

333

Thus the image triangle J/i, 2/2,2/3 is similar to the original triangle a; 1, 0:2, 3:3; and since this holds for an arbitrary triangle, the map / i has to be a similarity; so it is a Mobius transformation. Therefore, / = t^^ o fi o s^^ is a Mobius transformation as well. • R e m a r k . Note that we did not suppose / to be surjective in the proof of Proposition 15; hence the surjectivity is a consequence of injectivity together with the invariance of the absolute cross ratio for maps on the sphere. For n = 1, starting from the last formula a straightforward computation shows that / i is an affine map of the Euclidean line. We recommend that the reader proves the next corollary as an exercise; otherwise, see the paper [111] by J. B. Wilker: Corollary 16. Consider two families Mi = {xi)i^j, M^ = {yi)iei of points in the Mobius space S*". Then there is a Mobius transformation g G G„+i satisfying g{xi) = y^ for all i ^ I if and only if the absolute cross ratios of any four corresponding points coincide, i.e.

for all index quadruples whose associated points in Mi are different. If such a map g exists, then it is uniquely determined if and only if [Mi] = S", i.e. if the family Mi is not contained in a hypersphere or hyperplane. • To prove uniqueness we may apply Proposition 4.1 and Corollary 4.3. In the following example we refine the criterion in the corollary for the dimensions n= 1,2. Example 6. Let ^ be a projective scale on S'^ = P^(R) or S"^ = P^(C). Then by Definition 1.4.1, the cross ratio satisfies ^ = (^, 1;0, 00), where the points are described by their parameter values. This implies the following identities for the absolute cross ratio |e,l;0,oo| = | - e , l ; 0 , o o | = |e,l;0,oo| = | - e , l ; 0 , o o | = |e|, where, obviously, the last two relations are only relevant for non-real parameters ^. Looking at the first identity we see that, different than for the original cross ratio, it is not enough to know the absolute cross ratio for only one permutation of the four different points to ensure Mobius equivalence of the quadruples. In fact, the quadruple (^, 1,0, 00) is not equivalent to the quadruple (—^, 1,0,00), since the cross ratios are different and, in the case K = C, are not conjugate either, if Re(^) ^ 0. Proposition 1.4.1, (1.4.12), and (1.4.14) imply that the cross ratios of all permutations of the quadruple (xi, X2, X3, X4) are determined once the cross ratio of one of them is known. Note that this is false for the absolute cross ratio; it is easy to find examples showing that 1^1 = \r]\ does, in general, not imply |1 — CI = |1 — '?l- Nevertheless, if both the equations

334

2 Cayley-Klein Geometries | y i , y 2 ; y 3 . y 4 l = 1^31, 332; 033, 0341

(32)

I y i , y 3 ; y 2 . y 4 l = |a;i, 333; 032,0341

(33)

hold, then according to Proposition 1.4.1 the corresponding equations are also satisfied by the absolute cross ratios for aU corresponding permutations of the quadruples. Using CoroUary 16 we thus conclude t h a t the quadruples (a3i, 032, 033, 034) and (y^, y2, y3, VA) are Mobius equivalent. Hence it suffices to verify conditions (32) and (33) for a single arrangement of any four different points in M i . Note also the following exercise. T h e first case, ^ = r], occurs if b o t h the quadruples are m a p p e d into one another by a projective transformation (26), i.e. one preserving the cross ratio as weU as the orientation, whereas the second case ^ = f] means t h a t both are related by a Mobius transformation of S*^ reversing the orientation, hence one of the form (27); in the following we suppose ad — be = I. D Exercise 1 1 . Consider 5,?? e C and prove that the relations \^\ = |??| as well as |1 — 51 = |1 — ri\ both hold if and only ii ^ = rj or ^ = ij. Exercise 12. Let Xi e S^,i = 1,...,4, be four different points on the sphere. a) Prove that there is a Mobius transformation g E G3 leaving the points 0:1,0:4, fixed and interchanging the others, (7(0:2) = X3, gixs.) = X2, if and only if |a:i, 0:2; 0:3,0:41 = 10:1,0:3; 0:2,0:4!.

(34)

- b) Take the four points to be in general position. Denote hy s : S^ ^ S^ the inversion in the circle through 0:2,0:3,0:4, and 0:1 = s(o:i). If (34) holds, then the circles Si through 0:2,0:3,0:4 and IJ2 through 0:1,0:1,0:4 intersect orthogonally. The points 0:2,0:3 are transformed into one another by the inversion in the circle IJ2T h e relation between the cross ratio of four points in the Riemann sphere and the stationary invariants for the pair of 0-spheres determining the cross ratio as defined in Section 2 is described in the following proposition: P r o p o s i t i o n 17. Let S\ = (031,032), 6*2 = (033,034) C S*^ be two different 0-spheres in the Riemann sphere, and let r/ = (o3i, 032; 033, 034) be the cross ratio they determine. Then the stationary invariants for the pair of spheres defined in Section 2 are ai[bi,b2)

= -.

-77^ > a2[bi,b2) |?7 — 1 | ^

=-.

77^. I"*? — 11

(35)

Moreover, 1. Q.\ and Q.2 are independent of the orientations in the 0-spheres. 2. ai = a2 if and only if 81(^82 consists of a single point; then ai = a2 = I. 3. a.\ = \> a.2 if and only if 81,82 lie in a common circle S*^ and separate each other.

2.7 Mobius Geometry

335

4- ai > a2 = 1 if and only if 5*1, 5*2 lie in a common circle S^ and are non-separating. 5. ai > I > a2 if and only if Si, S^ are in general position. P r o o f . To prove (35) we transform the Riemann sphere into the pseudoEuchdean model for the Mobius space using the parameter representation by complex numbers as described above. So we view it as the set of generators for the isotropic cone in four-dimensional pseudo-Euclidean space. Then each 0-sphere corresponds to a two-dimensional pseudo-Euclidean subspace, and hence the orthogonal complement of t h a t is a two-dimensional Euclidean subspace. Now we express the operator introduced in Proposition 5 in terms of the complex parameters and compute its eigenvalues; this leads to (35)^. Reversing the order of the points in 5*1 or 5*2 the cross ratio rj is replaced by ry^^ according to (1.4.7); hence the expressions (35) for a i , a.2 do not change their value proving 1. Since the 0-spheres are not equal, at least three of the points 031, 032, 333, X/^ havc to bc different. By Lemma 11 we may suppose without loss of generality t h a t the points are 032 = 1, 033 = 0, 034 = OO-

As usual, they are identified with the corresponding numbers in the Riemann sphere. Since T] = {xi,X2]

X3, X4) = (ry, 1; 0, 0 0 ) ,

we then have xi = r]. T h e conditions in 2-4 occur if and only if at least one of the eigenvalues is equal to 1. But this is the case if and only if the deficit is equal to 1, i.e. if the four points all lie in one circle. It is easy to verify t h a t r/ is real for the special values we are considering. So we proved 5.; in fact, the four points are in general position if and only if r/ does not lie on the real axis (see also Exercise 10). This was Case 1 in Section 2. To prove 2 note t h a t a i = a2 if and only if ry = 0 or ry = 00; because of 6*1 ^ 6*2 these are the only possibilities for a non-empty intersection 6*1 n 6*2 = {0} or 6*1 n 6*2 = {00}. In b o t h cases we have a i = a2 = 1; this corresponds to Case 2 in Section 2. Statements 3 and 4 are proved by exploiting the respective conditions: T h a t in 3 amounts to ?y < 0, and t h a t in 4 is equivalent to ?y > 0. In 3 the condition a i = 1 just means t h a t the deficit is 1. This is Case 3.a from Section 2; the eigenvalue a i from Lemma 7 has to be identified with our a 2 , since Condition A is actually only satisfied for the circle. FinaUy, statement 4 again corresponds to Case 1 from Section 2, this time referring to the circle (the real axis). • CoroUary 18. Four different points in the Riemann man circle if and only if their C R is real.

sphere S'2

m a com•

'' The computations were done using the software Mathematica by Stephen Wolfram [114]. They are an interesting example showing the far-reaching capabilities of this program to perform extensive symbolic calculations. A corresponding notebook riemsph.nb: "The Riemann Sphere" can be downloaded from the homepage http://www-irm.mathematik.hu-berlin.de/~sulanke.

336

2 Cayley-Klein Geometries

E x a m p l e 7. We now pose the following question: Let a i , a 2 be two nonnegative real numbers. Is there a pair of 0-spheres in the Riemann sphere S"^ whose stationary invariants are these numbers? Leaving the trivial Case 2 aside, by Proposition 17 these numbers necessarily have to satisfy the following conditions: «1 > «2 > 0, a i > 1 > a2(36) First we determine the possible values rj of the C R in equation (35) for the given « ! , a2- Set ai+a2 -2

2 V ( a i - 1)(1 - a^) .

Then, in general, ??+, ??- and their conjugates fy^, f\- all are different solutions for (35); see the Mathematica notebook mentioned in Footnote 3. Since all complex numbers r\ ^ 0, l , o o can occur as the cross ratio of four different points, corresponding pairs of 0-spheres exist. Concerning the geometry of the situation we again suppose 032 = 1,033 = 0, and 034 = 00. T h e unit circle with center 0 can then be characterized in a Mobius invariant way as follows: Consider the fourth harmonic point y for 0:2, 0:3, 0:4 on the circle Ti\ determined by these points; in our special position this is the point —1 on the real axis. Then determine the circle Z'2 through y, 0:2 t h a t is orthogonal to T,\. T h e following can be proved: Under the inversion in T12 the point x\ = rj^ is transformed into the point r/-; on the other hand, complex conjugation is the involution in the real axis. Since these involutions m a p the 0-sphere (033,034) = (0,00) into itself and the point 032 = 1 remains fixed, it is clear t h a t the pairs of 0-spheres (??, 1), (0, 00) formed using the four solutions r/ = rj^,rj-,fj^,fjare Mobius equivalent. Hence they have the same stationary invariants. • Exercise 1 3 . Prove the assertions formulated in Corollary 18 and Example 7. Show, moreover: a) There are less than four solutions rj for (35) if and only if a i = 1, or a2 = 1, or a2 = 0; this again is the case if and only if a:i belongs to i7i Ui72. ?? = — 1 is the only solution if and only if a i = 1 and a.2 = 0, i.e. if the throw (xi,X2,X3.,X4) is harmonic. - b) The complex numbers ??+,??-,??+,??- lie on a circle . In the Riemann sphere hyperspheres are circles; everything said before concerning general hyperspheres holds in particular for them. T h e only Mobius invariant for two circles is their Coxeter invariant (14). According to (11) the space-like unit vector of a circle with center m G C and radius r is u{m,r)

= ( e o ( l - r ^ + | m p ) + e i R e ( m ) + e2lm(m) + e 3 ( l + r ^ - | m p ) ) / 2 r .

(38)

Correspondingly, the results formulated in Exercise 5 also hold for pairs of hues as weU as pairs consisting of a hue and a circle. The foUowing exercise shows t h a t a pair consisting of a point and a circle has no Mobius invariant at all:

2.7 Mobius Geometry

337

Exercise 14. Prove that the Mobius group G„+i acts transitively on the set of pairs {{IJ,a)\aeS", SCS" a hypersphere, a <^ S} as well as on the set of pairs {iS,a)\a

eS",

S CS"

& hypersphere, a e S}.

Hint. Consider S as the absolute of an n-dimensional hyperbolic geometry. For the sphere S"^ it finally remains to consider pairs {S, 5*°), where Z' is a circle, and 5*° = (zi, 2:2), zi ^ Z2, denotes a 0-sphere. On the set of such pairs {S, S°) with S° C S the Mobius group G3 acts transitively, since the isotropy group of S contains the Mobius group G2 and G3 acts transitively on the set of circles. We may thus restrict the discussion to the case t h a t 5*° is not contained in S; this assumption is the same as requiring Condition A to be satisfied. T h e general theory in Section 2 implies t h a t all we have to determine is a single stationary invariant, and t h a t can be computed as follows: Consider the fourdimensional pseudo-Euclidean vector space on which the Mobius geometry of the sphere S"^ is based. Let n = n{IJ) denote one of the space-hke unit vectors describing the points of S by x=[f]eIJ

^=> {n{IJ), p) = 0 and (p, p) = 0,

e.g. the one defined by (38). Note t h a t n is only determined up to sign. Below, let bi, ^2 be two orthogonal space-like unit vectors, whose associated circles b o t h contain the 0-sphere 5*°; in short, we will speak of an orthonormal basis for the pencil of circles, i.e. the family of all circles in S*^ containing S^. Then invcpp(i:, S°) = (n, bif

+ (n, 62)2

(39)

is independent of the choice of n just described as well as of the orthonormal basis; it is the single stationary invariant of the pair {S, S^) we were looking for. In fact, the matrix of the double projection described in Section 2 just reduces to the one component (39). Exercise 1 5 . Prove: a) All the pairs {IJ,S°) with IJnS° a single point are Mobius equivalent. - b) The stationary invariant of such a pair is 1, and its deficit is zero. (Hint. As in Exercise 14 one can show that each configuration with the stated property is Mobius equivalent to {So, {0,1}), where So denotes the unit circle with center 0.) - c) For S° C S the stationary invariant and the deficit both are 1. Now we want to express the Mobius invariant invcpp{IJ,S°) in terms of the Euclidean invariants for the pair {S, S^); we will derive a formula similar t o (14) for the Coxeter invariant also in this case. P r o p o s i t i o n 19. Let zi, z^ G C, zi 7^ z^, he two points in the complex number plane, and let IJ('m,r) C C be the circle with center m G C and radius r > 0. Consider the triangle (rn, zi, Z2) and denote its side lengths as follows:

338

2 Cayley-Klein Geometries a = \z-i - Z2\, b = \z2 -m\,

Moreover, stationary below:

c= \z\ - m\.

(40)

denote by a the angle of this triangle at the vertex m. Then the Mobius invariant of the pair is computed by either of the formulas

invcpp(i:(m,r),(zi,Z2)) =

r** — 2 r ^ 6 c c o s a + 6^c^ 5-^ ,

,^, , , ,, (r^ — 6c)^ i n v c p p ( i : ( m , r ) , (zi, Z2)) = ^ ^ ^ +

26c(l—cosa) ^ 5 '-.

(41) , , (42)

Obviously,completing the square in (41) yields (42). To prove the assertion consider the vector (11) for the circle IJ{m,r). Find an orthonormal basis for the Euclidean vector subspace of V corresponding to the point pair (zi, Z2), which is orthogonal to the isotropic vectors representing the points. Using this, the invariant can be computed by formula (39). Then the quantities occurring in this expression have to be interpreted. T h e details can be found in the M a t h e m a t i c a notebook mentioned in footnote 3. • Now we want to derive a third expression for the invariant invcpp, making the relation to the Coxeter invariant apparent. To do so, we represent the 0-sphere 5*° = (-21,-22) as weU as the higher-dimensional spheres in centerradius form: S°{m^,r^,u)

= {m^ + r^&",m^

- r ^ e ' " ) , m^eC,ue

R , r^ > 0;

(43)

m^ is the center, r^ the radius, and u an angle parameter determining the direction of S^ in the complex plane. Then P r o p o s i t i o n 20. For the stationary 0-sphere the following relation holds

Mobius

invariant

of a circle and a

i n v c p p ( i : ( m , r ) , 5 ° ( m „ r „ « ) ) = ( " ' + / ' ~ ^ ' ) ^ + (^f^fZrr^ r

(44)

here d = [m — mz\ denotes the distance of the centers, and [3 is the angle between the vector m — m^ and the normal to the connecting line z\\l z^- • Looking at Formula (44), it is remarkable t h a t the first term on the righthand side is nothing but the square of the Coxeter invariant for the circles IJ('m,r) and IJ('mz,rz), whereas the second t e r m coincides with the square of the projection of TO —TO^onto the normal of S^{mz,rz,u). In the next section we will describe a far-reaching generalization of this formula and hence omit to prove it. Of course, these formulas may be applied analogously to corresponding configurations in a Euclidean plane. T h e proof of Proposition 20 can also be found in the notebook mentioned in footnote 3. This notebook contains useful tools for solving the following exercises as well as designing or modifying the accompanying illustrations.

2.7 Mobius Geometry

339

Exercise 16. Consider tlie pencil of circles formed by tlie circles containing a given 0-sphere. Obviously, the Mobius group acts transitively on the set of all these pencils. Hence we may as well suppose that the pencil is just the set of all circles through the 0-sphere S° = (—1,1). The centers rn of these circles lie on the imaginary axis; let t := Im(m) be chosen as the parameter for the circles in the pencil. Then the radius of the corresponding circle is \/T~fW. Denote the circle of the pencil corresponding to the parameter value t by bc(t). Prove: a) The space-like vector bc(t) := {0, 0, —^=,—^=},

t / 0,

describes the circle bc(t) in the pencil as the set of all points a: = [p], whose coordinate vector p satisfies the equations (bc(t),j:) = 0, (?,?)= 0. - b) The real axis also belongs to the pencil of circles in question. It has, however, no representation of the form explained in a). Which space-like vector determines it? - c) Through each point z / ± 1 in the Riemann sphere there is a unique circle in the pencil. Find the corresponding parameter value t for this circle in the case Im(z) / 0, and with it center and radius. For real points as well as the point oo the real axis is this uniquely determined circle in the pencil. - d) For each circle ca in the complex number plane with center a and radius ra, 0 < ra < oo, there is a circle bc(t) in this pencil intersecting ca orthogonally. Determine such a circle. e) If ca = bc{s) itself is a circle in this pencil, then assigning the uniquely determined orthogonal circle bc{t) = bc{s) to it defines a fixed-point free involution on the pencil of circles. Describe this involution in the form t = F{s). Obviously, the unit circle with center 0 is in involution with the real axis; verify this by a computation.

Exercise 17. Prove: a) For each pencil of circles in the Riemann sphere there is a one-parameter family of circles orthogonally intersecting every circle in the pencil. These circles are called the orthogonal trajectories of the pencil. Determine the orthogonal trajectories for the pencil considered in Exercise 16. - b) Find the isotropy group H oi 'd 0-sphere. - c) Let M{S°) denote the pencil of circles through S°, and let M^{S°) be the set of its orthogonal trajectories. Prove that H acts transitively on the product set M(S°) x M^(S°). Exercise 18. Let M{S°) be the pencil of circles for 5° = (-1,1) (cf. Exercise 16), and let S^ {rn, r) be a test circle with center rn and radius r not containing S°. Denote by a = invcpp(S'^(m,r), 5°) the stationary invariant (39) for this pair. Prove that there is a circle bc(t) e M{S°) meeting the test circle S^{m,r) if and only if a > 1; for a = 1 there is a unique such circle, and for a > 1 there exist two of these circles. Compute the parameter value t for these circles in the case a > 1. Figure 2.49 shows the situation described in Exercise 18. The test circles are blue and the circles in the pencil meeting it, if there exist any, are violet. The red circle is the one in the pencil intersecting the test circle orthogonally; the test circles are no orthogonal trajectories for the pencil. The green circle is the one in the pencil for which the Coxeter invariant (14) with respect to the test circle is extremal. Since the stationary invariant of two circles / is the square of their Coxeter invariant, it

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Fig. 2.48. A pencil of circles and its orthogonal trajectories.

Fig. 2.49. Existence of circles in a pencil meeting a test circle.

attains its maximum for the green circle. If a > 1, as in the figure on the left, then the test circle does not separate the points of S°, and the pencil contains two circles meeting one another. The figure in the middle depicts the case a = 1; the test circle contains a point of S , and there is a unique touching circle in the pencil, which, at the same time, realizes the maximal value a = 1 of the stationary invariant. Finally, in the figure on the right we have a < 1; the test circle separates the points of

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341

S°, each circle of the pencil meets the test circle, and the green circle realizes the smallest intersection angle occurring. Exercise 19. Consider a pair {L,S°) consisting of an arbitrary (real) line L and a 0-sphere S° in the complex plane. Proof a result analogous to Proposition 20. Exercise 20. Replace the 0-sphere S° = { — 1,1} by the 0-sphere S° = {0, oo} in Exercises 16, 17, and 18. Then the pencil of circles is replaced by the pencil of all lines through 0. Solve the exercises in this case, and draw the resulting configurations. 2 . 7 . 4 M o b i u s Invariants a n d E u c l i d e a n I n v a r i a n t s In the Formulas (14) and (44) we expressed the Mobius invariant for a pair of hyperspheres and a pair {U^, S^) consisting of a circle and a 0-sphere in terms of their Euclidean invariants, this lead us to the Coxeter invariant. Now we want to derive a similar expression for pairs {U"^^, S*') consisting of a hypersphere and an /-sphere, 0 < / < n, in the n-dimensional Euclidean space E". In fact, according to Proposition 5 these pairs are also described by a single Mobius invariant up to Mobius equivalence. Euclidian invariants of such a pair are the radii r, R of the sphere S*' and the hypersphere as weU as the distance d of their centers. In the case I = n — 1 these already provide a complete system of Euclidean invariants. For / < n — 1 we still need to introduce a fourth quantity to obtain a complete system of invariants, e.g. the angle between the connecting line of the centers and the (/ + l)-plane determined by the /-sphere, or, what amounts to the same, the distance f> from the center m of the hypersphere S = IJ{m, R) to this (/ + l)-plane. We choose the pseudoorthonormal basis (e^) for the associated (n + 2)-dimensional vector space so t h a t the /-sphere appears as the intersection of the hypersphere IJ{o,r) of radius r > 0 with the origin o as its center and the coordinate hyperplanes through the origin determined by the basis vectors e^, K = / + 2 , . . . , n as their normal vectors. Hence, from the Euclidean point of view, S*' is the /-sphere of radius r with the origin as its center lying in the coordinate-(/ + l)-plane determined by the first / + 1 coordinates x " , a = l , . . . , / + l. Then the vectors u i := (eo(l - r^) + e „ + i ( l + r'^))/2r,

U2 = e;+2, • • •, Un-i = e„

(45)

are an orthonormal basis for the Euclidean subspace U C V determining the /-sphere S*'; by (11) the vector Ui is the space-hke unit vector belonging to the hypersphere considered. Likewise by (11), the vector w = (eo(l - i ? 2 + d 2 ) _ ^ 2 m + e „ + i ( l + i ? 2 - d 2 ) ) / 2 i ? determines the hypersphere IJ{m,R); here m is the position vector of the center m with respect to o, and d = |m| is the distance of the centers. Now we apply Proposition 5. The subspace W associated with the hypersphere IJ{m, R) is one-dimensional. Hence the operator A reduces to a single component, which thus is the invariant

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2 Cayley-Klein Geometries

a{IJ{m,R),S')

:=ai

to be computed. A straightforward computation starting from Definition (6.70) of the operator A leads to the formula aiSim, R), S') = IJ:i[{ro, u^f = C^^±^^f

+ (|)2,

(46)

which contains (44) as a special case; note that p = dcos f3. Hence we proved: Proposition 21. The only stationary Mohius invariant for pairs (Z'"^^,S'') consisting of a hypersphere S"^^ and an I-sphere S\ Q < I < n, in the n-dimensional Euclidean space E"^ is determined by (46); here R denotes the radius of S"^^, r that of S\ d is the distance of the centers, and p is the length of the projection of the vector connecting the centers onto the suhspace orthogonal to the {I + \)-plane of S\ p = Q in the case I = n — 1. D Next we want to express the Mobius invariants of a circle pair (Co, Ci) in the three-dimensional Euclidean space E in terms of their Euclidean invariants. So let rui be the centers, let rj be the radii, and let rij be the position vectors for the circles Cj, « = 0,1, i.e. the unit normal vectors for the planes in which they he. We consider the circles as non-oriented; hence the position vectors are unique only up to multiplication by ± 1 . Moreover, let t) = m i — mo denote the vector connecting their centers. It is easy to see that the following six quantities form a complete system of Euclidean invariants for the pair of circles (Co,Ci): 1. 2. 3. 4.

The radii ro, ri of the circles. The angle a between the circle planes: cos a = |(no,ni)|, 0 < a < 7r/2. The distance d= |t)| of the centers. The angles /3,7 between the position vectors for the circles and the vector f connecting their centers; for d = 0 these angles remain undefined.

According to Proposition 5 a circle pair has two stationary invariants «i > «2 > 0, namely the eigenvectors for the matrix of their double projection A. Instead of the eigenvalues we consider the determinant and the trace of this matrix, which again are a complete system of invariants equivalent to the eigenvalues in Mobius geometry. If Oi, O2 and bi, 62 are orthonormal bases for the Euchdean subspaces W ,U of the pseudo-Euclidean vector space V^ determining the circles Co, Ci, then the coordinates of A are n—l

a.ij = ^{ai,bk){aj,bk),

i,j = l,...,n-m,

(47)

fc=i

where in the case considered here we have to set n = 3 and I = rn = I. Obviously, det v4 = aiia22 — 012 = aia2, tT A = an + 022 = a i + a2-

(48)

Conversely, this provides a direct way to compute the eigenvalues. We prove

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343

P r o p o s i t i o n 22. The Mobius invariants (48) for a circle pair Co,Ci in the Euclidean space E are expressed in terms of the Euclidean invariants for the pair according to the following formulas: ((rg + rf — £•) cosa + 2£- cos/3cos7) det A = ^^ ^ ' '^ ^ , rlrl Ariri tiA

(49)

d** + Tg + rf + Ar1r\ + 2r1r\ cos 2 a + 2£-{r1 cos 2/3 + r\ cos 27)

Ar^rf (50)

W i t h o u t loss of generality, we may suppose for the proof of this proposition t h a t the circle Cg lies in the x, y-plane of a Euclidean coordinate system with the origin as its center. Then by (11) and (15) the vectors Di = (eo(l - rl) + 64(1 + r^))/2rg, D2 = Cs form an orthonormal basis for the vector space W of the circle Cg; here the first vector defines the sphere of radius rg with its center at the origin, and the second one the x, y-plane. The position vector for Cg is ng = 63. The position vector of the second circle Ci is an arbitrary unit vector in the Euclidean space, say rii = ei cos u cos v + e2 sin u cos v + cs, sin w. We may choose the coordinates in the x, y-plane in such a way t h a t the center mi of Ci has the position vector f = Cix + Csz. Again we represent the circle Ci as the intersection of the sphere with radius r i and center m i with the plane of the circle, i.e. the plane through m i orthogonal to n i . Then (11) and (15) define the vectors bi = (eo(l -rl+

d^) + 2t) + e4(l + r? -

d^))/2ru

b2 = n i + (eo - e4)(f,ni), which form an orthonormal basis for the vector space U of C i . Starting from these expressions the matrix A and its determinant as well as trace can be computed. Having calculated the Euclidean invariants, c o s a = (ng,ni) = sinw, dcos f3 = (ng,t)) = z, d cos 7 = (rii, f) = X cos u cos v + z sin v, the coordinates x,z,u,v can be eliminated from det A and ti A. This leads to the formulas (49) and (50). Since the terms involving the trigonometric functions of /3 and 7 all contain the factor d, we can do without a definition for d = 0 and still have the formulas in this case. To compute the details we again applied Stephen Wolfram's Mathematica [114], cf. the notebook mcircles.nb on the homepage of R. Sulanke. •

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2 Cayley-Klein Geometries

2.7.5 Three-Dimensional Mobius Geometry In this section we want to add some observations concerning circle geometry to the discussion of the geometry in the Mobius space S^. Even if, as beings hving in the three-dimensional, we are unable to directly image a threedimensional sphere, we inevitably view an n-sphere as a hypersurface in an (n + l)-dimensional space. The dimension three, however, has the advantage that we can pass to the familiar three-dimensional Euclidean space via stereographic projection (see the hint in Exercise 1). Since this transformation is conformal, it preserves all Mobius geometric properties at least locally, and thus renders them accessible to our immediate imagination as well as its graphic representation. The geometry of spheres S"^ C S^ and pairs {S"^, S*'), / = 0,1, is the special case n = 3 for the hyperspheres considered time and again in the preceding sections, cf. in particular Proposition 21. Up to now, for circles in S^ we only have Proposition 22, which, on the one hand, allows to compute the Mobius invariants as functions of the Euclidean configuration of the circles. It does, however, not directly yield the stationary invariants for the pair of circles. The expressions obtained for them starting from the formulas proved in Proposition 22 are rather unwieldy. Nevertheless, the stationary invariants, in particular the maximal one a i , contain the important pieces of information concerning the position relations of the circle pair. Applying the vector product provides an alternative approach to these position relations. The series of Mathematica Notebooks called Spheres contains computational tools to study and visualize three-dimensional Mobius geometry. They can be found under the title Spheres in MathSource, or at the homepage of R. Sulanke. In the notations of Proposition 5 let Co = S{U), Ci = S{W) be two circles in S^, and let {oi, O2}, {bi, ^2} be orthonormal bases for the associated Euchdean subspaces W , U . Under these assumptions consider the vector product in the oriented pseudo-Euclidean vector space V defined according to Proposition 2.2.17 n ( C o , C i ) := oi X 02 X bi X 62.

(51)

We will call this the indicator of the circle pair. Then obviously Lemma 23. The indicator defined by (51) is uniquely determined up to the factor ±1 by the circles. If the circles are oriented, then reversing the orientation in one of them changes the sign of the indicator. The indicator is assigned to the circle pair in an equivariant way; for g (^ G4 nigCo,gC\) =

±gniCo,C\).

The factor —1 occurs, if g reverses the orientation of S^.

U

This lemma is a straightforward consequence of Proposition 2.2.18, it immediately implies

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345

Proposition 24. The scalar square of the indicator is a Mobius invariant for the pair of non-oriented circles Co, Ci. Here W{CQ,CI)

:= (n(Co,Ci),n(Co,Ci)) =0.1 + 0.2 -0.^0.2 - 1,

(52)

where o.\,o.2 denote the stationary invariants for the pair Co, Ci. P r o o f . We use formula (2.37) with a = —1. By Lemma 23 we may suppose that the bases are just the canonical bases defined in deriving Proposition 6.24. For them (oi, bi) = V«r, (a2, ^2) = 1/02, (ai, 62) = (a2, bi) = 0. Inserting this into the determinant we need to compute then proves the assertion. D Corollary 25. Let n = n(Co,Ci) be the indicator of the circles Co,Ci. there are the following mutually excluding possibilities:

Then

1. Ifn = 0, then the circles are not in general position, i.e., there is a sphere S'^ C S^ containing both circles (Condition A is not satisfied). 2. The indicator n is space-like: w(Co,Ci) > 0. Then we are in Case 1; the circles are in general position, disjoint, and can he separated by spheres. 3. The indicator is isotropic w{Co,Ci) = 0, n 7^ c Then we are in Case 2; the circles are in general position and intersect in a single point. 4. The indicator n is time-like: w(Co,Ci) < 0. Then we are in Case 3; the circles are disjoint and linked. P r o o f . By Proposition 2.18, 3., Situation 1 occurs if and only if the vectors Oi, O2, bi, ^2 are hnearly independent. If the indicator n is space-hke, then U +W = [n]^ is pseudo-Euchdean and has dimension 4; the assertion in 2 follows as in Lemma 6. If n is isotropic, then so is t/ -]-W = [U n W)^, hence U C\W = [n]; the indicator represents the single point of intersection of the circles, and 3 holds. The proof of the assertion in 4 foUows from the proof of Lemma 7. • Example 8. Let Co,Ci be a circle pair with maximal stationary invariant a i = 0. Then, because of 0 < a2 < a i , we also have a2 = 0, and the matrix A of the double projection is the zero matrix. Such circle pairs are called orthogonal. A particular example for this situation is the pair consisting of the z-axis, represented as the intersection of the coordinate planes, i.e. with the subspace U = [ei, £2], and the unit circle in the x, y-plane with its center at the origin, represented as the intersection of this plane with the unit sphere around the origin, i.e. W = [63,64]. Each sphere containing the unit circle orthogonally intersects every sphere, i.e. plane containing the z-axis. This is apparently clear and can also easily be verified: The space-like unit vectors determining these spheres lie in t/ or W , respectively, and these spaces

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2 Cayley-Klein Geometries

are orthogonal. The scalar square of the indicator is w = —1; the circles are linked. Each pair of orthogonal circles is Mobius equivalent to the pair just described. • Exercise 2 1 . The notion defined in Example 8 is not to be confused with that of two perpendicular (or orthogonally intersecting) circles! Prove that two pairs of perpendicular circles in general position have equal stationary invariants a i = l , a 2 = 0 , and hence are Mobius equivalent. Two spheres each containing one of these circles can intersect at an arbitrary angle. Find an example for this situation and perform the corresponding calculations based on the associated Euclidean subspaces. Moreover, compute the stationary invariants for two perpendicular circles not satisfying Condition A; these again are a i = 1, a2 = 0. Since now the rank condition is violated, such circle pairs obviously are not Mobius equivalent to the one considered above. E x a m p l e 9. The Mathematica notebook mcircles.nb in the series Spheres contains the description of normal forms for circle pairs. For each pair of stationary Mobius invariants a circle pair with just these invariants is specified. As described in Section 6 for pairs of hyperbolic subspaces, starting from the eigenvectors of the operator A one can adapt an orthonormal frame for the pseudo-Euclidean vector space V to the circle pair, in which the defining subspaces of the circles are spanned by vectors whose coefficients only depend on the invariants. Now we consider the case of linked circles. Since the Mobius group acts transitively on circles, we may suppose t h a t the circle Co = S{U) of the pair is the unit circle in the x, y-plane with center at the origin. Then the vectors 63,64 span the defining space U for Co- Let the second circle Ci = C(a, h) of the pair be determined by the Euclidean subspace W

:= [01,02], Oi(a) := ei s i n a + e a c o s a , 02(6) := e2 sin6 + 64 cos6.

(53)

Obviously, C i ( 0 , 0 ) = Co, so t h a t we have to exclude this case. It is easy to conclude from Formula (47) t h a t the matrix A of the double projection has diagonal form in the constructed bases; its diagonal entries are the eigenvalues « ! = cos^ a, 0.2 = cos^ 6, 0 < a < 6 < 7r/2, 6 7^ 0. W i t h the stated bounds we obtain a unique representative for all possible stationary invariants of linked circles. T h e indicator of the circle pair is the time-like vector n{Co,C{a,b))

= eo sin^[a] sin^ [6],

a^O.

For a = 0 the circles C(0, 6) lie in the x, y-plane. Condition A is violated, and the indicator is the null vector. The Euclidean d a t a of the circle C{a, h) are T h e center: m((a, 6) = — e2 sin 6, The radius: r(a, 6) = 1/cos 6, The position vector: p(a, 6) = ei sin a + 63 cos a.

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347

Fig. 2 . 5 1 . The function _F(0.6, 0.6, s,i).

For b = 7r/2 this describes lines through the origin in the x, z-plane intersecting the X-axis at the angle a. T h e cosine of the intersection angle of two spheres each containing one of the circles is F{a, b, s, t) = (e3 cost + 64 sint, Oi(a) cos s + 02(&) sin s) = cos a cos s cost + cos 6 s i n s s i n t . Figure 2.50 shows an example for the graph of such a function. In the case of equal eigenvalues, i.e. for a = 6, we obtain a family of cosine functions with shifted phases, F{a, a, s,t) = c o s a c o s ( s — t ) .

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2 Cayley-Klein Geometries

cf. Figure 2.51. Figure 2.52 shows a part of the surface generated by the circles C(a, a), 0 < a < 7r/3. We caUed circle pairs with equal eigenvalues isogonal. This is not to mean t h a t the spheres containing the circles intersect at a constant angle. •

Fig. 2.52. The surface {C(a, a)|0 < a < 7r/3}.

Exercise 22. Prove that the circle pair C{a,a),C(b,b) is also isogonal for a ^ b, and compute its stationary invariant. Thus the generating circles of the surface in Figure 2.52 are isogonally hnked. In three-dimensional Mobius geometry we still have to consider the pairs (C, 5*°) consisting of a circle and a 0-sphere. Again we need two stationary Mobius invariants to characterize these pairs up to Mobius equivalence. The reader may consult the Mathematica notebook pairs.nb from the series Spheres for a discussion of these invariants and simple examples. Note also the following

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349

Exercise 23. In the n-sphere S", n > 2, we consider tlie sets Mo := {{C,b)\b eC,C

CS"

Ml := {(C, b)\be S"\C,C

circle}, C S" circle}.

Prove that the Mobius group acts transitively on each of these sets, and represent them as quotient spaces (cf. Appendix A.3). 2.7.6 O r b i t s , C y c l i d s of D u p i n , a n d L o x o d r o m e s In this section we want to describe orbits for subgroups of the Mobius group G4 in the three-dimensional Mobius space S^.

Fig. 2.53. The torus t)(s,i,7r/7).

E x a m p l e 10. Tori. Consider the three-dimensional Mobius space S^ = G4/CE{3), cf. (1), which we view, as described in the hint in Exercise 1, as the unit hypersphere in the four-dimensional Euclidean hyperplane E defined by the equation — (eo, p) = 1. T h e orientation preserving elements of the Mobius group are the pseudo-orthogonal transformations leaving the space as well as the time orientation invariant; they form the special pseudoorthogonal group SO{l, 4). This contains the group S'0(4) of orientation preserving isometrics of S*^ as the isotropy group of the vector eo G V^. We decompose the four-dimensional Euclidean vector space W oi E orthogonal t o eo into the two two-dimensional orthogonal components t / i = [ei, e2], t/2 = N , e4], VF^ = t / i e t / 2 .

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2 Cayley-Klein Geometries

In each of these components we consider the rotation group isomorphic to SO{2). In the fixed pseudo-orthonormal basis the matrices of the subgroup in G4 generated by theses rotations have the block form dd(s,t)=lod{s) [0

0 ] withd(^)=(''°^'^~^''^'^]. 0 dit)) V^in^cos^

(54) ;

This, obviously Abelian group is called the torus group T^. The orbits T^p in S^ are circles, if p G Co + Ui or p G Co + U2, and surfaces, called tori , for all other unit vectors; the orbit of the point x(a) with the position vector p(a) = eo + ei cos a + Cs, sin a consists of the points with position vector p(s, t, a) = eo + (ei coss + e2 sins) cos a + (ea cost + 64 sin t) sin a.

(55)

By stereographic projection of S^ onto the equatorial space one obtains for 0 < a < 7r/2 surfaces of revolution t)(s, t, a) = ((ei coss + e2 sins) cos a + 63 cost sin a)/(1 — sin t sin a),

(56)

which are generated by letting a circle in the x, z-plane with its center at (1/cos a, 0,0) and radius t a n a revolve around the z-axis. They are caUed tori as well. Figure 2.53 shows such a torus. Note that this particular one is the orbit for a subgroup of the Mobius group, but not for a subgroup of the Euclidean group of E . Applying inversions in arbitrary spheres leads to Mobius equivalent surfaces, which are also called tori. Figure 2.54 shows such a torus obtained by inversion. D Example 11. Circular Cylinder. In the Euclidean space E circular cylinders are usually described as the orbits of the Abelian group A^ = SO{2) x R; here SO{2) acts as the group of rotations in the x, y-plane, and R acts as the group of translations in the direction of the z-axis. Of course, as a subgroup of the Euchdean group A^ is also a subgroup of the Mobius group, so that, as the orbit of the point with position vector eir, r > 0, the resulting surface in the Euclidean space E^ whose points have the position vectors }{s,t,r)

= (ei coss + e2 sins)r + est

is also an orbit in the sense of Mobius geometry. Applying the inverse stereographic projection leads to corresponding orbits in the Mobius space S^. For r = 0 the orbit reduces to the z-axis. It is easy to prove that all circular cylinders are Mobius equivalent. The generating lines of a circular cylinder are all parallel, from the point of view of Mobius geometry, we have to consider them as circles meeting at 00 in the Mobius space S^ obtained from

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351

Fig. 2.54. A torus generated by inversion.

Euclidean space by adjoining just this point, oo is the only fixed point for all Mobius transformations from A^. Applying the inversion in a sphere whose center does not lie on it, the circular cylinder is transformed into a surface t h a t is generated by circles, the images of the generating lines, which meet at the center of the inversion sphere. Figure 2.55 shows such a surface. •

Fig. 2.55. Image of a circular cylinder under an inversion.

E x a m p l e 12. Circular C o n e . Analogous to the preceding example, we again start from the Euchdean space E . In the conformal group CE{^) we consider

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2 Cayley-Klein Geometries

the Abelian subgroup B^ = SO{2) x R*; here SO{2) again acts as the group of rotations in the x, y-plane, whereas the muhiphcative group R* of real numbers acts as the group of dilations in E^ Note that negative values t G R are not excluded. The origin o and the point oo are the only fixed points for all transformations from B^. The latter is a subgroup of the isotropy group of the Mobius space S^, cf. (1). The orbit of a point from the unit circle in the X, z-plane consists of the points with position vectors c(s, t, a) = ((ei cos s + e2 sin s) cos a + Cs, sin a)t

(57)

For a = 0 the orbit is the x,y-plane itself, for a = 7r/2 it is the z-axis, and for the remaining values of a we obtain the complete circular cone generated by all lines through the origin with slope angle a, where the origin, being a fixed point, has to be omitted. Again applying the inversion in a sphere whose center does not lie on the circular cone and is different from the origin, the generating lines are transformed into circles meeting at the center of the inversion sphere as well as the image of the origin under this reflection. Figure 2.56 shows such a surface. •

Fig. 2.56. Image of a circular cone under an inversion. The surfaces discussed in the last three examples have much in common. Since, in the sense of Mobius geometry, lines are also circles, the surfaces under consideration are Mobius equivalent to surfaces of revolution generated by circles. On each of them there are two families of circles generating the surface, one consisting of the generators for the surface of revolution, and the other formed by their orthogonal trajectories. In differential geometry they were already studied in 1822 by C. Dupin, and after him they are named eyelids of Dupin. Depending on the type of Euclidean surface to which a

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353

eyelid of Dupin is Mobius equivalent, it is said to be of toroidal, cylindrical, or conical type. In the last two cases, the fixed points are frequently considered as singularities of the surface; note, however, t h a t they do not belong to the surface defined as an orbit. It is clear t h a t eyelids of Dupin with different type cannot be Mobius equivalent. T h e eyelids of toroidal and conical type, defined in a Euclidean context, each depend on one parameter: In the first case this is the ratio of the radius of the generating circle to the distance from the center t o the axis of revolution; in the second it is the opening angle of the cone. In contrast to this, all Dupin surfaces of cylindrical type are Mobius equivalent. (Proofs as exercises!) T h e eyelids of Dupin have numerous generalizations occurring in connection with classification results in differential geometry as well as Lie sphere geometry. The book [31] by T h o m a s E. Cecil, which appeared in 1992, provides an excellent introduction into this topic; it describes the historical development, contains numerous references to the literature, and new, interesting results. T h e first textbook on sphere geometry in dimensions 2 and 3 is the book [15] by W. Blaschke and G. Thomsen. It represents a summary of the results obtained in Mobius geometry and Lie sphere geometry up to the time of its appearance.^ Lie sphere geometry has its origin in Lie's theory of contact transformations. The algebraic foundation of the n-dimensional Lie sphere geometry is based on the (n + 3)-dimensional pseudo-Euclidean vector space of index 2. It contains Mobius geometry, and hence also the Euclidean, hyperbolic, and elliptic ones, as geometries defined by certain subgroups. In fact, n-dimensional hyperbolic geometry is obtained from Mobius geometry by fixing a hypersphere, and then, having chosen a scale, identifying one of the hemispheres it bounds with n-dimensional hyperbolic space. Analogously one obtains the Euclidean and the elliptic geometries as subgeometries of Mobius geometry. E x a m p l e 13. L o x o d r o m e s . As a preparation for the following example, the spiral surfaces, we now want to introduce a family of curves, also interesting in itself, in the Euclidean plane E , and hence in the sphere S*^, which are themselves orbits. To this end, we consider the one-parameter subgroup B{a) = {'^/a{t)\t G R } C CE{2) of the conformal Euchdean group of the plane, whose elements are defined in an orthonormal standard basis by the matrices ,,/ c o s t — s i n t \ „• .,, i , T-. 7a(t) := . , , e"'^ with a = const, a, t G R . ' V s m t cost /

/ro\ (58) ^ '

^ In general, the reader should be careful with respect to historical annotations in mathematical textbooks (including the present one). Frequently, the experts don't have the necessary time and patience to completely track down the sources and study them in detail. E.g., in [31] the co-author G. Thomsen of the book [15] is neither mentioned in the text nor in the bibliography; not even his papers are quoted.

354

2 Cayley-Klein Geometries

Fig. 2.57. A loxodrome in the plane.

The corresponding transformations are rotations simultaneously coupled with a dilation of the plane, where the coupling is determined by the parameter a; for a = 0 we obtain the rotation group SO{2) of the plane. T h e only fixed points of all transformations 7a (i) different from the identity are the origin and the point oo. The orbits of a point XQ = eixo on the x-axis under the group H(a) are the spirals called loxodromes with the position vector ?a(^) •= (^1 cost + e2 sint)xo e"* mit a=^ 0, XQ =^ 0,

(59)

and, in the case a = 0, circles, cf. Figure 2.57. The curves in the sphere resulting from applying the inverse stereographic projection to a loxodrome are called hkewise; Figure 2.58 shows such a curve with the coordinate net on the sphere. A ship following a constant compass course moves along a loxodrome or, in the case a = 0, on a circle of latitude. In fact, the family of meridians of S"^ (with the origin as South and the point oo as North Pole) is transformed into itself under the transformations from H{a). Since the same is t r u e of the orbit and as these transformations are conformal, the angle at which the (tangent to the) loxodrome intersects the meridian is constant along the loxodrome. •

E x a m p l e 14. T h e paper by R. Sulanke [101] deals with the classification of homogeneous surfaces in Mobius space S*^, i.e. with surfaces which are orbits for subgroups of the Mobius group G^. In addition to planes, spheres, and the eyelids of Dupin this also includes the spiral cylinders, which we want to describe in this example. In [101] it was shown using a differential-geometric method developed by E. C a r t a n t h a t this hst comprises aU the homogeneous surfaces in S*^. T h e spiral cylinders arise as surfaces in the Euchdean space

2.7 Mobius Geometry

355

Fig. 2.58. A loxodrome on the sphere.

E by erecting the perpendicular cylinder on the plane of a loxodrome, cf. Figure 2.59. By (59) we thus have t h a t

..d

w Fig. 2.59. A spiral cylinder.

356

2 Cayley-Klein Geometries

is the position vector for the spiral cyhnder with the parameter a ^ 0. The spiral cylinders are special cases of the spiral surfaces already introduced by S. Lie. They arise by subjecting a curve in the x, z-plane to the Mobius transformations determined by the spiral group

(

cost — sint 0 \ sint cost 0 I e"* with a = const, a,t G R;

(60)

0 0 l) for a = 0 we obtain the well-known surfaces of revolution. If the generating curve is parallel to the z-axis, then we obtain the spiral cylinder. The subgroup M^ C G4 generating the spiral cyhnder as its orbit is the group depending on the parameter a generated by the transformations (60) and the parallel displacements in the direction of the z-axis; its elements are affine maps in E^ of the form fa{s,t) : a; = [y] G £;2 ^

y = [7„(t)j:+ egs].

(61)

It is easy to verify that for a 7^ 0 the group M^ is not Abelian; in fact, /a(s,t)o/„(so,to) = /a(soe"*+s,to + t)The orbit of the point [eixo] on the x-axis under M^ leads to a parameter representation of a spiral cylinder that is slightly different from the one above. Obviously, every point on the z-axis has the z-axis itself as its orbit. The only common fixed point of all transformations from M^ is the point 00. In Euclidean space the spiral cylinder 2.59 does actuaUy not look too interesting; one obtains a nicer and, from the point of view of Mobius geometry, more accurate image of part of the surface by considering its behavior at infinity. More precisely, reflecting a neighborhood of 00 by an inversion into the flnite region yields Figure 2.60. There, the generating lines of the spiral cylinder appear as circles meeting at the center of the inversion sphere. •

2.7 Mobius Geometry

Fig. 2.60. Inversion of a spiral cylinder.

357

358

2 Cayley-Klein Geometries

2.8 Projective Symplectic Geometry A projective symplectic geometry is understood to be the geometry of a projective space p 2 n - i yj^jgj. ^Yie action of the projective symplectic group PSp^ (cf. Section 1.3) in the sense of F . Klein's Erlanger P r o g r a m m . By Proposition 1.8.5 and the definition of the symplectic groups in this section we will always have to consider a vector space of even dimension, V ^ " , over a field K with a bilinear, alternating, non-degenerate scalar product (,); let F be the non-degenerate null system defined by this scalar product. The projective symplectic group PSp^ was already defined in 1.3 as the isotropy group of the null system F. According to Proposition 1.8.5 all projective symplectic geometries of equal dimension over the same field K are isomorphic. As the Examples 1.1, 1.2 show, the isotropy group PSp^ of F depends upon the algebraic properties of the field K. In general, the isotropy group of F in the group of all projectivities PL^n-i is larger t h a n the group of projectivities induced by the hnear maps g G S'p(V^"). According to Example 1.2 both groups coincide in the complex projective symplectic geometry; on the other hand. Example 1.1 shows t h a t in the real case i f = R apart from the projectivities induced by the linear symplectic transformations g G Sp(n, R ) there are in addition the anti-symplectic ones, whose inducing linear transformations reverse the sign of the scalar product. T h e m a p SQ from (1.21), cf. also (1.22), is an example for this type of projectivities. To have a consistent projective point of view it is here necessary to verify, each time, the invariance with respect to the transformation SQ as well. In the foUowing we take as the proper projective symplectic group PgSp^ the group of projectivities induced by the linear transformations g G S'p(V^"); we thus have

PoSp^=p{Sp{V^^))(lPSp^, and, see Example 1.1, for the real geometry PSp^

= PoSp„

U SoiPoSpJ,

if = R.

(1)

By Example 2.3 the group PgSp^ acts transitively on the projective space P " ^ , which, in t h a t context, was called the projective symplectic space. As a rule, we wiU use the symplectic bases defined in Section 1.3; the linear symplectic transformations are then described by matrices of the form (1.20). Choosing the point Xo = [a2n] as the origin for projective symplectic space, its isotropy group H C PgSp^ is generated by all linear transformations with matrices of the form ( 4 ) ^ ' ^ P ( " ' ^)

with 4 „ = 0 for i< 2n, all + 0.

(2)

E x a m p l e 1. As in Section 5 we consider the covering of the real projective space by the (2n — l)-sphere, cf. (5.4), which again is identified with the set of directions, i.e. the oriented one-dimensional subspaces of the vector space V " .

2.8 Projective Symplectic Geometry

359

The symplectic group S'p(n, R) acts transitively on S*^"^^, and the isotropy group of the direction determined by 02n is the subgroup of H described by matrices satisfying a^n > 0. The invariants of this transformation group form the subject dealt with in spherical symplectic geometry. • 2.8.1 Symplectic Transvections It is remarkable fact that a statement analogous to Proposition 1.4 concerning the generation of the special hnear groups by transvections also holds for the symplectic groups. Obviously, the transvections to be considered here are those preserving the symplectic scalar product; we call them symplectic transvections. Lemma 1. A transvection p i-^ p + hu){f} of the symplectic vector space V " is symplectic if and only if cu{f) = c{b,f); hence each symplectic transvection can be written in the form t(f, ^-,(j;) := ? + bc{b,f) with b eV,

ce K.

(3)

P r o o f . For symplectic vector spaces each transformation of the form (3) is a transvection; the requirement cu{b) = 0 (cf. Exercise 1.12) presents no extra condition, since b = [b] e b belongs to its own polar. If b = o or c = 0, then (3) is the identical map, which we call the trivial transvection. To characterize symplecticity we consider an arbitrary transvection f oi V. By Exercise 1.12 it has the form /(p) = p + buj{^), where u; e V' belongs to the annihilator of b. Inserting the map / into the scalar product we see that it is a symplectic transvection if and only if for all p, t) G V (b,t))^(p)-(b,p)^(t)) = 0. For each non-trivial transvection we have b 7^ o. Since the scalar product is non-degenerate, we find a vector t)Q e V with (b, t)o) ^ 0. Hence the formula above implies the assertion u;(p) =c{b,^)

with c =

uj{t)o)/{b,t)o).D

For each symplectic transvection t,^ , the polar of [b] is the hyperplane of fixed points for t.j, ,. For fixed b the set comprising these transformations is an Abelian subgroup of the symplectic group isomorphic to [K, +]; actually, as is easily verified, *(b,a) ° * ( b , c ) = * ( b , a + c ) -

Now we prove the already announced Proposition 2. Each symplectic transformation g G Sp{n,K) can he represented as the product of finitely many symplectic transvections.

360

2 Cayley-Klein Geometries

P r o o f . We will proceed by induction. The assertion for n = 1 is nothing but the beginning step of the induction proving Proposition 1.4; in fact, Sp{l,K) = SL{2,K). Without loss of generality, we may suppose that g leaves a vector o 7^ o fixed. To see this, we have to show that the group G generated by the transvections acts transitively on the set of vectors different from the null vector. If p, t) G V are two vectors spanning a symplectic subspace, i.e. satisfying (p, t)) ^ 0, then the map (3) with b = t)—p and c = l/(t), p) is a transvection with *(b,c)(?) = ? + ( t ) - ? ) ( t ) - ? . ? ) / ( * ) . ? ) = t ) -

If, however, (p, t)) = 0, then there is a vector c e V with (p, c) 7^ 0 and (t), c) ^ 0: Projectively this condition means that we have to find a point c = [c] for which c ^ [p]^ as well as c ^ [t)]^, i.e. which neither belongs to the polar of a; = [p] nor to that of y = [t)]. If the vectors p, t) are linearly dependent, then the polars coincide, and every point c of the complement satisfies the condition. If the vectors are linearly independent, then n > 1, and the polars intersect in the isotropic projective (2n — 3)-plane x^ A y ^ = (a; V y ) ^ . Now, in each projective space there always is a point that does not lie in the union of two non-intersecting hyperplanes. It suffices to choose a point in each of the two hyperplanes that does not belong to their intersection; on the connecting hue of these necessarily different points there is at least one more point, which then cannot belong to any of the hyperplanes. According to what we proved before we thus find symplectic transvections ti, t2 with ti(t)) = c and t2(c) = p; so, if gi(p) = t) 7^ p, then gi = ^2 o ti o gr has the fixed point p. Let now g{a) = 0, b be a vector with (0, b) = 1 and b' = g{b). We want to show that there are transvections hi, h^ also leaving 0 fixed and, moreover, satisfying /12 ° hi{b') = b; then the transformation g2 := h2 o hi o g leaves each vector in the symplectic subspace U = [0, b] spanned by 0, b fixed. Thus it suffices to prove that symplectic transformations with this property can be generated by finitely many symplectic transvections. If (b, b') ^ 0, then the maps ^^ = *(b-b',c) •^i*^ ^ = !/(''> t'') and the trivial transvection h^ each meets the requirement; to see this, note that (0, b) = (0, b') = 1. The case b = b' is trivial. If b 7^ b' and (b, b') = 0, then because of (b', 0 + b) 7^ 0 and using what was just proved we find a symplectic transvection hi mapping the pair (0, b') into the pair (0, 0 + b). As (b, 0 + b) 7^ 0, in the same way we find a transvection /12 mapping (0, 0 + b) to (0, b); but this is what was to be proved. Now it is easy to complete the induction. We suppose that g\u = idjy and consider the orthogonal decomposition into the symplectic subspaces V = U -\- W with W = U , both invariant under g. By the induction hypothesis we can represent the symplectic transformation (7||;j/ as the product of transvections in W. Let t ^ denote such a transvection, then

2.8 Projective Symplectic Geometry

361

is a transvection in V, and the product of the factors for g\\^ extended this way yields the representation for g as the product of finitely many symplectic transvections we were looking for. D Since each transvection has determinant 1, Proposition 2 immediately implies the following, which was already announced in (1.19), Exercise 1.3, C o r o l l a r y 3. Every symplectic nant 1.

transformation

g G Sp(n,

K)

has

determi•

2.8.2 S u b s p a c e s We will commence this section with the classification of subspaces W^ C V ^ " in a symplectic vectors space. At the same time, its projective interpretation wiU lead to the symplectic classification of A;-planes. Restricting the scalar product to the subspace W we obtain an alternating bilinear form. But according to Lemma 1.8.4 these are classified by their even rank 2r. T h e number d := k — 2r is called the defect of this bilinear form; the terms rank and defect wiU also apply to the corresponding subspaces. Then P r o p o s i t i o n 4. The rank 2r and the defect d of a k-dimensional W in a symplectic vector space V " satisfy the conditions 0
+ d
2r + d=k.

subspace

(4)

Let M2n,2r,d dcnotc the set of all subspaces with rank 2r and defect d. Then M2n,2r,d is not empty if and only if (4) holds. The symplectic group Sp{V " ) acts transitively on each of the sets M2n,2r,dP r o o f . According to Lemma 1.8.4 we find a basis O i , . . . , Or, bi, • • •, brr W whose vectors have the scalar products (Oi,bi) = - ( b i , O i ) = 1 , « = l , . . . , r ; ( 0 i , b j ) = (Oi,Oj) = {bi,bj)

=0,i

^ j; (5) a linearly independent vector sequence like this will be called symplectic. T h e inequality in (4) is an immediate consequence of Corollary 2.6; in fact, the subspace [ b i , . . . , br+d] is an (r + (i)-dimensional, totally isotropic subspace. The inequality in (4) follows from 2r + d = dim W. Let (Oj),« = 1 , . . . , 2n, be a symplectic basis; setting bj := a„+j,j = I,... ,r + d, we obtain a symplectic vector sequence O i , . . . , Or, b i , . . . , br+d, spanning a A;-dimensional subspace with rank 2r and defect d. This again is possible if and only if (4) holds. Since subspaces of equal dimension with equal rank are isomorphic, the transitivity statement follows from Corollary 2.11 of E. W i t t ' s Theorem 2.3. • Exercise 1. Prove that each symplectic vector sequence in a symplectic vector space can be completed to a symplectic basis. Conclude from this the transitivity statement in Proposition 4 without applying E. Witt's Theorem.

362

2 Cayley-Klein Geometries

The maximal dimension of any totally isotropic subspace in a symplectic vector space V " is n; because of their application in mechanics these subspaces, satisfying r = 0 as well as d = n, are also called Lagrange subspaces, see e.g. A. T. Fomenko [38]. Because of dimVt^ = 2n — diinW, the Gra£mann manifold G'2n,n of a symplectic vector space is mapped involutively onto itself by the nuU system F. The Lagrange subspaces are the elements fixed by this map. Interpreting subspaces projectively leads to a corresponding statement in projective geometry. For example, the polar of a line a = x V y in the three-dimensional projective symplectic space P is again a line, which appears as the support of the line pencil for the polar planes corresponding to the points of a: F{x V y) = {xV y)^ = x^ A y^ = F{x) A F{y). E x a m p l e 2. Consider the (2n — l)-dimensional projective space P "^ over the field K. Its points correspond to the one-dimensional subspaces of the associated symplectic vector space V ". Proposition 4 imphes r = 0, d = 1, and again the transitivity of the group PgSp^ on P "^ . For a hyperplane the dimension of the associated vector space is A; = 2n — 1; then (4) implies d = I, r = n — I, and by Proposition 4 the group PgSp^ acts transitively on the set of hyperplanes. The set of projective hues b = x\/ y splits into two orbits: The first is the linear line complex ^p corresponding to the symplectic polarity F, cf. (1.8.21) consisting of the isotropic lines, for which r = 0 and d = 2. And the second is its complement in the Grafimann manifold -P2n-i,i of all lines containing the symplectic lines, for which r = 1 and d = 0. The isotropic lines through the point a are the lines a\/ x connecting a to the points in its polar different from a: a; G a . All other lines through a are symplectic. The foUowing proposition shows that instead of the null system F one may also consider the linear line complex Rp of its isotropic lines as the absolute for the projective symplectic space. • P r o p o s i t i o n 5. A projectivity g G PL^n-i of the projective symplectic space P "^ maps the abso " absolute line complex ^p to itself if and only if g ^ PSp^ is projective symplectic. P r o o f . The condition is obviously necessary: each symplectic projectivity g G PSpn maps Rp bijectively onto itself. Let (Oj) be a symplectic basis, and let Oj = [Oj], i = l , . . . , 2 n , be the vertices of the corresponding (2n — l)-simplex. Moreover, let <; be a projectivity induced by the linear map a G GL{V ") mapping ^p into itself, and denote by bj = a{iXi) the images of the basis vectors. If (Oj, Oj) = 0, then (bj, bj) = 0 has to hold as weU, since by assumption the image of the edge g{ai V aj) = g{ai) V g{aj) = [hi, bj] belongs to Rp. For the scalar products of the image vectors this yields

2.8 Projective Symplectic Geometry {bi,bn+i)

= -{bn+i,bi)

= i^i,i = l,...,n;

{bi,bj)

= 0 else.

363 (6)

Since F has rank 2n, the determinant involving these scalar products is different from zero. This implies t h a t all yUj are different from zero. Setting Cj := bi and c„+j = b„+j//ij for « = 1 , . . . , n, the {Cj),j = 1 , . . . , 2n, form a symplectic basis. Let h G PgSp^ denote the symplectic projectivity induced by the symplectic transformation c t h a t transforms the basis (Cj) into (Oj). T h e projectivity hog induced by coa then also transforms Rp into itself. We show t h a t f = hog is a symplectic projectivity; consequently, this also holds for g = h^^of. The m a p / is induced by 6 = coa; the definition of these maps imply 6(0i) = c{bi) = c(ci) = Oi, H^n+i)

= c{b„+i)

= c(c„+j)/ij =

(7) ttn+il^i

(8)

for « = 1 , . . . , n. We assert t h a t in (8) aU /Xj have to coincide. Then the proof is complete; in fact, it is easily verified t h a t in this case we have for arbitrary vectors p, t) G F ^ " (6(y),6(t)))=M(?,t)); hence the m a p b is conformal symplectic, cf. Lemma 1.1. Consider, e.g., the vectors Oi, O2. Using (7) and (8) we conclude from (oi + 02, a„+i - a„+2) = 0 t h a t the images under the m a p b satisfy (6(oi + 02), 6(a„+i - a„+2)) = (ai + 02, fln+iMi - a„+2M2) = Mi - M2 = 0; this is just the assertion.

D

2.8.3 Triangles In this section we are going to classify triangles in the projective symplectic space p 2 n - i ^ First, let the scalar domain be an arbitrary field K; the complete classification, however, will only be carried out for i f = R , C . We again use the notation for the vertices and sides familiar from school mathematics. Let the triangle A = {A, B, C) be non-degenerate, and suppose t h a t its sides satisfy a = By C,b = Cy A,c = Ay B. The number k of isotropic sides provides a first coarse classification of triangles; the possible values are A; = 0 , 1 , 2, 3. T h e simplest case is covered by the following exercise: Exercise 2. Two triangles with three isotropic sides are symplectically congruent. This case can only occur in projective symplectic spaces P^ of dimensions N > 5.

364

2 Cayley-Klein Geometries

If at least one of the triangle's sides is not isotropic, then for the plane H := A\/ B y C spanned by the triangle the parameters are r = d = \, i.e. the rank of the scalar product restricted to the corresponding vector space is equal to two. Since the symplectic group acts transitively on the set of planes of this type we may suppose, without loss of generality, that the triangles in question lie in a common plane; moreover, we may assume that this plane lies in a three-dimensional projective space. So from now on we take N to be three. The pole p = H belongs to H; a hue h C H is isotropic if and only if p e h. Consider now the case k = 2; let c be the only non-isotropic side of the triangle. Since both the sides b, a are isotropic, p = C has to be their point of intersection. We may thus represent A, B, C as follows by a symplectic vector sequence A = [ e i ] , B = [e3],C=[e2]. Then we complete this vector sequence to a symplectic basis by adding suitable vectors. In a similar way, for each triangle of this type we construct a symplectic basis with the same properties; the symplectic transformation mapping these bases into one another provides a symplectic congruence between the triangles. These considerations imply: Proposition 6. Two triangles in the projective symplectic space P each having precisely two isotropic sides are symplectically congruent. Moreover, two arbitrary pairs of isotropic lines each meeting at a point and spanning a plane that is not totally isotropic are symplectically congruent. Hence there are two classes of pairs of intersecting isotropic lines: line pairs spanning a totally isotropic plane and those spanning a plane with r = 1. In the threedimensional projective symplectic space the first case cannot occur. • Now consider the case k = 1; let, say, the side b = Ay C ha isotropic and suppose that both the others are not isotropic. Then the pole p hes on the side 6, and we have p ^ C, since otherwise the side a were also isotropic. Choose a symplectic vector sequence adapted to the triangle in the following way: A = [ e i ] , B = [e3],p = [e2]. (9) Since C ^ b = Ayp, C ^ p, there is a representative c for C with the basis representation c = ei + 627. The relation C ^ A implies 7 7^ 0; replacing t^ by e27 and again denoting the result by e2, in this symplectic vector sequence we have A = [ e i ] , B = [e3],C=[ei + e2], and as in the case A; = 2 we obtain Proposition 7. Two triangles in the projective symplectic space P containing precisely one isotropic side are symplectically congruent.

each •

Constructing in a suitable way for the line pair in question a triangle with precisely one isotropic side one obtains

2.8 Projective Symplectic Geometry

365

Corollary 8. Two pairs of different intersecting lines in which corresponding lines have the same type are symplectically congruent. P r o o f . If both lines are isotropic, then the assertion is true by Proposition 6 and Exercise 2. Consider two intersecting symplectic hnes a, c intersecting in B. Then the pole p of the plane they determine lies on none of these lines. Then each line b of the plane pencil with center p not passing through B intersects both lines; let A = 6 A c, C = a A 6 be the points of intersection. The resulting triangle has precisely one isotropic side. Proceeding similarly for the second line pair, Proposition 7 implies the assertion. The case of one isotropic and one symplectic hue is treated analogously. • Now we consider the case that all sides of the triangle are symplectic. In order to adapt the symplectic frame we first take a symplectic vector sequence satisfying (9). Let c be a vector representing the point C. Its basis representation c = ei7i + e272 + 6373 has the property that all its components are different from zero; 72 = 0 would imply C ^ Ay B, and if 71 or 73 were zero, then the triangle would have an isotropic side. The representatives of A and B are only determined up to a transformation ei = tijj,, h = t'ilii. The condition 71 = 73 for the transformed coefficients leads to the equation 71/73 = M^) which, in general, has no solution. For if = C, however, it is always solvable, and otherwise only if 71/73 is a square. If this is the case, then by suitably normalizing the representatives t^ for p we may arrange all the coefficients in this basis representation for c to have the same value 7 := 71 = 73; the transition to the representative c/7 then leads to the following canonical representation for c (returning to the old notations): c = ei + e2 + e3.

(10)

In real symplectic geometry the situation 71/73 < 0 may well occur; in this case a transformation similar to the one above leads to the canonical representation c = ei + e 2 - e 3 . (11) In general, one can construct a canonical representation in which the third coefficient belongs to a complete system of representatives for the cosets in the quotient group K*/K*"^. The above considerations imply: Proposition 9. In the complex projective symplectic space P all triangles without isotropic sides are symplectically congruent. In real projective symplectic geometry there are precisely two classes of symplectically congruent triangles with exclusively symplectic sides with respect to the group PoSp^; they are distinguished by the relation (10) or (11) satisfied by the representative for C in the canonically adapted frame. With respect to the full real symplectic group all triangles with exclusively symplectic sides are congruent.

366

2 Cayley-Klein Geometries

P r o o f . The asserted congruence follows from the equality of the canonical forms by completing the symplectic vector sequence to a symplectic basis. The existence of congruent triangles which are not congruent with respect to the real proper symplectic group PgSp^ can be shown as follows: Consider the triangles {A,B,C), {A,B,C) in the real P^, where we assume that, in an arbitrarily fixed symplectic frame, A = [ei], B = [eg], C = [ei + ea + eg], C=[ti

+ t2- eg].

Let a be a linear transformation inducing the projectivity g satisfying {g{A),g{B),g{C)) = {A,B,C).i:\ien a(ei) = eia, a{t2) = e2/3,0(63) = 637, a(ei + e2 + 63) = (ei + e2 - e3)A, where all occurring coefficients are different from zero. The second condition has to be satisfied, since a symplectic transformation leaves a plane together with its poles fixed. Inserting the first three equations into the last one leads to a = /3 = - 7 = A, i.e. (a(ei), 0(63)) = -A^, and hence a cannot lie in Sp{n, R). The linear transformation defined by s(ei) = ei, s(e2) = e2, s(e3) = -63, s(e4) =

-CA

reverses the sign of the scalar product, and hence induces a map in the projective symplectic group PSp(n, R) thus proving symplectic congruence of the triangles. • 2.8.4 Skew Lines The symplectic congruence of pairs of intersecting lines was thoroughly discussed in the previous section. This leaves the interesting task of classifying pairs of non-intersecting, i.e. skew lines. We start by considering pairs of symplectic lines. Any two skew lines always span a three-dimensional projective subspace; if the dimension of the whole space is N > 5, and if at least one of the lines is symplectic, then this subspace may be symplectic or isotropic; in the latter case its rank and defect are equal to 2. Example 3. Consider a 5-dimensional projective space. Let (ej), « = 1 , . . . , 6, be a symplectic basis for the associated vector space. Then the hues a = [ei] V [63], 6 = [ei + 62] V [e3 + 64] are symplectic, and the projective subspace they span has rank and defect both equal to 2. D The expression defining an invariant for symplectic lines presented in the following lemma can be found in the book [95] by B. A. Rosenfeld:

2.8 Projective Symplectic Geometry

367

Lemma 10. Let g^ = [p] V [t)], §2 = [D] V [vo] he two symplectic lines in the N-dimensional projective symplectic space P . Then the number '^"^^3^,9,)-

^^^^^^^^^^

(12)

only depends on the lines and does not depend on the points or vectors representing them; consequently, it is a symplectic invariant for the pair of symplectic lines. Obviously, sym(gi,g2) = sym(g2,9i)-

(13)

P r o o f . Under a transformation of the bases p, t) and D, tn for the vector spaces associated with the hnes the denominator as well as the numerator are multiplied by the determinant of the transformation matrix. The invariance follows from the invariance of the scalar products. • Example 4. As in most cases up to now the vector representing a point x will be denoted by the corresponding Fraktur letter below, x = [p]. Let, moreover, (ej), « = ! , . . . , 2n, be a symplectic basis to which the coordinate simplex (ej) corresponds. For N = 2n — 1 = 3 the coordinate tetrahedron has precisely two symplectic sides which are polar to one another: F{ei Vea) = 62 V 64. Equation (12) immediately implies sym(ei V e3,,F{ei V 63) = 0. Since the symplectic group acts transitively on the set of symplectic hnes, the following holds for a symplectic line g in general: sym(g,F(g))=0.

(14)

The invariant syin(gi,g2) is, however, also equal to zero if 92 intersects the polar of gi- Because of the transitivity just mentioned, we may always suppose g^ = ei V 63. So let 62 be the point of intersection of ^2 = ^2 V a with the polar of gi] here a is an arbitrary point not belonging to the polar plane ei V 62 V 63 = F{e2). Because of (ei, 62) = (63, 62) = 0 the definition (12) immediately implies the assertion. Note that for pairs of skew lines not symplectically congruent the invariants sym may coincide; in fact, a pair of lines which are not polar can never be congruent to a pair of polar lines. Soon we will learn that, at least in the three-dimensional projective space, this can only happen for the values zero and one of the invariant. It is easy to prove that sym{g,g) = l (15) for each symplectic line g. The invariant syin(gi,g2) is, however, also equal to one if g2 intersects gi- We may suppose that gi = ei Ve3 and ^2 = ^3 Va, a ^ -F(e3); equation (12) implies the assertion. Using (11) one can show that

368

2 Cayley-Klein Geometries

for the line pair in 5-dimensional space spanning a plane of rank and defect 2 described in Example 3 the invariant is equal to 1. So, in summary, we constructed three mutually not congruent symplectic line pairs whose invariant is equal to one; they differ, however, in the rank or defect of the subspaces they span. D T h e following exercise shows t h a t the invariant sym is analogous to the cosine of the stationary angles for subspaces in the metric geometries. Exercise 3 . Let g,hhe two symplectic lines. The subspaces U,W to them define orthogonal decompositions

corresponding

Consider the projections pi : U ^ W, p2 : W ^ U they define. Prove that the linear endomorphism p2 o pi satisfies: P2opi = sym{g, h) idjj . T h e following proposition shows t h a t there are non-congruent line pairs with the same invariant only for the values zero or one of the invariant. P r o p o s i t i o n 11. Let (91,92), (hi, h2) be pairs of symplectic jective symplectic space P , N = 2n — 1. If their invariants 7 •.= sym{9-^,92)

lines in the proare equal:

= sym(/ii,/i2),

and J / 7 7^ 0 , 1 , then the line pairs are symplectically

congruent.

P r o o f . According to Example 4 the line 92 can neither intersect 9^ nor its polar F(9i). Hence 9i,92 span a three-dimensional projective subspace. By the following lemma this subspace cannot be isotropic, hence it has rank 4: L e m m a 12. Let {9^,92), (/ii,/i2) he two pairs of symplectic lines each spanning a three-dimensional projective subspace H , M with defect (and rank) 2. Then sym{9-^,92) = sym(/ii,/i2) = 1, and the line pairs are symplectically

congruent.

P r o o f . The assumption of the lemma imply N > 5. Once again we choose the vertices of the coordinate simplex for P ^ so t h a t 9^ = e i V e „ + i . Let k be the defect line corresponding to the defect subspace of the vector space W associated with H . Since the hues ^1,92 ^'"^ symplectic, they do not intersect k. T h e planes fe V e i and fe V e „ + i intersect the hue 92 at the points ai and 03, respectively, so t h a t 92 = a i Vaa; these points have to be different, since they cannot lie in k: in fact, 92 is symplectic. Now we determine the points of intersection

2.8 Projective Symplectic Geometry 6 = fe A (ei V a i ) , c = fe A (e„+i V 03).

369 (16)

They exist, since the hnes defining t h e m are different lines in a projective plane. In addition, they have to be different themselves; if we had b = c, then the lines fli,fl2 would belong to the plane e i V e „ + i V b, which contradicted the assumption. To obtain further vertices of the symplectic frame we now set 62 := b, 63 := c. T h e representatives Oi, O3, e2, C3 are normalized to satisfy Oi = ei + e2, 03 = e3 + e„+i. Then (oi, O3) = 1, and (12) implies sym(g^,g2) = 1- The same also holds for the other line pair, for which we find vectors ej, i = 1,2, 3, 5, with analogous position relations. Completing each of these vector sequences to a symplectic base for the vector space associated with P ^ the symplectic transformation they determine maps the line pairs into one another. D Now we return to the proof of Proposition 11. Since the subspaces spanned by the pairs are now three-dimensional as well as symplectic, and as the symplectic group acts transitively on the set of such subspaces, we may suppose t h a t all lines in question lie in the same projective space P^, i.e. iV = 3. As in the proof of the lemma, starting from g^ = e i V 63 and the hue k = F{gi) polar to it we construct points a i , 03 G §2 and b,c e k according to (16). Since, this time, k is symplectic and polar to Qi, we now set 62 := b and 64 := c. T h e points (cj), i = I,... ,A, are the vertices of a coordinate tetrahedron; for arbitrary representatives of the points we have by construction oi = ei/3i + e2/32, 03 = 6371 + 6472Here all these coefficients are different from zero, since otherwise §2 would intersect the hue Qi or its polar 62 V 64; according to Example 4 this would imply 7 = 0 or 7 = 1. Replacing t2 by ^2132/Pi and returning to the former notation we then have oi = (ei + e2)/3i, and dividing Oi by /3i we arrive at a representative for a i satisfying Oi = ei + e2. Hence, up to a common factor, which we may set equal to one, the vectors Oi, ei, e2 are fixed. Since the basis (ej) is symplectic, the vectors Cs and e4 are also uniquely determined; we normalize 03 so t h a t 1 = (oi, 03) = (ei + e2, e37i + 6472) = 71 + 72Inserting this into (12) leads to: 7 = sym(gi,g2) = (ei,ei + e2)(e3,e37i + 64(1 - 7 1 ) ) - (ei,e37i + ^4(1 - 7 i ) ) ( e 3 , e i + 62) = 71W h a t we proved so far is summarized in the following lemma:

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2 Cayley-Klein Geometries

Lemma 13. Let {91,92) be a pair of symplectic lines with invariant 7 7^ 0,1. Then these lines span a three-dimensional symplectic subspace. There is a symplectic basis for the associated vector space V " such that 9i=eiV

62 and 92 = [ei + e„+i] V [e„+i7 + e„+2(l - 7)]-

(17) D

Bases of this kind will be called canonically adapted. Now the proof of Proposition 12 is immediate: If (ej) is a basis canonically adapted to the line pair hi,h2, then, since the coordinates of the vectors determining the line pairs are equal, the transformation defined by (/(ej) = li,i = 1,... 2n, maps these, and hence the line pairs as well, into one another. D AU that remains is to consider a few special cases. First we discuss pairs of symplectic lines with invariant zero. Proposition 14. The invariant sym(g]^,g2) of two symplectic lines 9i,92 C P vanishes if and only if 92 meets the polar F{9i), 92 A F{9i) ^ o. In this case there are two classes of congruent line pairs: 92 C F{9i) and 92 A F{9i) is a single point. P r o o f . Again we start from 9^ = ei\/ 63. If 92 = ai\/ 03, then for any of its points we have [OlS + 03t] € 9 2 A-F(fli) if and only if (s,t) is a non-trivial solution of the following homogeneous system of equations: (ei, oi)s + (ei, 03)t = 0, (e3, ai)s + (e3, 03)* = 0.

(18)

Such a solutions exists if and only if the determinant of the system vanishes, but this is just the numerator of the invariant (12). If all coefficients are equal to zero, then 92 C F{gi); otherwise, the solution consists of a single point. Obviously, in the first case two pairs of symplectic lines are always symplectically congruent; in fact, two such hnes occur as the edges of symplectic coordinate simplices. In the second case we obtain a canonically adapted coordinate simplex by first setting 62 = ^2 ^ F{9I)J then choosing one more point e„+2 on the line 92, and, in dimensions larger than 3, completing the basis symplectically in a suitable way. • Now we consider the case sym(g^,g2) = 1. It is then enough to consider the situation that the subspace spanned by the skew lines is symplectic, i.e. has defect zero; the case rank = defect = 2 was already treated in Lemma 12. In the three-dimensional space P the situation is particularly symmetric: Exercise 4. Let 0^,02 C P^ be symplectic lines. Prove sym(gi,g2) + sym(-F(gi),g2) = 1-

2.8 Projective Symplectic Geometry

371

Conclude from this that sym(
372

2 Cayley-Klein Geometries

P r o p o s i t i o n 15. Two pairs of isotropic lines in the N-dimensional symplectic space are symplectically congruent if and only if they span suhspaces with equal rank and defect, respectively.

projective projective •

Exercise 5. Prove tlie assertion of Proposition 15 in the cases where the defect of the subspace spanned by the lines is positive, and determine the minimal dimension of a projective space in which skew line pairs spanning a subspace of defect two or four, respectively, can occur. Find examples for such line pairs. E x a m p l e 6. Now we consider the case t h a t one of the hnes is symplectic, and the other one is isotropic. We take any symplectic basis of V^ and set g^ = e i V e 3 , oi = e2 + e3, 02 = ei + e4, (19) then Qi and §2 = aiM a^ form such a pair; the defect of the projective subspace spanned by fli,fl2 i^ equal to zero. Conversely, we will show t h a t for an arbitrary line pair P i , P 2 with these properties there is a symplectic basis in which g^ = eiM 63 and §2 = aiM 02; here 01,02 are points represented by the vectors (19). This again implies the symplectic congruence of all such pairs. • P r o p o s i t i o n 16. Let P i , P 2 ^^ two skew lines in the N-dimensional projective symplectic space, Qi symplectic, §2 isotropic, together spanning a threedimensional symplectic projective subspace. Then they are symplectically congruent to the pair described in Example 6. P r o o f . As earlier, in the examples with rank four and defect zero, we may suppose iV = 3. First we choose two points 61,63 G Qi and represent t h e m so t h a t (ei, 63) = 1. T h e line §2 cannot lie in the polar of any point x ^ g^, because of §2 C F{x) and, since x e g^ imphes F{g-^) C F{x), we then had two different hnes, the symplectic one -F(fli) and the isotropic one ^2 i^^ the plane F{x). They thus determined a point of intersection a = F{gi) A ^2But then the polar of a were F{a)=g,VF{g2)=giVg2, since ^2 is isotropic, i.e. ^2 = ^iQi)- ^^ 9IT92 ^^^ skew, they span the whole space and not just the plane F{a), so t h a t the equalities above show a contradiction. Hence the points of intersection a i := 92 A -F(63), 02 := ^2 ^

Fi^i)

are well-defined. These points are different; in fact, otherwise there were a point of intersection F{gi) A ^2 with a = ai = 02, whose existence we just excluded. Take representatives Oj = [Oj], i = 1,2,. Then (oi,ei) 7^0, (02,63) 7^0;

2.8 Projective Symplectic Geometry

373

actually, otherwise we had a i G F{ei)

A ^ ( 6 3 ) = F{ei V 63) = F ( g i ) ,

which was excluded above; analogously for 02. Thus we may normalize Oi, O2 so t h a t (oi,ei) = 1, (02,63) = 1. Now define e2 := ai - cs, U •= 02 - eiIt is easy to verify t h a t {Cj),j satisfying (19).

= 1 , . . . , 4 , is a symplectic basis obviously •

Exercise 6. Find an example for a skew line pair
§2 isotropic, congruent.

span•

Obviously, it is an interesting task to find symplectic invariants for pairs of subspaces with arbitrary dimension and to derive corresponding classification results. Approaches to treat this problem and particular results can be found in the paper [57] by I. M. Jaglom. Since the overaU situation is considerably more complicated t h a n in the Euclidean, elliptic, or hyperbolic geometries, we only discuss the sole remaining case for dimension 3, pairs consisting of a plane and a line. E x a m p l e 7. Let (A, h) be a pair formed by a plane A = A and a line h in the symplectic projective space P^. First suppose t h a t the line h is symplectic. Then the foUowing position relations have to be distinguished: 1. h C A,

2. h^ C A,

3. h and h^ b o t h intersect A.

Since a symplectic line h and its polar h are skew, they cannot b o t h lie in the plane A. T h e foUowing is easy to show: Two pairs of the kind in question are symplectically congruent if and only if they satisfy the same position relation. Obviously, the conditions are necessary. To show t h a t they are also sufficient we adapt, in each of the cases 1-3, a symplectic frame to the pair (A, h) in which the pair is fixed in a specific way only depending on the respective type of relation. Then the symplectic transformation mapping one of these frames to the other also transforms the pairs into one another. In Case 1 the assumption h C A immediately implies a := A e h , and hence the pole a of A is the point of intersection a = h A A . Choosing two

374

2 Cayley-Klein Geometries

points e i , 63 G h, then setting 62 := a, and finally fixing as 64 a point in h diiTerent from 62 we obtain an adapted frame. In the second case we proceed analogously by interchanging the roles oi h,h . Now consider the third case and set e i := hA A, e^ '•= h A A. Then e i V 62 C A is an isotropic line, and hence the pole a = A lies in e i V 62. Here a 7^ e i , a 7^ 62, since otherwise we would be in the first or the second case: In fact, a = A = ei e h would imply h C a^ = A. So if 0 is a vector representing the pole a, then we have 0 = eis + e2t with s,t 7^ 0, and by a suitable normalization we can arrange s = t = 1. Completing 61,62 by 63 G h and 64 G /i to a symplectic frame the last equation states t h a t it is adapted to A = a,h and hence also to the pair {A,h). This imphes the assertion. T h e case of an isotropic hue wiU be discussed in the next exercise. • Exercise 7. Consider pairs {A,h) consisting of a plane A and an isotropic line h in the three-dimensional projective symplectic space P ^ . Then the only possible position relations are 1. h
i.e.

(a(j:), t)) + (y, a(t))) = 0 for aU y, t) G F , cf. Section 2.10. RecaU the G r a m matrix of a symplectic basis.

(20)

2.8 Projective Symplectic Geometry

375

((e.e,))^(_0/^ and write the matrix of the endomorphism as a block matrix,

then condition (20) expressing the skew symmetry of a is equivalent to D = -A', B = B',C

= C,

(21)

where A' denotes the transposed matrix oi A. Lemma 2.20 and Exercise 2.17, d), immediately imply Lemma 18. Assigning the associated endomorphism to each symmetric bilinear form defines a linear Sp{V ")-isomorphism from the space of symmetric bilinear forms onto the space of skew symmetric endomorphisms on the symplectic vector space V ". D This reduces the classification of symmetric bihnear forms to that of skew symmetric endomorphisms. Propositions 2.30 and 2.33 then yield Corollary 19. If a is a splitting, skew symmetric endomorphism of the symplectic vector space V ", then V " can be represented as the direct sum of orthogonal subspaces, which are invariant under a: rln

= t^oe0(zi'^(A)ez\'^(-A)).

(22)

Here UQ denotes the root subspace for the eigenvalue 0, and the Jordan cells A''{X), Xy^O, are totally isotropic subspaces. According to the multiplicity of the cells Z\'^(A) of a, these subspaces may occur repeatedly. Moreover, they are combined pairwise forming neutral subspaces. • For the sake of simplicity we will mainly consider non-degenerate bilinear forms and quadrics in the sequel. Then UQ is trivial. We remark that the classification of nilpotent skew symmetric endomorphisms in a symplectic vector space whose only eigenvalue is A = 0, i.e. whose characteristic polynomial is Xa{x) = x^" (cf. Definition 1.5.8.2), presents quite some difficulty. We refer the reader to the literature already cited, [75] and [56]. Since the direct summands Z\''(A) © Z\''(—A) in (22) are orthogonal to one another, they again are symplectic vector spaces; hence it suffices to consider each summand separately. So let k = n, and take V'^'' = A''{X) © A''{-X) to be the Jordan decomposition for the skew symmetric operator a. Since a leaves the Jordan cell Z\'^(A) invariant, and its matrix with respect to a suitable basis has the form of a Jordan box, according to Proposition 2.34 there is only one extension of a as

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2 Cayley-Klein Geometries

a skew symmetric operator on V .In the basis constructed during the proof of this proposition it has the matrix K-) = ^ ' ( A ) := (^0^^^ _zi^(A)') ' ^^*^'

(23)

where now Z\''(A) also denotes the Jordan box (2.58) (cf. (21)). Now, by its very definition this basis is symplectic, and hence: Proposition 20. Two skew symmetric, splitting endomorphisms of a symplectic vector space whose determinants are different from zero are symplectically congruent if and only if, up to the order of the summands, they have the same Jordan decomposition; similarly, any two symmetric bilinear forms the associated endomorphisms of which have the stated property are symplectically congruent. • To prove this, apply Lemma 2.22 to the symplectic transformations between the summands defined as above using symplectic bases, which combine to form a symplectic transformation of the whole space. The matrix for the corresponding symmetric bilinear form on a single summand is obtained from (2.49):

B\X)=[^J^^y^'^^^^,X^Q.

(24)

Formulas (23) and (24), apphed to each direct summand of the decomposition (22) (with t/o = {o}) are canonical representations for the corresponding objects; in fact, they determine a complete system of invariants. In particular, as is well-known, each endomorphism of a complex vector space is splitting, so that by Proposition 20 the classification of the non-degenerate symmetric bilinear forms is complete in the case if = C. In this case we even have more generally Proposition 21. Two skew symmetric or symmetric endomorphisms, or two symplectic transformations of a complex symplectic vector space are symplectically congruent if and only if, up to the order of their summands, they have the same Jordan decomposition. • For the proof we refer to A. I. Maltzev [75], §VII.l, Section 100. In the next example we discuss the classification of skew symmetric endomorphisms in a four-dimensional complex symplectic vector space. Example 8. Let a G End(V ) be a skew symmetric endomorphism of a complex symplectic vector space. Then there are only the possibilities specified below: A. The endomorphism a is non-degenerate. 1. a is diagonalizahle. Then there is a symplectic basis in which the matrix

2.8 Projective Symplectic Geometry of a has the form A^{X) © A^{i^), elements this amounts to

i<)

377

A, /x 7^ 0; in a suitable labehing of the basis

/A 0 0

0

0 jj, 0

0

0 0 -A \0 0 0

0

\ X,l^y^O. -fij

The matrix of the corresponding symmetric bilinear form is

fo 0 xo\ (hj) =

0 0 0 jj,

A0 0 0 \0 fiO Oj

,

X,^i^O.

2. a is not diagonalizable. Then there is a symplectic basis in which the matrix of a has the form (23) with k = 2:

(«}) =

/A 1 0 \0

0 0 0 \ A 0 0 , A^O. 0 -A - 1 0 0 -Xj

The matrix of the corresponding symmetric bilinear form is

ih

/O 0 A l \ 0 0 0A AO 0 0 \1 A0 0/

Xy^O.

Hence according to Corollary (19) and Proposition 20 we described normal forms for all the non-degenerate skew symmetric endomorphisms in dimension 4. B . Suppose that the endomorphism a is degenerate. 1. a is diagonalizable. In this case, normal forms are obtained by specializing A . l ; for A = /i = 0 we have the zero matrix, and for A 7^ 0, /x = 0 it is a skew symmetric endomorphism of rank 2. 2 . a is not diagonalizable and has a non-zero eigenvalue X. Then

(aM

/AO 0 0 0 0 \0 1

0 0\ 0 0 -AO 0 0/

A^O,

is a normal form for endomorphisms of this kind; the J o r d a n box for the eigenvalue zero has order 2. Interchanging the second and third rows as well as the corresponding columns in the matrix above one obtains the J o r d a n normal form of a. This example shows t h a t the J o r d a n cells for the eigenvalue

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2 Cayley-Klein Geometries

zero are not necessarily isotropic and do not always occur in pairs. The matrix of the corresponding symmetric bilinear form is fO 0 0-1 (hj) = A 0 \0 0

A0\ 00 0 0 0 0/

X^O.

These matrices have rank three. 3 . a is not diagonalizable and does not have any non-zero eigenvalue; it has precisely one Jordan cell of order 2. T h e normal form is then constructed by setting A = 0 in Case B.2 just described. 4. a is not diagonalizable and does not have any non-zero eigenvalue; it decomposes into two Jordan cells of order 2.

/oooo\ (aM

000 1 1000

\ooooy is a normal form for endomorphisms of this kind. T h e corresponding bilinear form is diagonal of rank 2, its defect subspace is isotropic:

(-\ ooo\ (hj) =

0 000 0 000 \ 0 00 1/

5. a, is not diagonalizable and has a Jordan cell of order 4 with the eigenvalue zero. The J o r d a n normal form of the endomorphism defined by the matrix below in a symplectic basis has the stated form:

(aM

/ 0 1 0 0 -10 \ 0 0

0 0\ 0 0 0 0 -10/

The corresponding symmetric bilinear form has the matrix

/I000\ 00 10 (hj) = 0 100 \0000/ The following proposition (cf. A. I. Maltzev [75], §VII.3, Section 106) implies t h a t also for the eigenvalue zero J o r d a n cells of dimension 3 do not occur: D

2.8 Projective Symplectic Geometry

379

P r o p o s i t i o n 22. Let a be a skew symmetric, nilpotent endomorphism of the complex symplectic vector space V " . Then each of its Jordan cells with odd dimension occurs with even multiplicity. Jordan cells of even dimension k < 2n, however, may occur with arbitrary multiplicity. P r o o f . Let m be the maximal dimension of a J o r d a n cell occurring in the J o r d a n decomposition of a. If TO = 1, then a is the zero endomorphism, and the assertion is trivial. In all other cases a™^^ is different from the zero endomorphism; we will distinguish two possibilities: Case A. There is a vector b e V satisfying a™^^(b) 7^ o as well as (a™^^(b),b) = 0, and Case B: This does not hold. If, in particular, TO is even, then the endomorphism a™^^ is symmetric, and hence the corresponding bihnear form 6m_i(j;, t)) := (a™^^(j;), t)) is skew symmetric of the same positive rank as a™^^; so we are in Case A. In I. M. Jaglom [56] there is an ingenious proof for the following lemma: L e m m a 2 3 . Under the assumptions of Proposition 22 suppose, moreover, that a Jordan cell Z\™ of a with maximal dimension m satisfies the conditions characterizing Case A. Then there exists a further Jordan cell Z\™ of the same dimension such that the subspace Z\™ © Z\™ spanned by both is a symplectic subspace of dimension 2m. D More precisely, the proof even shows t h a t the J o r d a n cells may be chosen to be totaUy isotropic subspaces; the decomposition Z\™ © Z\™ has a symplectic basis in which the restriction of the endomorphism to this subspace is described by a matrix of the form (23) with A = 0 and k = m. Since the subspace is symplectic and invariant under a, the same also holds for its orthogonal complement. So the argument applies here as well. After a finite number of steps we either arrive at the result, or Case B has to occur. Then TO- = 2/ is even, and there can only be a single J o r d a n cell of dimension TO. In fact, if there were two of these J o r d a n ceUs with generating vectors Ci, C2 for which by assumption C i : = ( a ™ - i ( C i )