MORSE T H E O R Y BY
J. Milnor Based on lecture notes by M. SPIVAK and R. WELLS
PRINCETON, NEW JERSEY PRINCETON UNIVERSITY PRESS 1963
CONTENTS
Copyright 0 1963. by Princeton University Press All Rights Reserved L C Card 63-13729
PART I .
NON-DEGENERATE SMOOTH FUNCTIONS ON A MANIFOLD
. .
. $2.
. . . . . . . . . . . . . . . . . . . . . . D e f i n i t i o n s and Lemmas . . . . . . . . . . . . . . . . . . $3 . Homotopy Type i n Terms of C r i t i c a l Values . . . . . . . . $ 4 . Examples . . . . . . . . . . . . . . . . . . . . . . . . . $5 . The Morse I n e q u a l i t i e s . . . . . . . . . . . . . . . . . .
$1
I n t r o d u c t i o n.
0 6 . Manifolds i n Euclidean Space: Non-degenerate Functions . $7
.
PART I1
.
The k f s c h e t z Theorem on
1
4 12
25 28
The Existence of
. . . . . . . . . . . . . . . . Hyperplane Sections . . . . . . .
12
39
A WID COURSE I N RIEZ"NIAN GEOKETRY
.
Covariant D i f f e r e n t i a t i o n
.
The Curvature
$8 $9
$10
. Geodesics
. . . . . . . . . . . . . Tensor . . . . . . . . . . . . . . . .
and Completeness .
43 51
. . . . . . . . . . . .
55
.
PART I11 THE CALCULUS OF VARIATIONS APPLIED TO GEODESICS
. §12 . $11
$13
.
. 515 .
$14
Printed in the United States of America
. . . . . . . . . . . Path . . . . . . . . . . . . . . . . . . .
The Path Space of a Smooth Manifold
67
The Energy of a
70
The Hessian of t h e Energy Function a t a C r i t i c a l Path Jacobi F i e l d s :
The Index Theorem
. .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Null-space of
E*++ .
§16
.
A F i n i t e Dimensional Approximation t o
817
.
The Topology
$18
.
EXistence of
§I9
.
Some R e l a t i o n s Between Topology and Curvature
74
77 a3
. . . . . . . . aa of the Pull Path Space . . . . . . . . . . . 93 Non-conjugate Points . . . . . . . . . . . . 9a
V
QC
......
loo
CONTENTS PART IV. APPLICATIONS TO LIE GROUPS $20.
$21. $22.
$23. $24.
APPmIX.
I
IC SPACES
. . . . . . . . . . . . . . . . . . . . Symmetric Spaces . . . . . . . . . . . . Lie Groups a s Symmetric Spaces Whole Manifolds of Minimal Geodesics . . . . . . . . . The Bott P e r i o d i c i t y Theorem f o r t h e Unitary Group . . The P e r j o d i c i t y Theorem f o r t h e Orthogonal Group . . .
THE HGMOTOPY TYPE OF A MONOTONE UNION
109
. . . .
. . . . . . . .
112
118 124
13’1
PART I
149
NON-DEGENERATE SMOOTH FUNCTIONS ON A MANIFOLD.
$1.
Introduction.
I n this s e c t i o n we w i l l i l l u s t r a t e by a s p e c i f i c example t h e s i t u a t i o n that we w i l l i n v e s t i g a t e l a t e r for a r b i t r a r y manifolds. sider a torus
M,
tangent t o t h e plane
Diagram
Let above t h e
f(x)
5
a.
V
f:
M+R
Let u s conas i n d i c a t e d i n D i a g r a m 1 .
V,
1.
(R always denotes t h e r e a l numbers) be t h e height I@‘ be the s e t of a l l p o i n t s x E M such that
plane, and l e t
Then the following t h i n g s a r e t r u e : (1)
If
a
<
< f ( p ) , then M& i s < a < f ( q ) , then I@’ < a < f ( r ) , then @
0
(2)
If
f(P)
(3)
If
f(q)
(4)
If f ( r ) < a
vi
<
w
f ( s ) , then M” manifold of genus one having
6.
vacuous.
i s homeomorphic t o a 2- cell. i s homeomorphic t o a cylinder:
is homeomorphic t o a compact c i r c l e as boundary:
INTRODUCTION
§I.
NON-DEGENERATE FUNCTIONS
I.
2
3
The boundary
i.k (5)
If
f(s)
<
a,
then
M&
willbe denoted by
I n order t o describe t h e change i n Mi" a s a passes through one it is convenient to consider homotopy of t h e p o i n t s f ( p ) , f ( q ) , f ( r ) , f ( s ) I n terms of homotopy types: type r a t h e r than homeomorphism type. For as f a r as i s t h e operation of a t t a c h i n g a 0 - c e l l . (1) + (2) homotopy type i s concerned, the space
Ma,
f(p)
<
a
<
Here
"J"
+
g)
ca1 sum ( = d i s j o i n t union) of
Y
Sk-'
E
with
g(x)
6'
p o i n t and l e t
G3
Y.
E
S-l
=
j u s t the union of
is the operation of a t t a c h i n g a
(3)
E
Rk
g: Sk-'
( y with a k - c e l l attached by
x
f ( q ) , cannot be d i s -
: ((x(= ( 11
-+
Y
i s a continuous map then
Y
i s obtained by f i r s t t a k i n g t h e topologiand
ek,
and then i d e n t i f y i n g each
To take c a r e of the case
be vacuous, s o t h a t
M&
topy type of
p,q,r
1- cell:
(x,y)
both zero. that
n
x
2
-
f Y
2
p
constant
=
let
eo
and
a t which t h e homo-
s
we can choose -x
2
- y
2
,
s o that
(x,y)
and a t
q
and
r
f =
ar
and
x2 + y2 , a t
s o that
f
=
W (3)
(4)
-+
+
(5)
Let
Y
a
<
f(point)
<
b.
so
constant + f
Ma
at to
O u r f i r s t theorems w i l l g e n e r a l i z e these
i s a g a i n the operation of a t t a c h i n g a 1 - c e l l : REFEFENCES
For f u r t h e r information on Morse Theory, t h e following sources are extremely u s e f u l . M. Morse,
H. S e i f e r t and W. T h r e l f a l l ,
R. Bott,
(x
c~~
The s t a b l e homotopy of t h e c l a s s i c a l
R . Bott,
<
Annals of
Morse Theory and i t s a p p l i c a t i o n t o homotopy theory,
Lecture n o t e s by A. van de Ven (mimeogpaphed), University of
: \(XI\ 1 )
Bonn, 1960.
be t h e k - c e l l c o n s i s t i n g of a l l v e c t o r s i n Euclidean k-space w i t h l e n g t h
~J-OUPS,
Mathematics, Vol. 70 (19591, pp. 313-337.
be any t o p o l o g i c a l space, and l e t =
"Varlationsrechnung i m Grossen,"
published i n t h e United S t a t e s by Chelsea, New York, 1951.
is t h e operation of a t t a c h i n g a 2 - c e l l .
ek
"The c a l c u l u s of v a r i a t i o n s i n t h e large,'' American
Mathematical Society, New York, 1 9 3 4 .
The p r e c i s e d e f i n i t i o n of " a t t a c h i n g a k- cell" can be given as follows.
s
f a c t s f o r any d i f f e r e n t i a b l e f u n c t i o n on a manifold.
0 (4)
are
Note that the number of minus s i g n s I n the expression for
where
f.
If we choose any coordinate
each p o i n t is t h e dimension of t h e c e l l we must a t t a c h t o go from
Mb,
be a
with a 0 - c e l l attached i s
near these p o i n t s , then t h e d e r i v a t i v e s
At
.
o
=
changes, have a simple c h a r a c t e r i z a t i o n i n terms of
They a r e t h e c r i t i c a l p o i n t s of t h e function. system
Y
k
and a d i s j o i n t p o i n t .
A s one might expect, t h e p o i n t s
means "is of the same homotopy type a s . " (2)
(x
Y vg ek
tinguished from a 0 - c e l l :
-
. If
Sk-'
is t h e f u l l t o r u s .
=
5
1
I.
NON-DEGENERATE FUNCTIONS
since
has
f
The words "smooth" and " d i f f e r e n t i a b l e " w i l l be used interchange-
manifold
a t a point
M
smooth map w i t h
g(p)
point
p
T%
f,: 1
(x
q,
f
w i l l be denoted by
+TRAP)
,...,xn)
T%.
If
g: M + N
is a
-+
=
c
bj
(x' , . . . ,xn)
If
p$I-
ij
=
c b. a J
M.
A
s o t h e matrix
U
of
p
(i a2f -( p ) )
f
f(p)
If
a
the s e t of a l l p o i n t s
i s not a c r i t i c a l value of
function theorem t h a t f-'(a)
r e p r e s e n t s the b i l i n e a r f u n c t i o n
t o r space such that
i s a smocth submanifold of
p
a.
f(x)
then i t follows from the i m p l i c i t
f
on which
f,,
H
M.
i s c a l l e d non-degenerate i f and only i f the
f o r every
on
f
index of
f,,
The
of a b i l i n e a r f u n c t i o n a l
the subspace c o n s i s t i n g of a l l The p o i n t
if and only if
on
p
v
p f,,
V
E
such that
H(v,w)
f,,
on
T%
has n u l l i t y equal t o
0.
f
at
p
I t can be checked d i r e c t l y that non-degeneracy does not T h i s w i l l follow a l s o from the following
LEMMA 2 . 1 . borhood V
i s a c r i t i c a l p o i n t of f we d e f i n e a symmetric b i l i n e a r on TMp, c a l l e d t h e Hessian of f a t p . I f v,W E TMp
Let of
f
f(Xl,.
be a Cm f u n c t i o n i n a convex neighi n Rn, with f ( 0 ) = 0 . Then
. .,xn)
=
i=l
then
v
f,,(v,w)
and =
w
have extensions
v" and
t o vector f i e l d s .
for some s u i t a b l e
We l e t *
v"P ( w " ( f ) ) , whe1.e v"P i s , of course, j u s t v .
this i s symmetric and well-defined.
We must show that I t i s symmetric because
v"P ( i j ( f ) ) - $ p ( v ( f ) ) where
$
[?,%I
i s t h e Poisson bracket of
PROOF : =
[v",?lp(f)
v" and %,
= 0
and where [?,$lp(f)
= 0
~
* Here G ( f )
denotes t h e d i r e c t i o n a l d e r i v a t i v e of
f
i n the direction
3.
=
o
The
f
Before s t a t i n g this lemma we f i r s t prove the
0
s-
Cm f u n c t i o n s
p.
can be completely
following:
intrinsic definition. If
on a vec-
w i l l be r e f e r r e d t o simply a s t h e index of
T%
described by this index.
functional
H,
i s obviously a non-degenerate c r i t i c a l
The Lemma of Morse shows t h a t t h e behaviour of
depend on t h e coordinate system.
t h e n u l l i t y of t h e b i l i n e a r
i s negative d e f i n i t e ; the n u l l i t y i s the dimension of the
w E V.
Point of
TMp.
index and
i s defined t o be t h e maximal dimension of a subspace of V
V,
space, i . e . ,
The boundary
matrix
i s non-singular.
with
f,,
a
r e s p e c t t o the b a s i s
f.
x E M
I@i s a smooth manifold-with-boundary.
A c r i t i c a l point
P'
now denotes a con-
this means that
i s c a l l e d a c r i t i c a l value of
M"
axj
bj
-1ax:a
v = C ai
axJ
ax
functional
We denote by
where
while
Then
We can now t a l k about t h e
The r e a l number
v" of v,
i s a l o c a l coordinate system and
we can t a k e
stant function.
i f t h e induced map I f we choose a l o c a l coordinate system
i n a neighborhood
It i s now c l e a r l y well-defined since
TNq.
be a smooth r e a l valued f u n c t i o n on a manifold
i s zero.
w
then t h e induced l i n e a r map of tangcnt spaces
i s c a l l e d a c r i t i c a l p o i n t of
M
E
=
& : TMp
w i l l be denoted by
Now l e t
p
I
The tangent space of a smooth
Cm.
i s symmetric.
f,,
$ (w"(f)) = v(w"(f)) i s independent of t h e extension P Gn(?(f)) i s independent of w".
D e f i n i t i o n s and Lemmas.
ably t o mean d i f f e r e n t i a b l e of c l a s s
5
as a c r i t i c a l point.
p
Therefore $2.
DEFINITIONS AND LEMMAS
§2.
gi defined i n V,
with
6
I.
mm
Therefore, applying 2 . 1 t o t h e g j
(Lemma of Morse). L e t p be a non-degenerate c r i t i c a l p o i n t f o r f. Then t h e r e i s a l o c a l coordinate 1 system ( 7 ,.. .,yn) i n a neighborhood U of p with i y ( p ) = 0 for a l l i and such that t h e i d e n t i t y 2.2
1 2
...
f(p) - (y ) -
f =
holds throughout
- (yX)* + (yX+l)* +
where
U,
...
g j ( x l ,*
f at
x
then
p.
(z’,
...,z n ) , f(q)
f
at
p.
It follows that
hij.
...,xn)
t
=
. . .,xn) .
xixjhij ( x , ,
i,j = 1
f,
We f i r s t show that i f t h e r e i s any such expression f o r
must be t h e index of
we have
,xn) =
f o r c e r t a i n smooth f u n c t i o n s f(xl,
PROOF:
a .
We can assume that
For any coordinate system
Kij
and then have
=
Eji
hij
and
f
f(p)
-
... -
(z’(q))2-
... +
( z h ( q ) ) 2 + ( z X + ’ ( q ) ) 2+
i s equal t o
(z”(q)I2
( 1 a2f
(o)),
= Z X X F Ii j
l(h 2 ij+ hji)J Moreover t h e matrix (li ( 0 ) )
.
i j
and hence i s non-singular.
There i s a non-singular transformation o f t h e coordinate functions ( - 2 2
aZi azJ
if
i = j l h ,
if
i = j
otherwise
0
>
which gives us t h e d e s i r e d expression f o r
,
h
borhood of
,
0.
f,
i n a perhaps smaller neigh-
To see this we j u s t i m i t a t e the u s u a l d i a g o n a l i z a t i o n proof
f o r quadratic forms.
with r e s p e c t t o t h e b a s i s
f,,
which shows that the matrix r e p r e s e n t i n g
(See f o r example, BirkhoffandMacLane, “A survey of
modern a l g e b r a , “ p. 271 .)
The key s t e p can be described as follows.
u l , ...,un
Suppose by induction that t h e r e e x i s t coordinates
a neighborhood
2*.
\
\
Therefore t h e r e i s a subspace of t i v e d e f i n i t e , and a subspace
on which
V
T%
.
throughout
of dimension T%
n-A
where
X
where
i s nega-
f,,
f,,
i s positive
of dimension g r e a t e r than
Therefore
X
i s t h e index of
p
i s t h e o r i g i n of Rn
By 2.1 we can w r i t e
+
(UrJ2
and that
where t h e m a t r i c e s
V,
1
..
UiUjHij (5, * 9 % ) i,jIr (%j(ul,...,un)) a r e symmetric. After
a l i n e a r change i n the last Let
g ( u l , ...,%)
borhood
U, C U,
vi = ui
( y , . . . , y n,
We now show that a s u i t a b l e coordinate system Obviously we can assume that
... 5
n-r+l coordinates we may assume t h a t Krr(o) f denote t h e square r o o t of ( q r ( u l , un) 1 . T h i s w i l l
...,
be a smooth, non-zero f u n c t i o n of
X
f,,.
1
U,;
of
0.
ul,.
. .,un
throughout some smaller neigh-
Now introduce new coordinates
v1,
...,vn
for i f r
f(p)
=
f(0) =
-n
It follows from t h e i n v e r s e f u n c t i o n theorem that
It is e a s i l y v e r i f i e d that (xl,.
..,xn)
i n some neighborhood of
c r i t i c a l point: gj(o)
=
0.
af -p) ax
Since = 0
.
0
by
exists.
vl,
...,vn
w i l l serve as
coordinate f u n c t i o n s w i t h i n some s u f f i c i e n t l y small neighborhood
for
in
s o that
0
I
were negative d e f i n i t e then this subspace would i n t e r s e c t
which i s c l e a r l y impossible.
of
U,
f = + (u1)2
I ‘1.
of dimension
If t h e r e were a subspace of
f,,
=
2 Z F - S
then we have
definite.
Eiij
since we can w r i t e
hji,
=
if
=
7
i=l”
+ (y92
i s t h e index of
X
DPINITIONS AND LEMMAS
§2.
NON-DEGENERATE FUNCTIONS
i s assumed t o be a
f
can be expressed as
U3
of
0.
0.
P2.
I.
8
thoughoutu3'
m LEMMAS
9
his completes t h e induction; and proves Lemma 2 . 2 .
COROlL&3y 2 . 3
Non-degenerate c r i t i c a l p o i n t s a r e i s o l a t e d .
Examples of degenerate c r i t i c a l p o i n t s ( f o r f'unctions on R2)
DEFINITIONS
NON-DEGENERATE FUNCTIONS
R
and
a r e given below, together with p i c t u r e s of t h e i r graphs.
7 (a) f ( x )
=
(d)
f(x,y) = x2.
The set of c r i t i c a l p o i n t s , a l l of which
a r e degenerate, i s the
(b)
x3. The o r i g i n
i s a degenerate c r i t i c a l p o i n t .
F(x)
= e-'ix2sin2(1/X)
x axis,
which i s a sub-manifold of R 2 .
.
The o r i g i n i s a degenerate, and non- isolated, c r i t i c a l p o i n t .
(e)
f(x,y) = x2y2.
The set of c r i t i c a l p o i n t s , a l l of which are
degenerate, c o n s i s t s of the Union of the n o t even a sub-manifold of
x and g a x i s , which i s
R2.
W e conclude this s e c t i o n with a d i s c u s s i o n of 1-parameter groups of diffecpnorphisms.
The reader i s r e f e r r e d t o K. Nomizu,"Lie Groups and D i f f e r -
e n t i a l Oeometrg," f o r more d e t a i l s . A 1-parameter group
-
R e a l p a r t of
(x + i ~ ) ~ .
f(x,y)
=
(0.0)
i s a degenerate c r i t i c a l p o i n t ( a "monkey saddle").
x3
3xy2
=
of diffeomorphisms of a manifold
M
is a
C"
Thus for each p o i n t of
such that 1)
t
for each cpt(q)
=
t,s
E
on
X
M
cpt:
M
+
M
R we have
cpt+s
= cpt
number
defined by
i s a diffeomorphism of
a 1-parameter p o u p
Given
a vector f i e l d
R the map
E
cp(t,q)
for a l l
2)
f
DEFINITIONS AND LEMMAS
92.
NON-DEGENERATE FUNCTIONS
I.
10
0
M
t h e r e e x i s t s a neighborhood
M
dTt(9)
-=x The compact s e t
numbers
E.
U. Let Setting
6
i s s a i d t o generate the group
q
cp,
E
>
E~
cpt(q)
PROOF:
for q
q
-
t
c(t)
M
= k(E0/2)
dc
by t h e i d e n t i t y
dc =(f)
=
't
h .
k
~
2
= 'E0/2
O
'pE0/2
.
only necessary t o r e p l a c e (Compare $ 8 . )
Now l e t
cp
cp
be a 1-parameter group of diffeomorphisms, generated by t h e v e c t o r f i e l d
X.
i t follows that this d i f f e r e n -
K,
qt(q)
i s defined for a l l v a l u e s of
Eo/2
t.
It1
for
cpt
q
t h e curve
completes the proof of Lemma
t
+
RFMARK: cannot be omitted.
-a€--= Xcpt(q) '
where
p
=
cpt(q).
cpo(q)
=
q.
and l e t
But i t i s w e l l known that such a d i f f e r e n t i a l equation,
l o c a l l y , has a unique s o l u t i o n which depends smoothly on the i n i t i a l condi(Compare Graves, "The Theory of Functions of Real V a r i a b l e s i ' p . 166. ,un, t h e d i f f e r e n t i a l equaNote that, i n terms of l o c a l coordinates u' , tion.
i
2
E
for a l l
and
Further-
l t ( , 1 s ) , l t + s 1< c O . . Any
~
r
t
number
with
<
Irl
can
.
c0/2
set '
*.*
'E0/2
i s iterated by
'p-E0/2
' 'r
'E0/2
k
times.
iterated
If -k
k
<
it i s
0
times.
Thus
It i s not d i f f i c u l t t o v e r i f y that = cpt
cpt+s
cp,
,
cpt
is
qt
This
2.4
&
. ..
=
xi(,'
,...,un),
X
The hypothesis that
X
vanishes outside of a compact s e t
For example l e t M be t h e open u n i t i n t e r v a l
be the standard v e c t o r f i e l d
d
on
M.
generate any 1-parameter group of diffeomorphisms of
T h i s i s t r u e since
t i o n t a k e s on t h e more f a m i l i a r form:
E~
cpt(4)
s a t i s f i e s t h e d i f f e r e n t i a l equation dcpt(q)
with i n i t i a l condition
<
p/ l u 2s a remainder
w e l l defined, smooth, and s a t i s f i e s the condition Then f o r each f i x e d
It(
for
providing that
cp,
0
~
0,
where the transformation
TMc(t)
lim f ( t + h ) - f ( t )
h-r
with
+ r
i t i s convenient t o d e f i n e t h e v e l o c i t y v e c t o r
aT
cpt
=
be expressed as a m u l t i p l e of E
6
i s a diffeomorphism.
It only remains t o d e f i n e
If t
E.
denote t h e smallest of t h e corresponding
0
=
cpt+s
mt
Therefore each such
Given any smooth curve
<
q
T h i s s o l u t i o n i s smooth as a f u n c t i o n of b o t h v a r i a b l e s .
M.
more, i t i s c l e a r t h a t LEMMA 2 . 4 . A smooth v e c t o r f i e l d on M which vanishes outside of a compact s e t K C M generates a unique 1 parameter group of diffeomorphisms of M.
It1
U,
=
K can be covered by a f i n i t e number of such
t i a l equation has a unique s o l u t i o n X
rpo(q)
cpt(9)'
has a unique smooth s o l u t i o n for q neighborhoods
let
T h i s vector f i e l d
and a
U
s o that t h e d i f f e r e n t i a l equation
0
cp,
For every smooth r e a l valued f u n c t i o n
as follows.
>
onto i t s e l f ,
of diffeomorphisms of M we define
cp
E
11
i
= 1
,...,n . )
Then
M.
X
(0,l)
does not
C R,
I.
12
NON-DEGENERATE FUNCTIONS
HOMOTQPY TYPE
$3. C:
R
+
$
i s a curve w i t h v e l o c i t y vector
M
13
note t h e i d e n t i t y
.
< dc E , g r a d f >d (=f o cT) $3.
Homotopy Type i n Terms of C r i t i c a l Values.
Throughout t h i s s e c t i o n , i f manifold
Let
i s a r e a l valued f u n c t i o n on a
f
M&
.
= f-’(- m , a l
= (p E M : f(p)
5 a)
.
1 / < g r a d f , grad f>
be a smooth f u n c t i o n which i s equal t o
throughout thecompact s e t f - l [ a , b ] ; and which vanishes Then t h e v e c t o r f i e l d
X,
defined by
f be a smooth r e a l valued f u n c t i o n on a manifold M. Let a < b and suppose that t h e s e t f - l [ a , b l , c o n s i s t i n g of a l l p E M w i t h a L f ( p ) L b , i s compact, and contains no c r i t i c a l p o i n t s of f. Then M& i s diffeomorphic t o Mb. Furthermore, i s a deformation r e t r a c t of Mb, so t h a t t h e i n c l u s i o n map Ma Mb i s a homotopy equivalence.
THEOREM 3 . 1
M+R
outside of a compact neighborhood of this s e t .
we l e t
M,
p:
Xq
Let
=
p ( q ) (grad
s a t i s f i e s the conditions of Lema 2 . 4 .
Hence
f)s
X
generates a 1-parameter
group of diffeomorphisms
vt: M For fixed
q E M
+
M.
consider the f u n c t i o n
t + f(cpt(q)).
If Tt(q)
-+
l i e s i n the s e t The i d e a of t h e proof is t o push n a l t r a j e c t o r i e s of t h e hypersurfaces
Mb
M&
down t o
f = constant.
f-’[a,bl,
then
<-,
along the orthogo-
. d f ( Vd t ( q )) t = dcpt(q) grad f >
(Compare D i a g r a m 2 . )
=
<X,
grad f >
=
+ 1.
Thus t h e correspondence
t
\
i s l i n e a r with d e r i v a t i v e
-*
f(vt(9))
as long as
+1
f ( c p t ( q ) ) l i e s between
Now consider the diffeomorphism
M&
diffeomorphically onto
Mb.
M + M.
\
I I
I
I
I
I
D i a g r a m 2.
I
Y
r t : M~
4
~b
and
b.
Clearly this c a r r i e s
T h i s proves the f i r s t half of
Define a 1-parameter family of maps
a
7.1.
I.
14
NON-DEGENERATE FUNCTIONS
HOMOTOPY TYPE
§ 3.
Choosing a s u i t a b l e c e l l in@;
e x C H,
15
a d i r e c t argument
( i . e . , push-
i n along the h o r i z o n t a l l i n e s ) w i l l show t h a t
M C - E ~ ex i s a deformation r e t r a c t of M ~ u- H.~ F i n a l l y , by applying 3 . 1 t o the f u n c t i o n F and the region F-1 [ C - E , C + E I we w i l l see that M ~ u- H~ i s a deformation r e t r a c t
of MC+& . T h i s w i l l complete t h e proof. Choose a coordinate system 90
=
holds throughout
c -
(U1l2-
...
Choose
>
E
(UX+l)2,
=
...
=
u”(p)
p
...
+
The region
p o i n t s other than
f-’[ c - E
MC-E
i s heavily shaded.
= f - l (-m,C-E
= 0
.
,C+E
1
i s compact and contains no c r i t i c a l
The image of
U
under the diffeomorphic
. . . ,un) :
imbedding
U -Rn
“u Now d e f i n e
We w i l l introduce a new f u n c t i o n
coincides w i t h t h e h e i g h t f u n c t i o n
f
except that
F
F: M
<
f
-+
R which
i n a s m a l l neighMC -E
borhood of
p.
Thus t h e r e g i o n
gether w i t h a region
H
near
p.
contains the closed b a l l .
1
F-’ ( - m , c - ~ l w i l l c o n s i s t of I n D i a g r a m 4,
H
to-
ek
1
,..., u n ) : c
( u ’ ) ~ +. . . + ( u ‘ ) ~ 5
E
and
.
( u i ) 2 5 2El
t o be t h e s e t of p o i n t s i n
U with
u’+~=
...
=
un
= 0.
The r e s u l t i n g s i t u a t i o n i s i l l u s t r a t e d schematically i n D i a g r a m 5 .
i s the h o r i z o n t a l l y
shaded region.
fr
D i a g r a m 4.
p
w i l l have coordinates
p.
( u1 , The region
of
(Un)2
The i d e a of t h e proof of this theorem i s i n d i c a t e d i n D i a g r a m 4, f o r the s p e c i a l case of the height f u n c t i o n on a t o r u s .
U
s u f f i c i e n t l y s m a l l s o that
0
(1 )
(2)
- (uA)2 +
Thus t h e c r i t i c a l p o i n t
U.
u1 ( p ) THEOREM 3 . 2 . L e t f : M + R be a smooth function, and l e t p be a non-degenerate c r i t i c a l p o i n t w i t h index A . Sett i n g f ( p ) = c , suppose that f - ’ [ c - E , c + E ~ i s compact, and contains no c r i t i c a l p o i n t of f other then p, f o r some E > 0 . Then, f o r a l l s u f f i c i e n t l y s m a l l E , t h e s e t MC+E has t h e homotopy type of MCFE with a k - c e l l a t t a c h e d .
i n a neighborhood
that the i d e n t i t y f
Diagram 3 .
u 1 ,..., un
c’
f
C+E
I Diagram 5.
f . C i E
‘f.c
16
u1
=
...
ux
=
and G;
b a l l of r a d i u s
.
and f - ' ( c + E )
respectively;
= 0
un
= 0
F
and
i s h e a v i l y shaded; t h e region
MC-€
z o n t a l dark l i n e through
p
f-' ( c - E )
Within this e l l i p s o i d we have
x
We must prove t h a t
formation r e t r a c t of M ' + ~ . Construct a new smooth f u n c t i o n
F: M
-
R
MCeE
u
The h o r i -
PROOF:
C+E
a r e t h e same as those of
F
f.
Note that
so that e x e x i s a de-
as follows.
c-c+q < c+ ls+q 2 -<
The c r i t i c a l p o i n t s of
ASSERTION 2.
aF
&',
i s p r e c i s e l y t h e boundary
a s required.
f =
T h i s completes t h e proof.
.
e
5
F
f-l [c-E,c]
i s l i g h t l y dotted.
f-'[c,c+el
represents the c e l l
e h n MC-E
coincide.
the c i r c l e r e p r e s e n t s t h e boundary of t h e
i s heavily d o t t e d ; and t h e region
i s attached t o MC-E
=
t h e hyperbolas r e p r e s e n t the hypersurfaces
The r e g i o n
Note that
. ..
ux+l=
The coordinate l i n e s r e p r e s e n t the planes
HOMOTOPY TYPE
13.
NON-DEGENERATE FUNCTIONS
I.
=
-
-1
Llf(E+2q)
<
0
Since
Let
p:R-R where t h e covectors f u n c t i o n s a t i s f y i n g the conditions
be a C"
I
U(0)
>
E
u(r)
=
o
<
-1
where
.
~ ' ( r =)
neighborhood
U,
F
for
pI(r)
r 2 2~ o for a l l
F coincide w i t h
w i t h i n t h i s coordinate neighborhood.
f
o u t s i d e of the coordinate
Therefore
F
is a
Hence
q = ( uX + 1 ) 2
f
=
-
c
5 + q;
F(q) for a l l
q
E
c
-
f-l
.
[C-E,C+EI
It can contain no c r i t i c a l p o i n t s of
1.
= C
-
P(0)
<
C
-
.
E
contains no c r i t i c a l p o i n t s .
ASSERTION 3.
Together w i t h 3 . 1 this
The r e g i o n
F-l
(-w,c-E]
i s a deformation r e t r a c t of
It w i l l be convenient t o denote this region (u52 p
+
E
u H;
where
H denotes t h e closure of
F-' (-m,c-E 1 by
F-'(-m,C-El
-
.
(Unl2
REMARK: described as
S(q) + q(q) - P(C(q) + 211(9))
The region
MC-E
I n t h e terminology of
Smale, t h e r e g i o n
w i t h a "handle" a t t a c h e d .
that the manifold-with-boundary
F-' ( - m , c + ~ I coincides with t h e r e g i o n
MC-E
u
H
MC-E
5 + 2q
1. 2a
the functions
f
and
u
is
H
I t follows from Theorem
i s diffeomorphic t o
MC+E
.
3.1
This
(Compare 3. h l e , Generalized PoincarB's conjecture i n dimensions g r e a t e r than f o u r , h
~,c+E 1.
Outside of t h e e l l i p s o i d
F
But
f a c t i s important i n Smale's theory of d i f f e r e n t i a b l e manifolds.
= f-l(-
PROOF:
c
MC+E
U.
ASSERTION MCfE
... + ...
+
By Assertion 1 together
Proves t h e following.
so that: =
p.
o t h e r than t h e o r i g i n .
F- 1 [ c - E , c + E ] .
this region i s compact.
F-'[c-E,C+EI
[o,m)
(u')2 +
U
we see that
F(p)
M.
by
f
F-'[C-E,C+EI
It i s e a s i l y v e r i f i e d that
u-
6
r,
I t i s convenient t o d e f i n e two functions 5,q:
F
Z?(U'+')~+...+~(U~)~)
w e l l defined smooth f u n c t i o n throughout
Then
are simultaneously zero only a t t h e o r i g i n ,
dq
has no c r i t i c a l p o i n t s i n
with t h e i n e q u a l i t y
except p o s s i b l y
...+( u ' ) ~ +
V ( ( U ' ) ~ +
=
F
and
Now consider t h e region
and l e t
f -
=
Now l e t
it follows that
dk
l
s of Mathematics, V o l . 74 (1961), pp. 391-406.)
18
I.
Now consider t h e c e l l
eh
t(q)
Note that
ex
03. HOMOTOPY TYPE
NON-DEGENERATE FUNCTIONS
c o n s i s t i n g of all p o i n t s
5 E,
q
with
q(9) = 0 .
i s contained i n the "handle"
H.
I n f a c t , since
we have F(q) but
f(q)
2
c-e
for q
E
e
h
5
F(P)
<
C-E
.
D i a g r a m 7.
Thus
rl
ro maps t h e e n t i r e region i n t o
i s the i d e n t i t y and
f a c t t h a t each rt maps e q u a l i t y aF > 0.
ex.
The
F - ' ( - m , c - ~ l i n t o i t s e l f , follows from the i n -
'I
CASE 2 .
Within the region
I.
E
q +
E
let
rt
correspond t o
the transformation ( u1 )...,u")
where the number
st
rl
=
f - l ( C- E) .
remain continuous as with that of Case The region
The p r e s e n t s i t u a t i o n i s i l l u s t r a t e d i n D i a g r a m 6 . MC-"
i s heavily shaded;
and t h e region
t h e handle
F-l [ C - E , C + E ]
ASSERTION 4 . PROOF:
MC-€
i s shaded with v e r t i c a l arrows;
H
i s a deformation r e t r a c t of
A deformation r e t r a c t i o n
r t : MC-€
u H
-.
MC-€
M~-"
u
H
H. is
More p r e c i s e l y i n d i c a t e d schematically by the v e r t i c a l arrows i n D i a g r a m 6 . l e t rt be t h e i d e n t i t y outside of U; and d e f i n e r t w i t h i n U as f o l lows.
It i n necessary t o d i s t i n g u i s h t h r e e cases as i n d i c a t e d i n Diagram 7. CASE 1 .
Within t h e region
formation (U')
...,un,
-+
(u',.
5
5E
2 .
1
I.
(l-t)((I.-E)/rl)''2
.
ro maps the e n t i r e region i n t o the
The reader should v e r i f y that the functions --c E ,
when
I.
q
-+
stui Note that this d e f i n i t i o n coincides
0.
= E.
Within t h e r e g i o n
be the i d e n t i t y .
, . . . )StU n )
11
+
E
5 ( i . e . , within
MC-€)
let
T h i s coincides with the preceeding d e f i n i t i o n when
E = q + E .
i s dotted.
ex
CASE
rt
t +
i s again t h e i d e n t i t y , and
hypersurface Diagram 6.
(u1 ,...)uA ,stu h + l
i s defined by
[0,11
E
St
Thus
-+
let
rt
correspond t o the t r a n s -
. . , u A , t u h + l , ... ,tun, .
T h i s completes t h e proof that Of
F-l ( - m , C + E I .
Together with
MC-€u ex
i s a deformation r e t r a c t
Assertion 3, i t completes t h e proof of
Theorem 3 . 2 . R E m K 3.3.
More g e n e r a l l y suppose t h a t t h e r e are
...
k
non-degenerate
c r i t i c a l points p , , ...,pk w i t h indices x l , ,xk i n f - ' ( c ) . Then a s i m i l a r proof shows that MC+" has t h e homotopy type of M C - E ~ e x 1 u . . . U e 'k
.
REMARK MC
the s e t
2.4.
(Compare D i a g r a m 8 . )
ex
A
PROOF:
simple modification of t h e proof of 3.2 shows t h a t
i s a l s o a deformation r e t r a c t of
deformation r e t r a c t of
MC-"
93.
I . NON-DEGENERATE FUNCTIONS
20
MCCE.
In fact
MC
k(x)
is a
which i s a deformation r e t r a c t of
F-' (-m,cI,
MC+&.
Combining this f a c t with 3 . 2 we see e a s i l y that
i s a deformation r e t r a c t of
Define
MC.
Here
= 2
k(tu)
=
t
v2-2t(u)
t
compositions
k
kP
and
X
O < t L s 1, for 1 5 t I,
denotes the homotopy between
rpt
E
for
~
and
9,
with t h e unit vector
i s defined by s i m i l a r formulas.
Thus
x
for
k(tu)
Pk
21
by t h e formulas
x
=
product of t h e s c a l a r
I
k
HOMOTOPY TYPE
-A
U
u
E
E
and
'pl;
~
&A
tu
denotes the
u. A corresponding map
It i s now not d i f f i c u l t t o v e r i f y that the
a r e homotopic t o the r e s p e c t i v e i d e n t i t y maps.
i s a homotopy equivalence.
For f u r t h e r d e t a i l s t h e reader i s r e f e r r e d t o , L e m a 5 of J . H. C . Whitehead,
On Simply Connected 4-Dimensional Polyhedra, Commentarii Math.
H e l v e t i c i , Vol. 2 2 ( 1 9 4 9 ) , pp. 48-92.
LEMMA 3 . 7 . Let q : 6 ' - X be a n a t t a c h i n g map. Any extends t o a homotopy homotopy equivalence f : X - Y equivalence F : x u e A + Y uf'P e A
I
.
'P
Diagram 8:
MC
i s heavily shaded, and
F-'[c,c+E]
i s dotted. PROOF:
THEOREM 3 . 5 . If f i s a d i f f e r e n t i a b l e f u n c t i o n on a manifold M w i t h no degenerate c r i t i c a l p o i n t s , and i f each Ma i s compact, then M has the homotopy type of a CW-complex, with one c e l l of dimension A for each c r i t i c a l p o i n t of index A .
g: Y --cX
i
Fie'
=
ex
-x
Since
gfcp
GIY
i s homotopic t o
=
and d e f i n e
f
u eA gf'P g, G ( e A =
9,
(Whitehead) L e t 'po and 'p, be nnotopic maps from t h e sphere 6' t o X. Then t h e i d e n t i t y map of X extends t o a homotopy equivalence k:Xue'+Xue 'PO
'P1
A
u e x --cXu ex gf'P m we w i l l f i r s t prove that t h e composition
.
GF:
x
ex. - X 'P
is homotopic t o t h e i d e n t i t y map.
identity.
i t follows from 3 . 6 that t h e r e i s
k: X
LEMMA 3 . 6 .
F
.
identity
a homotopy equivalence
with a c e l l attached.
Define
f
f'P
bg t h e corresponding conditions
The proof w i l l be based on two l e m a s concerning a t o p o l o g i c a l X
=
G: Y
a t o r i a l Homotopy I , B u l l e t i n of the American Mathematical Society, Vol. 55,
space
FIX
be a homotopy i n v e r s e t o
CW-complex see J . H. C . Whitehead, Combin-
( 1 9 4 9 ) , Pp. 21 3-245.)
P.Hilton.)
by t h e conditions
k t (For t h e d e f i n i t i o n of
(Following an unpublished paper by
.
eA 'P
Let
ht
be a homotopy between k, G, and
s p e c i f i c d e f i n i t i o n s of kGF(x)
=
F,
gf(X)
kGF(tu)
= 2tU
kGF(tu)
=
h2-2tcp(u)
and t h e i d e n t i t y .
gf
Since
G(Fk)
i n v e r s e , i t follows t h a t Step 3 .
for x
E
for'o
5 t L ~ ,u
X, 1
1
Step 2 .
Using the
note t h a t
for
t
1,
HOMOTOPY TYPE
03.
NON-DEGENERATE FUNCTIONS
I.
22
u
a x
E
e
E
e
.h
Since
a l s o , i t follows t h a t
,
2
qT:
x
-
ex cp
ex
F(kG)
h,(x)
=
q,(tu)
=
q,(tu)
=
for
2
x
for
tu
h2-2t+Tcp(u)
kt
, o L t L Tl t T, X
E
5t 5
for
1,
-A e
u u
.x
e
E
,
c
2
The proof that
If
a map
R,
i s a 2-sided
(or L)
PROOF :
has a l e f t homotopy inverse
F
then
2
F
L
and a
E
2
c2
(cil
>
a.
=
<
c3
< ...
tme
identity,
(LF)R"R.
h'
h
0
0
x
a
<
c,.
Suppose
x
u..
e J(') cpj(c)
3.2,
and 3 . 3 ,
f o r c e r t a i n maps
.U
h':
M&
which proves that
R
2
LF
2
identity
,
K
6 j
-
(xj-l)
- skeleton of
u
eh l
The r e l a -
h: MC-E
-
K,
has
. ., c p j ( c )
-
M&.
where
V...u
e j(')
i s a CW-complex, and has t h e same homotopy
by Lemmas 3 . 6 ,
3.7.
If
M
e' has
the homotopy type of a
i s compact this completes the proof.
If
M,
SO
M&
M
i s not com-
e,then a i s a deformation
t h e proof i s a g a i n complete.
I f t h e r e a r e i n f i n i t e l y many c r i t i c a l p o i n t s then t h e above cons t r u c t i o n gives us an i n f i n i t e sequence of homotopy equivalences
tion kGF a s s e r t s that
F
2
Ma
identity
has a l e f t homotopy i n v e r s e ; and a s i m i l a r proof shows that
Step 1 .
Since
i n v e r s e , i t follows that
k(GF)
vI K1
has a l e f t homotopy i n v e r s e . 2
i d e n t i t y , and
(GF)k 2 i d e n t i t y .
k
i s known t o have a l e f t
K
*j(c)
as M'+",
r e t r a c t of
The proof of Lemma 3.7 can now be completed as follows.
ql,.
K.
Proof similar t o that of Theorem 3.1 shows t h a t t h e s e t
i s a 2-sided i n v e r s e .
MCtE
i s homotopic by c e l l u l a r approximation t o
cpj
Pact, b u t all c r i t i c a l p o i n t s l i e i n one of the compact s e t s RF
CW-complex.
A
m-complex.
Consequently
be t h e c r i t i c a l
has no c l u s t e r p o i n t since each
By Theorems 3 . 1 ,
By induction i t follows that each LzL(FR)
T h i s completes the
i s of the homotopy ,type of a
M C - E ~ ex l
qj:
Then
imply that
G
<
c,
i s vacuous f o r
Ma
ci
a map
$1
FR
kG as l e f t inverse
i s s m a l l enough, and t h e r e i s a homotopy equivalence
i s a homotopy equivalence; and
homotopy i n v e r s e .
identity,
has
is a CW-complex.
The r e l a t i o n s
LF
Let
'p1
Then each
right homotopy i n v e r s e
R
and that
be t h e smallest
formal, based on t h e following. ASSERTION.
F
We have assumed that t h e r e i s a homotopy equivalence
i s a homotopy equivalence w i l l now be purely
F
M&
The s e t
.
has a l e f t homotopy i n v e r s e .
F
i d e n t i t y , and
The sequence
the homotopy type of
when Therefore
i s known t o have a l e f t
i s a homotopy equivalence.
F
M -R.
f:
M& i s compact. a 4 c , , c 2 , c 3 ,...
cp
i s now defined by the formula qT(x)
G
proof of 3.7.
.
values of
x
i d e n t i t y , and
(Fk)G 2 i d e n t i t y .
PROOF OF THEOREM 3 . 5 . The required homotopy
23
each extending the previous one.
C Ma2 C Ma
vl
C
...
C
...
vl
C K2 C K3
,
Let K denote the union of t h e Ki i n the d i r e c t l i m i t topology, i . e . , t h e f i n e s t p o s s i b l e compatible topology, and
I.
24
let
-
g: M
-moupg - i n
be t h e l i m i t map.
K
Then
g
25
induces isomorphisms of homotopy
a l l dimensions.
W e need only apply Theorem 1 of Combinatorial [Whitehead’s homotopy I t o conclude that g i s a homotopy equivalence.
theorem s t a t e s that i f any map
EXAMPLES
§4.
NON-DEGENERATE FUNCTIONS
M
-+
equivalence.
and
M
K
94.
are b o t h dominated by CW-complexes, then
Examples.
A s an a p p l i c a t i o n of t h e theorems of $ 3 we shall prove:
K which induces isomorphisms of homotopy groups i s a homotopy Certainly
K
i s dominated by i t s e l f .
To prove that
dominated by a W-complex i t i s only necessary t o consider of tubular neighborhood i n some Euclidean space.]
M
THEOREM 4 . 1 (Reeb). If M i s a compact manifold and f i s a d i f f e r e n t i a b l e f u n c t i o n on M with only two c r i t i c a l p o i n t s , b o t h of which are non-degenerate, then M i s homeomorphic t o a sphere.
is
as a r e t r a c t T h i s completes t h e proof M
of Theorem 3.5.
of a f i n i t e
has t h e homotopy type
W e have a l s o proved that each
REMARK.
CW-complex, with one c e l l of dimension
p o i n t of index
A
in
Ma.
T h i s i s t r u e even i f
a
X
f o r each c r i t i c a l
i s a c r i t i c a l value.
(Compare Remark 3.4. )
PROOF: T h i s follows from Theorem 3.1 together w i t h the Lemma of Morse ( 9 2 . 2 ) . points. If
e
The two c r i t i c a l p o i n t s must be t h e minimum and maximum
Say that
f(p) =
i s t h e m i m i m u m and
i s small enough then the sets
closed n - c e l l s by 9 2 . 2 .
M
0
But
I6
ME
between
M
and
REMARK degenerate.
M’-€
1
f - l [ o , ~ l and
i s homeomorphic t o
i s the union of two closed n - c e l l s ,
along t h e i r common boundary.
=
f(q) =
and
M’-€
i s t h e maximum. f-l[i-E,i]
by 9 3 . 1 .
f-l[l-~,ll,
are Thus
matched
It i s now easy t o c o n s t r u c t a homeomorphism
Sn. 1.
The theorem remains t r u e even i f the c r i t i c a l p o i n t s are
However, t h e proof i s more d i f f i c u l t .
(Cmpare Milnor,
Sommes
de varietes d i f f e r e n t i a b l e s e t s t r u c t u r e s d i f f e r e n t i a b l e s des sph&res, B u l l . SOC. Math. de France,Vol. 8 7 ( 1 9 5 9 1 , pp. 439- 444; Theorem l ( 3 ) ; o r R . Rosen, A weak form of t h e star conjecture f o r manifolds, Abstract 570- 28, Notices h e r . Math Soc.,Vol. 7 ( 1 9 6 0 ) , p. 380; Lemma I . ) REMARK 2 .
It i s n o t t r u e that
M must be diffeomorphic t o
Sn with
i t s u s u a l d i f f e r e n t i a b l e structure.(Compare: Milnor, On manifolds homeomorP N c t o t h e 7-sphere, Annals of Mathematics, V o l . 64 ( 1 9 5 6 ) , pp, 399-405. I n this paper a 7-sphere w i t h a non-standard d i f f e r e n t i a b l e s t r u c t u r e i s Proved t o be t o p o l o g i c a l l y
S7 by f i n d i n g a f u n c t i o n on i t w i t h two non-
J4.
degenerate c r i t i c a l p o i n t s . )
A s another a p p l i c a t i o n of t h e previous theorems we note that i f a n n-manifold b s a d i f f e r e n t i a b l e f u n c t i o n on i t with only t h r e e c r i t i c a l p o i n t s then they have index
~
n 'and
n/2
(by Poincark d u a l i t y ) , and t h e
has t h e homotopy type of an n/2-sphere with an n - c e l l attached.
&fold T
0,
Ss i t u a t i o n i s studied i n forthcoming papers by J . Eel19 and N . Kuiper.
Such a function e x i s t s for example on t h e r e a l or complex p r o j e c t i v e plane. Let
equivalence c l a s s e s of clzj l 2
=
1.
..., zn)
( n + l ) - t u p l e s (z,,
c O , c l , ..., cn
. .,zn)
by ( z o : z l : . . . : z n ) .
f on CPn by t h e i d e n t i t y
a r e d i s t i n c t r e a l constants.
I n order t o determine the c r i t i c a l p o i n t s of following l o c a l coordinate system.
Let
f,
consider the
be t h e set of
U,
( z o : z l : ... :'n)
Then
a r e t h e required coordinate f u n c t i o n s , mapping
=
xj
diffeomorphically onto
Clearly
t h e open u n i t b a l l i n R2". IZjI2
U,
2 +
Yj
2
1z0l2
= 1
-
c
(Xj 2
+ yj2)
so t h a t
throughout t h e coordinate neighborhood
f within
U,
Thus t h e only c r i t i c a l p o i n t of
U,.
l i e s a t the center point p,
of t h e coordinate system.
=
(1:o:o:
... :o)
A t this p o i n t
index equal t o twice t h e number of
j
f i s non-degenerate; and has
with
cj
<
co.
S i m i l a r l y one can consider other coordinate systems centered a t t h e points p,
=
(0:1:0:
... :o),
...,p,
=
(0:0:
... : 0 : 1 ) .
27
It follows that index of f a t
P O , P , , . . .,Pn a r e the Only c r i t i c a l p o i n t s of f . The pk i s equal t o twice the number of j with c . < ck. Thus every p o s s i b l e even index between 0 and 2n occurs e x a c t Jl y once. Theorem 3 . 5 : cpn
has the homotopy type of a CW-complex of t h e form e0 u e 2 u e 4 u..."
e2n
.
It follows t h a t t h e i n t e g r a l homology groups o f
CPn
CPn as
of complex numbers, with
Denote t h e equivalence c l a s s of (z,,.
Define a r e a l valued f u n c t i o n
where
We w i l l t h i n k of
CPn be complex p r o j e c t i v e n-space.
EXAMPLES
Hi(CPn;Z)
=
Z 0
f o r i = 0 , 2 , 4 ,..., 2n f o r other values o f i
a r e given by
Let
be such t h a t
The Morse I n e q u a l i t i e s .
be a compact manifold
M
contains e x a c t l y
Mai
Then I n Morse's o r i g i n a l treatment of this s u b j e c t , Theorem 3 . 5 was not available.
The r e l a t i o n s h i p between t h e topolo@p of
M
and
29
a d i f f e r e n t i a b l e function
f
M with i s o l a t e d , non-degenerate, c r i t i c a l p o i n t s .
on 55.
THE: MORSE INEQUALITIES
95.
N O N - D E G E ~ T EFUNCTIONS
I.
28
exiM a i - i
i J M a i - i ) = H,(Mai-i
9
and t h e c r i t i c a l
where
p o i n t s of a r e a l valued f u n c t i o n on M were described i n s t e a d i n terms of a c o l l e c t i o n of i n e q u a l i t i e s . T h i s s e c t i o n w i l l describe this o r i e i n a l
a1
c r i t i c a l p o i n t s , and
i
a
H,(M
Let
M
(...(
a
)
i s t h e index of t h e
Xi
H,(e xi, & I i )
=
by excision,
c o e f f i c i e n t group i n dimension
S i s subadditive i f whenever
the integers.
S(X,y) + S(Y,Z).
I f e q u a l i t y holds,
f o r any p a i r
(X,Y)
we have
S(X,Z)
5
Applying
(1)
to
0 =
Ma'
Xi
otherwise.
0
C . . .C M&k = M
S
with
= R,
we have
S i s called additive.
A s an example, given any f i e l d
R,(X,Y)
X3Y3 2
{
=
S be a f u n c t i o n from c e r t a i n p a i r s of spaces t o
Let
M.
=
c r i t i c a l point,
p o i n t of view. DEFINITION:
ak
as c o e f f i c i e n t group, l e t
F
=
Xth B e t t i number of
=
rank over
of
F
(X,Y)
H,(X,Y;F)
such that t h i s rank i s f i n i t e .
R,
where
,
C,
denotes t h e number of c r i t i c a l p o i n t s of index
formula t o the case
S
=
x
A.
Applying this
we have
i s subadditive, a s
i s e a s i l y seen by examining t h e following p o r t i o n of t h e exact sequence f o r (x,y,Z):
...
-
H,(Y,Z)
-
H,(X,Z)
The Euler c h a r a c t e r i s t i c
C
(-1)'
-.
%(X,Y)
+
Thus we have proven:
. .. I
i s a d d i t i v e , where
X(X,Y)
X(X,Y)
THEOREM 5 . 2 (Weak Morse I n e q u a l i t i e s ) .
=
number of c r i t i c a l p o i n t s of index f o l d M then
R,(X,Y).
Then
(2)
...
S be subadditive and l e t XoC C %. S(Xn,Xo)5 S(Xi,Xi-l). I f S i s a d d i t i v e then
LEMMA 5 . 1 .
Let
2
e q u a l i t y holds. PROOF:
R,(M)
(3)
C
i= 1
Induction on
n
= 2
CX
RX(M)
(-1)'
=
, C
If
denotes t h e
C,
on t h e compact mani-
and (-1)'
.
C,
S l i g h t l y sharper i n e q u a l i t i e s can be proven by t h e following argument. n.
For n
=
1 , e q u a l i t y holds t r i v i a l l y and
LEMMA 5 . 3 .
t h e case
5
h
i s t h e d e f i n i t i o n of [ s u b ] a d d i t i v i t y .
If t h e r e s u l t i s t r u e f o r n -
1,
then
S(Xn-, ,Xo)
The f u n c t i o n
SX i s subadditive, where
n- 1
5 c1
S(Xi,Xi-l).
-
s,(x,y)
=
PROOF:
Given a n exact sequence
Rx(X,Y)
RX-,(X,Y)
+ R,-,(X,Y)
-
+...?
Ro(X,Y)
.
LALBJ-C ... L ...+ D + O of v e c t o r spaces note t h a t t h e rank of t h e homomorphism of
i
i s equal t o t h e rank of
A.
Therefore,
h
p l u s t h e rank
I. NON-DEGENERATE FUNCTIONS
30
95.
THE MORSE INEQUALITIES 31
rank
h
Subtracting this from t h e e q u a l i t y above we o b t a i n t h e following:
=
rank
A - rank i
=
rank
A
rank
B + rank
j
=
rank
A - rank
B + rank
C
-
- rank
C O R O W Y 5.4.
k
Rb+l
... B + rank
A - rank
rank
=
C -
+...+ rank
D
a-
a
-
R,(Y,Z)
=
+
- R,-l(Y,Z)
... 2 0
C o l l e c t i n g terms, this means that
which completes the ? r o o f . Applying this subadditive f u n c t i o n S, t o the spaces a &k 0 C C M 2 C...C M
Ma1
we o b t a i n t h e Morse i n e q u a l i t i e s
or ( M ) +-.
- R,-l
R,(M)
( 4,)
. .+
5
Ro(M)
C,
-
C,-l
+
-. .+ C,.
These i n e q u a l i t i e s a r e d e f i n i t e l y sharper than t h e previous ones.
(4,)
I n f a c t , adding with
(4,-,)
for
and (4,-1),
>
A
n
one obtains ( Z h ) ;
and comparing
one obtains t h e e q u a l i t y
(4,)
(3).
A s an i l l u s t r a t i o n of t h e use o f t h e Morse i n e q u a l i t i e s , suppose
that
Cx+l
= 0.
(4,)
and
(4x+1),
Then
must also be zero.
Comparing t h e i n e q u a l i t i e s
we see that
R,
Now suppose that
RX+l
- R,-l
+-...+
R0
i s a l s o zero.
C,-l
C,
=
+ - . . . t C,
- C,-l
Then R,-l
= 0,
.
I
arid a similar argu-
ment shows t h a t Rh-2
-
R,-3
+-...+_ R,
=
Ch-2 - C,-3
+-...+
Co
.
.
C,-l
= 0
then
R,
=
C,
and
= 0.
space ( s e e 9 4 ) without making use of Theorem 3 . 5 .
%(Y,Z)
+ R,(X,Y)
R,(X,Z)
=
( O f course t h i s would a l s o follow from Theorem 3 . 5 . ) Note t h a t this c o r o l l a r y enables u s t o f i n d the homology . g o u p s of complex p r o j e c t i v e
we see that rank
R,-l
Ch+l
.
Hence t h e last expression i s 2 0. Now consider t h e homoiogy exact sequence of a t r i p l e X 2 Y 1 2 . Applying this computation t o t h e homomorphism
Hx+l(X,Y)
=
If
, §6.
NON-DEGENERATE FUNCTIONS
I.
32
56.
33
THEOREM 6.1 ( S a r d ) . If M, and M2 a r e d i f f e r e n t i a b l e manifolds having a countable b a s i s , of t h e same dimension, M, i s of c l a s s C 1 , then t h e image of t h e and f : M1 s e t of c r i t i c a l p o i n t s has measure 0 i n M,.
-
Manifolds i n Euclidean Space.
Although w e have s o f a r considered, on a manifold, only functions
A c r i t i c a l p o i n t of
which have no degenerate c r i t i c a l p o i n t s , we have n o t y e t even shown that
singular.
I n this s e c t i o n we w i l l construct many func-
such functions always e x i s t .
MANIFOLDS I N ETJCLIDM SPACE
f
i s a p o i n t where the Jacobian of
f
is
For a proof see de Rham, “Vari6tBs D i f f e r e n t i a b l e s , ‘I Hermann,
P a r i s , 1955, p . 1 0 .
t i o n s w i t h no degenerate c r i t i c a l p o i n t s , on any manifold embedded i n Rn. I n f a c t , if for f i x e d
5:
p cRn
define t h e f u n c t i o n M - R by p, t h e f u n c t i o n It w i l l t u r n out that f o r almost a l l ((p-q1I2.
5
$(q)
=
has
only non-degenerate c r i t i c a l p o i n t s .
Let
be a manifold of dimension
M C Rn
bedded i n R”.
Let
N C M x Rn
<
n,
d i f f e r e n t i a b l y em-
be defined by
=
vector bundle of
Let
E
M,
x
PROOF:
N
v
M.)
We have j u s t seen t h a t
i s a f o c a l p o i n t iff x E: N -R”.
perpendicular t o M a t 9 ) . I t i s not d i f f i c u l t t o show that N i s an n-dimensional manifold i s t h e t o t a l space of t h e normal d i f f e r e n t i a b l y embedded i n R2n. ( N N
[(q,v): q
k
COROLLARY 6 . 2 . For almost a l l not a f o c a l p o i n t of M.
E
Rn,
the point
x
i s an n-manifold.
is
The point
x
i s i n t h e h g e of t h e s e t of c r i t i c a l p o i n t s of
Therefore t h e s e t of f o c a l p o i n t s has measure
0.
For a b e t t e r understanding of the concept of f o c a l p o i n t , i t i s convenient t o introduce t h e “second fundamental form“ of a manifold i n Euclidean space.
We w i l l not attempt t o give an i n v a r i a n t d e f i n i t i o n ; b u t w i l l make
use of a f i x e d l o c a l coordinate system.
E: N + R n
be
E(q,v)
=
q +
V.
(E
i s t h e “endpoint“ mSp.) Let
ul,.
. .,uk
be coordinates f o r a r e g i o n of t h e &fold
Then t h e i n c l u s i o n map from x , ( u1
M
t o Rn
determines
n
M C Rn.
smooth functions
,...,uk) ,..., x,(u 1 ) . . . )uk) .
These functions w i l l be w r i t t e n b r i e f l y a s z ( u ’ ,
( x l , . . . ,%) . To be c o n s i s t e n t the p o i n t
q
E
. . .,uk)
M C Rn
where
-L
x
=
w i l l now be denoted by
-L
9.
The f i r s t fundamental form a s s o c i a t e d w i t h t h e coordinate system i s defined t o be t h e symmetric matrix of r e a l valued f u n c t i o n s DEFINITION. p
if
e
nullity
= p
q + v
>
0.
e
E
Rn
i s a f o c a l p o i n t of
(M,q)
with multiplicity
where (q,v) E N and t h e Jacobian of E a t The p o i n t e w i l l be c a l l e d a f o c a l p o i n t of
a f o c a l p o i n t of
(4,v)
has is
M if e
The second fundamental form
on t h e other hand, i s a symmetric matrix
Fij)
of v e c t o r valued f u n c t i o n s . (M,q)
f o r some
q
I n t u i t i v e l y , a f o c a l p o i n t of
E
M. M
i s a p o i n t i n Rn
where nearby
normals i n t e r s e c t . W e w i l l use the following theorem, which w e w i l l not Prove.
a%
It i s defined as follows. The vector be expressed as t h e sum of a v e c t o r tangent t o Define -c
v
yij
t o be t h e normal component of
which i s normal t o
M
at
t h e matrix au
a t a p o i n t of M can M aU and au a v e c t o r normal t o M.
3%
aU
.
Given any u n i t vector
I.
34
can be c a l l e d t h e "second fundamental form of d
V.
86.
NON-DEGENERATE FUNCTIONS
M at
z
vectors
i n the d i r e c t i o n
a; auk1 w1
aa?u l ' " ' l
+
MANIFOLDS I N EUCLIDEAN SPACE +
,wn-k
9 ' * '
we w i l l o b t a i n an
35
nxn
matrix whose
rank equals the rank of t h e Jacobian of
I1
This
It w i l l simplify t h e d i s c u s s i o n t o assume t h a t t h e Coordinates
nxn
E a t t h e corresponding p o i n t . matrix c l e a r l y has the following form
-+
g.l j r evaluated a t
mve been chosen t o that
men the eigenvalues of t h e matrix ,matures ,iprocals
K 1 , . . .,Kn
K;~,
of
.. .
M
3
at
3 . dij)
(
i s t h e i d e n t i t y matrix.
9,
are c a l l e d t h e p r i n c i p a l re-
\
of t h e s e p r i n c i p a l curvatures a r e c a l l e d t h e p r i n c i -
pi r a d i i of curvature.
(7. dij)
O f course it may happen that t h e matrix
w i l l be zero; and hence
I n this case one o r more of the
is singular.
7. The
i n t h e normal d i r e c t i o n
the corresponding r a d i i
K;'
w i l l not be defined.
Now consider t h e normal l i n e
z
c o n s i s t i n g of a l l
1
+ v i s a f i x e d u n i t v e c t o r orthogonal t o
M
at
identity
0
matrix
Thus the rank i s equal t o the rank of the upper l e f t hand block
.
Using
the i d e n t i t y + t?,
where
,
we see that this upper l e f t hand block i s j u s t the matrix LEbMA 6 . 3 . The f o c a l p o i n t s of (M,<) along P + c i s e l y the p o i n t s q + K;' 3, where 1 5 i 5 k, Thus t h e r e a r e a t most, k f o c a l p o i n t s of (M,;)
P,
a r e pre-
$#
0.
along
each being counted with i t s proper m u l t i p l i c i t y . +
k
1
Thus : +
k
1
PROOF: Choose n-k v e c t o r f i e l d s w l ( u ,..., u ),...,Wn-k(u , . . . , U ) -+ -+ along the manifold s o that w , , . . . jWn-k a r e unit v e c t o r s which a r e orthogok
1
nal t o each other and t o
t',
...,tn-k)
M.
We can introduce coordinates
(U ,...,U
1
on t h e manifold
N C M x Rn
as follows.
Let
(U
.
1.1
,
nf
tQ;a(ul,...,u
a=1
k ))
E
N
.
i I
Then the f u n c t i o n
E: N gives r i s e t o t h e correspondence
i s a f o c a l point of
'glj - tT;
i s singular, with n u l l i t y Now suppose that
I
,
+ ;t
(M,a with m u l t i p l i c i t y
i f and only i f t h e matrix
(*)
4
k 1 ,...,U , t ,..., I
tn-k) correspond t o t h e p o i n t (Z(u1 , .. ,u )
ASSERTION 6 . 4 .
l a r If and only if more t h e m u l t i p l i c i t y
(g
)
. -Ti$
1.1.
i s the i d e n t i t y matrix.
Then ( + ) i s sin&-
i s a n ieigenvalue j of the matrix (7. Ti,) . Furtheri s equal t o t h e m u l t i p l i c i t y of as eigenvalue.
$
IJ
T h i s completes the proof of Lemma 6 . 3 .
+Rn
Now for f i x e d
I
'i; E Rn l e t us study the f u n c t i o n = f: M + R
3
where with p a r t i a l d e r i v a t i v e s
2;
We have
*
5 + i; . T; .
i s equal t o
Setting
3
x + tv,
=
because this sum i s a t o p o l o g i c a l i n v a r i a n t
X(M)
The preceding c o r o l l a r y can be sharpened as follows.
as i n t h e proof of Lemma 6 . 3 , this becomes
The p o i n t E M i s a degenerate c r i t i c a l p o i n t i f and only i f i s a f o c a l p o i n t of of as c r i t i c a l p o i n t i s equal t o t h e multiThe n u l l i t y + p as f o c a l p o i n t . p l i c i t y of 6.5.
(M,T).
f=Lp
(Compare M. Morse, Combining t h i s r e s u l t with Corollary 6 . 2 t o SarG's theorem, we
Let
k
2o
be a compact s e t .
K C M
COROLLARY 6.8. Any bounded smooth f u n c t i o n f: M- R can be uniformly approximated by a smooth f u n c t i o n g which has no degenerate c r i t i c a l p o i n t s . Furthermore g can be chosen s o that t h e i - t h d e r i v a t i v e s of g on t h e compact s e t K uniformly approximate t h e corresponding d e r i v a t i v e s of f , f o r i 5 k.
Theref O r e :
of
M
( s e e Steen-
rod, "The Topology of F i b r e Bundles," 1 3 9 . 7 ) .
be an i n t e g e r and l e t
LEMMA
37
It follows that t h e sum of the i n d i c e s of any v e c t o r f i e l d on
The second p a r t i a l d e r i v a t i v e s a t a c r i t i c a l p o i n t a r e given by
- -
MANIFOLDS I N EXJCLIDEAN SPACE
§6.
NON-DEGENERATE FUNCTIONS
I.
The c r i t i c a l p o i n t s of a f u n c t i o n of n v a r i -
a b l e s , Transactions of t h e American Mathematical Society, V o l . 3 3 (1931), pp. 71-91 .)
immediately obtain: THEOREM 6 . 6 . For almost a l l measure 0 ) t h e f u n c t i o n
Lp:
p
E
PROOF:
( a l l b u t a s e t of
Rn
Choose some imbedding
h: M -.Rn
of
M
s e t of some euclidean space so t h a t the f i r s t coordinate
M- R
the given function
f. L e t
c
be a l a r g e number.
p
=
(-C+E1
as a bounded subh,
i s precisely Choose a p o i n t
has no degenerate c r i t i c a l p o i n t s .
T h i s theorem has s e v e r a l i n t e r e s t i n g consequences. c l o s e t o (-c,O,. . . , O ) degenerate; and s e t
C O R O W Y 6 . 7 . On any manifold M t h e r e e x i s t s a d i f f e r e n t i a b l e f u n c t i o n , with no degenerate c r i t i c a l p o i n t s , for which each Ma i s compact.
s i o n a l manifold R2n+1
Clearly, i f
A d i f f e r e n t i a b l e manifold has t h e homotopy type of
f
i s non-degenerate.
:
L.p
M
-
R i s non-
- c2 2c A s h o r t computation shows t h a t -
On a compact manifold
M
there i s a vector f i e l d
such that t h e sum of t h e i n d i c e s of the c r i t i c a l p o i n t s of
X
f
M. T h i s can be seen as follows: f o r on M we have x(M) = C ( - 1 ) ' C, where C,
i s t h e number of c r i t i c a l p o i n t s w i t h index t h e v e c t o r f i e l d grad
f
a t a p o i n t where
A.
f
But
(-1)'
has index
i s l a r g e and t h e
Ei
are small,
then
g
w i l l approximate
The above theory can a l s o be used t o describe t h e index of t h e function
equals
t h e Euler c h a r a c t e r i s t i c of
any d l f f e r e n t i a b l e f u n c t i o n
c
as required.
T h i s f o l l a r s from t h e above c o r o l l a r y and Theorem 3.5.
APPLICATION 2 .
X(M),
s o that the f u n c t i o n
M can be embedded d i f f e r e n t i a b l y a s . & closed subset of
APPLICATION 1 .
X
g
. .,En)
T h i s follows from Theorem 6 . 6 and t h e f a c t that an n-dimen-
( s e e Whitney, Geometric I n t e g r a t i o n Theory, p . 1 1 3 ) .
a CW-complex.
Rn
g(x) Clearly
PROOF:
E
,E2,.
i s the index of
at a c r i t i c a l point LEMMA 6.9. (Index theorem f o r Lp. T h e index of a t a non-degenerate c r i t i c a l p o i n t q E M i s equal t o the number of f o c a l p o i n t s of (M,q) which l i e on t h e s e p e n t from q t o p; each f o c a l p o i n t being counted w i t h i t s multiplicity.
5
I.
38
THE LEFSCHETZ T€EOREM
57.
NON-DEGENERATE FUNCTIONS
39
An analogous statement i n P a r t I11 ( t h e Morse Index Theorem) w i l l
be of fundamental importance. The Lefschetz Theorem on Hyperplane S e c t i o n s .
$7.
The index of t h e matrix
PROOF:
As a n a p p l i c a t i o n of t h e . i d e a s which have been developed, we w i l l prove some r e s u l t s concerning the topology of a l g e b r a i c v a r i e t i e s .
is equal t o t h e number of negative eigenvalues.
Assuming that
(gij)
the i d e n t i t y matrix, this i s equal t o t h e number of eigenvalues of are
2
1
.
is
(7 bij)
These
were o r i g i n a l l y proved by Lefschetz, using q u i t e d i f f e r e n t arguments. p r e s e n t version i s due t o Andreotti and
The
* Frankel .
THEOREM 7 . 1 . If M c C" i s a non-singular a f f i n e a l g e b r a i c v a r i e t y i n complex n-space with r e a l dimension 2k, then Hi(M;Z) = 0 f o r i > k.
Comparing this statement w i t h 6 . 3 , the conclusion follows.
This i s a consequence of the s t r o n g e r : THEOREM 7 . ? . A complex a n a l y t i c manifold M o f complex dimension k, b i a n a l y t i c a l l y embedded a s a closed subset of cn has t h e homotopy type of a k-dimensional CW-complex.
The proof w i l l be broken up i n t o s e v e r a l s t e p s . q u a d r a t i c form i n
k
complex v a r i a b l e s Q ( z1 ,..., k )
I f we s u b s t i t u t e
xh
+ iyh
zh,
for
o b t a i n a r e a l q u a d r a t i c form i n Q'(x',
...,x k , y ' , ...,yk )
ASSERTION 1 . then
-e
F i r s t consider a
If
e
2k =
=
.
l b h j zhzj
and then take t h e r e a l p a r t of r e a l variables:
r e a l p a r t of T b h j ( x h + i y h ) ( x d + i y j )
.-.
i s a n eigenvalue of
PROOF.
The i d e n t i t y
change of v a r i a b l e s .
&'
Q( i z '
,. . ., i z k )
=
can be transformed i n t o
. v,
Q' w i t h m u l t i p l i c i t y
i s a l s o a n eigenvalue w i t h the same m u l t i p l i c i t y
the q u a d r a t i c form
we
Q
p.
.
-&( z1 ,. . ,zk) shows that -&I by an orthogonal
Assertion 1 c l e a r l y follows.
*
See S. Lefschetz, " L ' a n a l y s i s s i t u s e t l a ggornetrie a l g-6 b r i a u e . " P a r i s and A . Andreotti and T . Frankel, The Lefschetz theorem on hyperplane s e c t i o n s , Annals of Mathematics, Vol. 69 '( 1 9 5 9 ) , pp. 71 3-71 7. 1924;
- 7
Now consider a complex manifold ded as a subset of
Cn. Let
M
41
$: M- R
which i s b i a n a l y t i c a l l y imbed-
be a p o i n t of
q
THE LEFSCHETZ THEOREM
17.
NON-DEGENERATE F7JNCTIONS
I.
40
has no degenerate c r i t i c a l p o i n t s .
M.
Since
i s a closed subset of
M
it
Cn,
i s c l e a r t h a t each s e t The f o c a l p o i n t s of
ASSERTION 2 .
(M,q)
along any normal l i n e
fi
@ I
q.
i n p a i r s which a r e s i t u a t e d symmetrically about
i s compact. q + tv
I n other words i f
i s a focal point,
then
q - tv
$‘[o,al
=
$
Now consider t h e index of
is a
which l i e on t h e l i n e segment from
f o c a l p o i n t with t h e same m u l t i p l i c i t y .
p
to
f o c a l p o i n t s along t h e f u l l l i n e through PROOF.
1
. ..
borhood of
q
s o that
determines
n
complex a n a l y t i c f u n c t i o n s w,
Let
v
z
Choose complex coordinates z (4)
=
w,(z
=
1
zk(q)
=
,..., z
k
1
,. . . ,zk
for
i n a neighM + Cn
M
buted symmetrically about
The i n c l u s i o n map
= 0.
and
a
1
=
,..., n .
be a fixed u n i t v e c t o r which i s orthogonal t o
at
q.
w
and
,..., zk)C,
lw,(zl
...,zk)
constant + Q(z’,
=
Q
denotes a homogeneous q u a d r a t i c function.
i s h since
i s orthogonal t o
v
Now s u b s t i t u t e system f o r
M;
v
*
=
Q’
at
q
k.
$
at
is
q
(The l i n e a r terms van-
s o as t o o b t a i n a r e a l coordinate
zh
of them can l i e between
p
I t follows that
2 k;
M
has t h e
which completes t h e proof
induces isomorphisms of homology groups i n dimensions l e s s than k-1. Furthermore, t h e induced homomorphism
1
H ~ - , ( v n P;Z)
for
v
=
r e a l p a r t of
..
k 1
constant + ~ ‘ ( x ’,, , x ,y
Clearly t h e q u a d r a t i c terms M
5
and these a r e d i s t r i -
V n P-+V
1
-
H~-~(v;z)
i s onto. PROOF.
.
Wac,
Q’
v.
proves Assertion
q
and
+ higher terms.
By Assertion 1
occur i n equal and opposite p a i r s . q + v
q(v,v
t h e eigenvalues of
Hence t h e f o c a l p o i n t s of
(M,q)
a l s o occur i n symnetric p a i r s .
This
2.
We a r e now ready t o prove 7.2. squared- distance f u n c t i o n
*
q(V,V
n P;Z)
= 0
for
(V,V n P)
r
5
it i s
k-1.
But the
Lefschetz d u a l i t y theorem a s s e r t s t h a t
determine t h e second fundamental form of
i n t h e normal d i r e c t i o n
along the l i n e through
,. . .,yk)
Using t h e exact sequence of t h e p a i r
c l e a r l y s u f f i c i e n t t o show that
T h i s f u n c t i o n has t h e r e a l power s e r i e s expansion *
k
q;
and consider t h e r e a l i n n e r product w
w
Hence a t most
+ higher terms,
M.)
xh + i y h
and
COROLLARY 7 . 3 ( L e f s c h e t z ) . L e t V be a n a l g e b r a i c v a r i e t y of complex dimension k which l i e s i n the complex p r o j e c t i v e Let P be a hyperplane i n CPn which contains space CP,. the s i n g u l a r p o i n t s (if any) of V. Then t h e i n c l u s i o n map
1
where
p
of 7 . 2 .
Consider
T h i s can be expanded as a complex power s e r i e s
v.
Accord-
(M,q) But t h e r e a r e a t most 2k
q.
homotopy type of a CW-complex of dimension
M
q.
q.
t h e Hermitian i n n e r product
of
q.
Thus t h e index of
),
a t a c r i t i c a l point
i n g t o 6 . 9 , this index i s equal t o the number of f o c a l p o i n t s of
Choose a p o i n t
p
€
Cn
SO
that t h e
But
V -(V n p)
CPn - P.
n P;Z)
=H
~ ~ - ~-(v ( vn P) ;z)
.
i s a non-singular a l g e b r a i c v a r i e t y i n t h e a f f i n e space
Hence i t follows from 7.2 that t h e l a s t group i s zero f o r T h i s r e s u l t can be sharpened as follows:
THEOREM 7.4 ( L e f s c h e t z ) . Under t h e hypothesis of t h e preceding c o r o l l a r y , t h e r e l a t i v e homotopy group a,(V,V n P) i s zero f o r r < k.
r
5
k-1.
NON-DEGENERATE FUNCTIONS
I.
42
43
The proof w i l l be based on t h e hypothesis that Sme ne igh-
PROOF.
borhood U of V n P can be deformed i n t o V n P w i t h i n V. hi^ can be proved, for example, using t h e theorem that a l g e b r a i c v a r i e t i e s can be triawlated.
Lp : V
I n place of the f u n c t i o n where f(x)
=
ir o
f
have index
no degenerate c r i t i c a l p o i n t s with homotopy type of
VE = f -
1
[ O , E ~
E
-c
R we x E
for
Lp 2
have index 2k
5 f <
w i l l use V
~
f:
V --c R
P,
PART I1
for x B p .
I/L~(X)
Since t h e c r i t i c a l p o i n t s of the c r i t i c a l p o i n t s of
- V n P
-
m.
k
=
k.
The f u n c t i o n
Therefore
A RAPID COURSE I N RIEMANNIAN GEOMETRY
i t follows that f
has §8.
V has t h e
with f i n i t e l y many c e l l s of dimension
The o b j e c t of P a r t I1 w i l l be t o give a rapid o u t l i n e of some b a s i c
2k
concepts of Riemannian geometry which w i l l be needed l a t e r .
attached. Choose
E
small enough s o
Then every map of the p a i r
r-cube.
ed i n t o a map (IF,F)
since
r
Covariant D i f f e r e n t i a t i o n
<
k,
-
that
VE C U .
Let
(Ir,fr) into
1'
denote t h e u n i t
(V,V n P)
can be deform-
#
#
mation the reader should consult Nomizu, geometry,"
For more i n f o r -
"Lie groups and d i f f e r e n t i a l
The Mathematical Society of Japan, 1956; Laugwitz, " D i f f e r e n t i a l -
geometrie," Teubner 1 9 6 0 ; or Helgason, " D i f f e r e n t i a l geometry and symmetric (VE,V n P) C ( u , V n P)
and hence can be deformed i n t o
spaces," Academic Press, 1 9 6 ? .
,
Let V
n P.
M
be a smooth manifold.
T h i s completes the DEFINITION.
proof. 'I
An a f f i n e connection a t a p o i n t Xp E T%
which a s s i g n s t o each tangent v e c t o r
p
E
M
i s a function
and t o each vector f i e l d
Y
a new tangent v e c t o r Xp k Y E T%
i
c a l l e d t h e convariant d e r i v a t i v e *
of
quired t o be b i l i n e a r a s a f u n c t i o n of f:
i s a r e a l valued function, and i f
Y X
This i s r e P' Furthermore, i f
i n the direction P
and
Y.
X
M-R
fl denotes t h e v e c t o r f i e l d
(fYIq = then
k
* Note
i s required t o s a t i s f y t h e i d e n t i t y
that our X b Y coincides with Nomizu's tended t o suggest t h a t t h e different:al operator Y.
",Y. X
The n o t a t i o n i s i n a c t s on the v e c t o r f i e l d
denotes t h e d i r e c t i o n a l d e r i v a t i v e of
%f
(As usual,
8 8.
GEOMETRY
11. R I E ” I A N
44
i n the d i r e c t i o n
f
where t h e symbol
yk ,i
COVARIANT DIFFERENTIATION
45
stands for t h e real valued f u n c t i o n
X -I
P
A global a f f i n e connection
function which a s s i g n s t o each
p
M
E
is a
(or b r i e f l y a connection) on M a n a f f i n e connection
kp
at
P,
If
1)
field
X I- Y ,
X
Y
and
a r e smooth v e c t o r f i e l d s on
M
then t h e vector
defined by t h e i d e n t i t y ( X I- Y),
=
xp
r ikj
Conversely, given any smooth real valued f u n c t i o n s
s a t i s f y i n g t h e following smoothness condition.
one can d e f i n e
X k Y
the conditions
(11,
(21,
u,
The r e s u l t c l e a r l y s a t i s f i e s
( 3 ) , (4), ( 5 ) .
Using t h e connection
,
kp Y
by t h e formula ( 6 ) .
on
I-
one can d e f i n e t h e covariant d e r i v a t i v e of
a v e c t o r f i e l d along a curve i n
M.
A parametrized curve i n
M
F i r s t some d e f i n i t i o n s .
must a l s o be smooth. Note that:
i s b i l i n e a r as a f u n c t i o n of X
X I- Y
(2)
(3)
(fx) I - Y
(4)
( X I- (fY)
Conditions ( 1 )
f ( X I- Y)
= =
,
,
(3)
Y
numbers t o
.
, ,
U C M,
u 1 ,. . . ,un
r ikj
valued f u n c t i o n s field
a
on
2
U.
I-
t h e connection on
U,
as follows.
Let X
on
a,
on
M
ak
denote t h e vector
aj
F o r any smooth func-
U
can be expressed
should d e f i n e a smooth f u n c t i o n on R . A s an example t h e v e l o c i t y vector f i e l d
c
Here
d
$
of t h e curve i s the
which i s defined by t h e r u l e
x dc
k
.
the correspondence
vector f i e l d along
xk a r e r e a l valued f u n c t i o n s on U.
TMc(t)
smooth r e a l
k=1
field
f
i s a f u n c t i o n which
t -Vtf n3
uniquely as
where t h e
c
=
c * dx
denotes the standard v e c t o r f i e l d on t h e r e a l numbers, and
I n p a r t i c u l a r t h e vector
can be expressed as denotes t h e homomorphism of tangent spaces induced by t h e map Diagram 9.)
These f u n c t i o n s
r ik3 determine t h e connection completely on U.
I n f a c t given v e c t o r f i e l d s expand
X I- Y
from t h e r e a l
defined on a coordinate
i s determined by
Then any v e c t o r f i e l d
along the curve
V
c
R a tangent v e c t o r
E
T h i s i s required t o be smooth i n t h e following sense: tion
I n terms of l o c a l coordinates
t
Vt
can be taken as the d e f i n i t i o n of
(4)
A vector f i e l d
.
a connection.
neighborhood
M.
a s s i g n s t o each
(Xf)Y + f ( X kY) (2)
and
i s a smooth f u n c t i o n
by t h e r u l e s
X (2),
lxiai
=
(3)
k
,
(4) ;
and
Y
=
L$aj
one can
y i e l d i n g t h e formula
i Diagram g
c.
(Compare
11. RIEMANNIAN GEOMETRY
46
Now suppose t h a t any v e c t o r f i e l d
M
along
V
LEMMA 8 . 2 .
i s provided w i t h a n a f f i n e connection. Then DV c determines a new v e c t o r f i e l d along c
c a l l e d the c o v a r i a n t d e r i v a t i v e of
V.
COVARIANT DIFFERENTIATION
§ 8.
at the point vector f i e l d
47
Given a curve c and a t a n g e n t v e c t o r V, c ( o ) , t h e r e i s one and o n l y one p a r a l l e l V a l o n g c which extends Vo.
The o p e r a t i o n DV
PROOF.
v - Z €
The d i f f e r e n t i a l e q u a t i o n s
i s c h a r a c t e r i z e d by t h e f o l l o w i n g three axioms.
a) b)
DV = +DW = .
D(V+W) =
If
f
D(fV) If
V
$ V + f =
=
at c)
R
i s a smooth r e a l valued f u n c t i o n on
DV
vk(0).
.
vk(t)
t,
for e a c h
then
V t = Yc(t) ( = t h e c o v a r i a n t d e r i v a t i v e of
that i s if dc i s e q u a l t o -m k Y
Y
on
M,
Since t h e s e e q u a t i o n s are l i n e a r , t h e s o l u t i o n s can be d e f i n e d for
Real V a r i a b l e s , "
c) V
-
p.
l a t i o n along
c.
Now suppose that DV
-m
u l ( t ) ,. .,un(t) field
where
V
v'
M,
of two v e c t o r s
Xp, Yp
i s a Riemannian manifold.
M
w i l l be denoted by A connection
1 on
The inner product
.
Yp>
<Xp,
i s compatible w i t h t h e Rieman-
M
and l e t nian metric i f p a r a l l e l t r a n s l a t i o n preserves inner products.
denote the c o o r d i n a t e s of t h e p o i n t
c(t).
The v e c t o r
f o r any parametrized curve
can be expressed u n i q u e l y i n t h e form
, . . . ,vn
(Compare Graves, "The Theory of F u n c t i o n s of
i s s a i d t o be obtained from Vo by p a r a l l e l t r a n s -
Vt
DEFINITION. Choose a l o c a l c o o r d i n a t e system f o r
t.
152.)
The v e c t o r
Y i n t h e d i r e c t i o n of t h e
LFDlMA 0 . 1 . There i s one and o n l y one o p e r a t i o n which s a t i s f i e s these three c o n d i t i o n s .
PROOF:
which are uniquely determined by the i n i t i a l v a l u e s
a l l r e l e v a n t v a l u e s of
i s induced by a v e c t o r f i e l d
v e l o c i t y v e c t o r of
have s o l u t i o n s
then
are r e a l v a l u e d f u n c t i o n s on R
s u b s e t of R), and
a , , ..., an
along
the i n n e r p r o d u c t
and any p a i r
are t h e s t a n d a r d v e c t o r f i e l d s on t h e co-
should be c o n s t a n t .
that
PROOF:
Choose p a r a l l e l v e c t o r f i e l d s
are orthonormal a t one p o i n t of
c
t i v e l y (where vi
w
lows that
V
= =
1
by t h i s formula, i t i s n o t d i f f i c u l t t o v e r i f y
that c o n d i t i o n s ( a ) , (b), and ( c ) A vector f i e l d
along
i f the c o v a r i a n t d e r i v a t i v e
DV
are s a t i s f i e d .
c
i s s a i d t o be a p a r a l l e l v e c t o r f i e l d
Theref o r e i s i d e n t i c a l l y zero.
which completes the p r o o f .
a
viwi
P,,
. . . , Pn
along
and hence a t e v e r y p o i n t of
t h e g i v e n f i e l d s V and W can be e x p r e s s e d as
Conversely, d e f i n i n g
I n o t h e r words,
of' p a r a l l e l v e c t o r f i e l d s
P, P '
LEMMA 8 . 3 . Suppose that t h e connection i s compatible w i t h the m e t r i c . L e t V, W be any two v e c t o r f i e l d s a l o n g c . Then
(or a n a p p r o p r i a t e open
I t f o l l o w s from ( a ) , ( b ) , and ( c )
o r d i n a t e neighborhood.
c,
c
1
viPi
and
2
c. wjPj
i s a r e a l valued f u n c t i o n on R) . and t h a t
c
which Then respec-
It f o l -
COROLLARY 8 . 4 . For any v e c t o r f i e l d s vector Xp E TMp:
xp < Y , Y ' >
<xp
=
COVARIANT DIFFERENTIATION
§ 8.
11. RIEMANNIAN GEOMETRY
48
Y,Y'
on
M
three quantities
and any
I- a j , a k > ,
k y,y;>
+
.
I- Y ' >
'
ai,a,>,and
a,
(There a r e only t h r e e such q u a n t i t i e s since
t
49
aj,ai>
aj
aj
=
1 a,
*
.)
These
equations can be solved uniquely; y i e l d i n g the f i r s t C h r i s t o f f e l i d e n t i t y Choose a curve
PROOF.
t
whose v e l o c i t y v e c t o r a t
c
= 0
is
X
connection
A
t
aj@;ik- a @ i j )
+
1
[X,YI
=
denotes t h e poison b r a c k e t
two v e c t o r f i e l d s . ) = 0
[X,YI
=
X(Yf) - Y(Xf)
Applying this i d e n t i t y t o t h e case
X
=
a,,
Y
=
(gPk) this y i e l d s t h e second C h r i s t o f -
of
aj '
one o b t a i n s t h e r e l a t i o n
r ikj - r jki
r iPj gQk . Multiplying
f e l identity
I
[X,Ylf
'
P by t h e i n v e r s e (gkP) of t h e matrix
(X t Y) - ( Y I- X )
[a1 ,a 3 1
7 1 (aigjk
=
i s c a l l e d symmetric i f i t satis-
f i e s the i d e n t i t y *
since
aj,ak>
The l e f t hand s i d e of t h i s i d e n t i t y i s equal t o
DEFINITION 8.5.
(A S usual,
'
P;
and apply 8.3.
Thus t h e connection i s uniquely determined by t h e m e t r i c . Conversely, d e f i n i n g
0.
=
k Conversely i f r i j = r j i then u s i n g formula ( 6 ) i t i s not d i f f i c u l t t o v e r i f y that t h e connection k i s symmetric throughout t h e coordinate neighborhood.
by t h i s formula, one can v e r i f y that the
r e s u l t i n g connection i s symmetric and compatible w i t h the m e t r i c .
This
completes the proof. An a l t e r n a t i v e c h a r a c t e r i z a t i o n of symmetry w i l l be very u s e f u l Consider a
later. (Fundamental lemma of Riemannian geometry.) LEMMA 8.6. A Riemnnian manifold possesses one and only one symmetric connection which i s compatible with i t s m e t r i c .
r iPj
"parametrized surface'' i n
s : R2 By a vector f i e l d (x,y)
6
s
along
V
M
-c
M:
that i s a smooth function
.
is meant a f u n c t i o n which a s s i g n s t o each
R* a tangent v e c t o r
(Compare Nomizu p . 76, Laugwitz p . 95.)
and s e t t i n g
< a j ' ak >
a, Permuting
i,j,
gJk
and
k
Applying 8 . 4 t o t h e v e c t o r f i e l d s
=
gjk
one obtains t h e i d e n t i t y
=
< a,
t aj,ak>
+
< aj,ai
t
ai,aj,a,,
ak> .
this g i v e s t h r e e l i n e a r equations r e l a t i n g t h e
tor fields
as
and
%
and
The following reformulation may (or may n o t ) seem more i n t u i t i v e . Define The " covariant second d e r i v a t i v e " of a r e a l valued f u n c t i o n f along two v e c t o r s $,Yp t o be t h e expression
xp(Yf)
-
'5
k Y)f
where Y denotes any v e c t o r f i e l d extending Y I t can be v e r i f i e d that P' ( C o m p a r e t h e proof of this expression does n o t depend on t h e choice of Y. LRmma 9.1 below.) Then t h e connection i s symmetric i f this second derivat i v e i s symmetric as a f u n c t i o n of Xp and Y P'
$
s, ;
a
and
s*
a
along
s.
&
a and give r i s e t o vecThese w i l l be denoted b r i e f l y by
A s examples, t h e two standard v e c t o r f i e l d s
yo,
*
.
TMS(x,Y)
YX,Y)
PROOF of uniqueness.
and c a l l e d t h e " v e l o c i t y v e c t o r f i e l d s " of
s.
For any smooth vector f i e l d V along s the covariant d e r i v a t i v e s 5 DV a r e new v e c t o r f i e d d s , constructed as follows. For each f i x e d
restricting
V
t o t h e curve x
-
S(X,Y0)
one obtains a v e c t o r f i e l d along this curve. respect t o
x
i s defined t o be
the e n t i r e parametrized s u r f a c e
(
)(x,y,)
I t s covariant d e r i v a t i v e with
.
T h i s defines
%
along
s.
A s examples, we can form t h e two covariant d e r i v a t i v e s of the two
L
50
,
RIEMANNIAN GEOMETRY
11.
§9. THE CURVATURE TENSOR
1
vector f i e l d s
as
and
as
.
The d e r i v a t i v e s
simply t h e a c c e l e r a t i o n v e c t o r s of s u i t a b l e t h e mixed d e r i v a t i v e s
X
as
5
and
-&
D
as
and
as
D
coordinate curves.
are However,
cannot be d e s c r i b e d s o simply.
§9. The c u r v a t u r e t e n s o r
LEMMA
8.7.
If
t h e connection i s symmetric t h e n
D
as
ax ay
=
D
as
ay ax
.
The Curvature Tensor
R
of a n a f f i n e connection
a,
e x t e n t t o which t h e second c o v a r i a n t d e r i v a t i v e metric i n R(X,Y)Z
PROOF.
51
and
i
by the
j.
Given v e c t o r f i e l d s
t-
( aJ.
t measures t h e t Z)
i s sym-
d e f i n e a new v e c t o r f i e l d
X,Y,Z
* identity
Express b o t h s i d e s i n terms of a l o c a l c o o r d i n a t e system, R(X,Y)Z
and compute.
=
-X t- ( Y 1 Z) + Y k ( X t- Z) + [X,YI t Z .
LEMMA 9 . 1 . The v a l u e of R ( X , Y ) Z a t a p o i n t p E M depends only on t h e v e c t o r s X , Y , Z a t this p o i n t P P P p and n o t on t h e i r v a l u e s a t nearby p o i n t s . F u r t h e r more t h e correspondence
-
XPJPJP
from
T$
x T%
to
x T$
R(XpJp)Zp
is tri- linear.
T$
B r i e f l y , t h i s lema can be expressed by s a y i n g that R i s a " t e n s o r . " PROOF: If
X
Clearly
R(X,Y)Z
i s r e p l a c e d by a m u l t i p l e
Y 1 ( X t- Z ) , i) ii)
iii)
[X,YI
i s a t r i - l i n e a r f u n c t i o n of X,Y,
fX t h e n t h e t h r e e terms
-X
and
Z.
t- ( Y I- Z ) ,
t- 2 are r e p l a c e d r e s p e c t i v e l y by
-fX t-(Y t-Z) ,
,
(Yf)(X t- Z) + fY 1 ( X t- Z)
z .
- ( Y f ) ( X t- Z) + f[X,YI t-
Adding these three terms one o b t a i n s t h e i d e n t i t y R(fX,Y)Z Corresponding i d e n t i t i e s f o r
Y
computations. Now suppose that
X
Then R(X,Y)Z
and
1
=
=
z
fR(X,Y)Z
=
Z
.
are e a s i l y o b t a i n e d by similar
Y
=
R(xiai,yjaj)
* Nomizu g i v e s
1 Yiaj ,
and
z
=
1
zkak.
(zkak)
R the o p p o s i t e sign. Our s i g n convention has the advant a g e that ( i n t h e Riemannian c a s e ) the inner p r o d u c t c o i n c i d e s w i t h t h e c l a s s i c a l symbol R hijk
'
Evaluating this expression a t (R(X,Y)Z)p
p
1
=
(
D
=
Henceforth we w i l l assume that
at
xi,yj,zk
p,
and
T h i s completes the proof.
I n conclusion we w i l l prove t h a t t h e t e n s o r
3
one can apply t h e two covariant d i f to
as as
D D
D D
*xv-x*v
.
R ( Z ' * b
=
PROOF:
and compute, making use of t h e i d e n t i t y
aj
I-
(ai
F
a,)
-
a,
t-
(aj F a,)
R(ai,aj)ak
=
s
D
E P and which has a given value vector f i e l d e x i s t s . P
p
I n g e n e r a l no such
Let
P(x,o)
P
(Xo,Y)
be a p a r a l l e l vector
= 0.
S(X0,Y)
i s i d e n t i c a l l y zero;
and
&P
and
[Y,Zl
+
- Y I - ( 2 I-X)
+
2 I - ( X I -Y )
Y 1z -
maining terms cancel i n p a i r s . P
everywhere along
s.
i s zero along the x- axis.
Now t h e i d e n t i t y
are all
Y I - ( X t-2)
z + x
F ( Y I-X) I - ( 2 I -Y )
0 .
=
z
k Y
=
[Y,ZI
=
0
.
Thus t h e upper l e f t term cancels the lower r i g h t term.
,
T h i s defines
[X,Zl
[X,YI,
But t h e symmetry of t h e connection implies that
For each
be a p a r a l l e l v e c t o r f i e l d along t h e curve
-
follows immediately from the
(1)
are t e n s o r s , it i s s u f f i c i e n t t o
k ( Y I-2)
-
However, i f the curvature t e n s o r happens t o be zero
b v i n g t h e right value for y
Y
-x
a t the o r i g i n .
p(o,o)
Y
P
.
Under this hypothesis w e must v e r i f y t h e i d e n t i t y
0,
=
(2)
prove ( 2 ) when t h e bracket products zero.
x- axis, s a t i s f y i n g the given i n i t i a l condition.
xo l e t
Clearly
z
can be constructed as follows.
f i e l d along t h e fixed
=
The skew-symmetry r e l a t i o n
Since a l l t h r e e terms of
which i s p a r a l l e l , i n the sense that
D
then
= 0
d e f i n i t i o n of R .
.
[ I t i s i n t e r e s t i n g t o ask whether one can c o n s t r u c t a vector f i e l d along
= 0
Express b o t h s i d e s i n terms of a l o c a l coordinate system,
PROOF:
P
The curvature t e n s o r of a Riemannian manifold
satisfies: R(X,Y)Z + R(Y,X)Z = 0 (1) R ( X , Y ) Z + R ( Y , Z ) X + R(2,X)Y (2) ( 3 ) + (4) < R ( X , Y ) Z , W > =
I n g e n e r a l t h e s e operators w i l l
V.
n o t commute with each o t h e r . umim9.2.
s a t i s f i e s four
symmetry r e l a t i o n s .
LEMMA 9 . 3 . s.
along and
R
R2-M.
s:
D
i s a Riemannian manifold, pro-
M
vided w i t h t h e unique symmetric connection which i s compatible with i t s metric.
Now consider a parametrized surface
f e r e n t i a t i o n operators
3.1
i s p a r a l l e l along
P
i s i d e n t i c a l l y zero;
X'(P)Yj(P)zk(P) ( R ( a i , a j ) a k \ ,
n o t on t h e i r values a t nearby p o i n t s .
V
53
&P
this implies t h a t
0,
Since 3% P ) ( X o , O ) and completes t h e proof that
one obtains t h e formula
WMch depends only on t h e values of the functions
Given any vector f i e l d
T B CURVATURE TENSOR
§9.
11. RIEMANNIAN GM)METRY
52
To prove ( 3 ) w e must show that t h e expression skew-symmetric i n
2
and
W.
Similarly the re-
T h i s proves ( 2 ) .
This i s c l e a r l y equivalent t o the a s s e r t i o n
that
D D
D D
3 x P - a x y P
D D implies that * x P
= 0.
=
R(,%&%
=
I n other words, the v e c t o r f i e l d
p a r a l l e l along t h e curves Y
-
0
D P
is
for a l l X,Y,Z.
S(X0,Y)
.
Again we may assume that
=
O
[X,Yl
=
0,
SO
i s equal t o
<- x
is
I- ( Y I- 2 ) + Y t- ( X t- Z ) , Z >
.
that
11. RIEMANNIAN GEOMETRY
54
§ 10.
GEODESICS AND COMPLETENESS
$10.
Geodesics and Completeness
55
In other words we must prove that the expression
< Y 1 (X 1 Z),Z> is symmetric in X Since
and Y.
[X,YI
=
0
Let M the expression YX
be a connected Riemannian manifold.
is symmetric in X DEFINITION. A parametrized path
and Y. Since the connection is compatible with the metric, we have
I -+M,
y:
x
=
2
Yx
= 2
(Y
(x
1
z,z >
where
hence
acceleration vector field
I-(X k Z),Z> +
But the right hand term is clearly symmetric in X
(Y
I denotes any interval of real numbers, is called a geodesic if the
k (X I- Z )
2
(x
k Z,Y I-
z> .
and Y. Therefore
vector field
8
&$
is identically zero. Thus the velocity
must be parallel along
y .
If
is a geodesic, then the
7
identity
,Z> is symmetric in X and Y; which proves property (3).
Property (4) may be proved from
(I),
(21,
(X. Y 2. W
and (3) as follows.
>
shows that the length constant along
Il$ll
=
<%, $>”’
of the velocity vector is
7. Introducing the arc-length function
s(t)
1 @lldt
=
+ constant
This statement can be rephrased as follows: The parameter
t along a
geodesic is a linear function of the arc-length. The parameter ally equal to the arc-length if and only if
l la d yT II
=
t
is actu-
1.
In terms of a local coordinate system with coordinates u1 ,. . . ,u“ a curve < R (Z.W)
Formula
(2)
t
-
The equation
X.Y>
7(t) E
&8
1
smooth functions u (t), .
M determines n
. . ,un(t).
for a geodesic then takes the form
asserts that the sum of the quantities at the vertices
of shaded triangle W
is zero. Similarly (making use of
(1)
and (3)) the
sum of the vertices of each of the other shaded triangles is zero. Adding these identities for the top two shaded triangles, and subtracting the
The existence of geodesics depends, therefore, on the solutions of a certain system of second order differential equations.
identities for the bottom ones, this means that twice the top vertex minus twice the bottom vertex is zero. This proves ( 4 ) ,
More generally consider any system of equations of the form
and completes the proof.
d2;’ dt2 +
Here u
stands for
(u’,
..., un)
=
T(<,-df; d< ) .
and
7
stands f o r an n-tuple of
functions, all defined throughout some neighborhood U
(zl,Tl) E R2”
.
of a point
Cm
11. RIEWW"NN GEOMETRY
GEODESICS AND COMPLETENESS
§ 10.
MISTENCE AND UNIQUENESS THEOREM 1 0 . 1 . There e x i s t s a neighborhood W of the p o i n t (T, ,Yl) and a number E > 0 s o that, for each (To,To) E W the differen-
numbers I1vII
<
E~
E
~
,
>
E
s o that:
~0
f o r each
p
E
and each
U
57
v
TMp w i t h
E
t h e r e i s a unique geodesic ( - 2 ~ ~ , 2 - +~M ~ )
yv:
s a t i s f y i n g the required i n i t i a l conditions. has a unique s o l u t i o n t - < ( t ) which i s defined for It\ < E, and s a t i s f i e s t h e i n i t i a l conditions
To o b t a i n t h e sharper statement i t i s only necessary t o observe that the d i f f e r e n t i a l equation for geodesics has t h e following homogeneity property.
c
Let
be any c o n s t a n t .
I f t h e parametrized curve
Furthermore, t h e s o l u t i o n depends smoothly on t h e i n i t i a l conditions. I n other words, t h e correspondence
Go,Yo,t ) from w x ( - E , E ) 2n+1 v a r i a b l e s .
to
is a
R~
3t ) ern f u n c t i o n
i s a geodesic, then the parametrized curve
+
of a l l
t
+
Y(Ct)
w i l l a l s o be a geodesic.
PROOF:
Introducing t h e new v a r i a b l e s
second order equations becomes a system of
v
i
=
dui
t h i s system of
Now suppose that
n
(tl
?n f i r s t order equations:
<
2
i s smaller than
E
<
I(v/E~II Hence we can define
and
t o be
yv(t)
( E 2 t ( <
2
.
yVlE2(~,t)
I/v((<
E
and
~
.
~
This proves l o . ? .
T h i s following n o t a t i o n w i l l be convenient.
The a s s e r t i o n then follows from Graves, "Theory of Functions of Real V a r i a b l e s , " p. 1 6 6 .
Then if
note t h a t
Let
v
E
TM 9
be a
tangent v e c t o r , and suppose that t h e r e e x i s t s a geodesic
(Compare our 5 2 . 4 . )
one obtains the following.
[O,ll
y:
Applying t h i s theorem t o t h e d i f f e r e n t i a l equation f o r geodesics,
- +M
s a t i s f y i n g the conditions Y(0)
LEMMA 1 0 . 2 . For every p o i n t po on a Riemannian manifold M t h e r e e x i s t s a neighborhood U of p, and a number E > 0 s o that: for each p E U and each tangent v e c t o r v E T% with length < E t h e r e i s a unique geodesic yV:
(-2,2)
Then t h e p o i n t
*
exponential
y(1)
E
M
=
9,
(,ody )
=
exp ( v )
w i l l be denoted by
of the tangent v e c t o r
v.
v. 4
The geodesic
and c a l l e d t h e y
can thus be des-
cribed by the formula
+M
r(t)
=
expq(tv)
.
s a t i s f y i n g t h e conditions YV(O)
PROOF.
=
P,
dYV
(0)
=
v
.
I f we were w i l l i n g t o r e p l a c e t h e i n t e r v a l
*
(-2,p)
by a n
a r b i t r a r i l y small i n t e r v a l , then t h i s statement would follow immediately from 1 0 . 1 .
To be more p r e c i s e ; t h e r e e x i s t s a neighborhood
U
of Po
The h i s t o r i c a l motivation for this terminology i s the following. I f M i s the ~ o u p of a l l n x n u n i t a r y matrices then the tangent space TMI a t the i d e n t i t y can be i d e n t i f i e d with t h e space of n x n skew-Hermitian matrices. The f u n c t i o n expI: TMI M -+
and
as defined above i s then given by t h e exponential power s e r i e s expI(A)
=
I + A + -&A2
+ &A3
+
... .
1
11. RIEMANNIAN GEOMETRY
58
-ma
10.2
says t b t
1n general,
exp ( v )
expq(v)
i s small enough.
IIvi/
i s not defined for l a r g e v e c t o r s
q
defined a t a l l ,
i s defined proving that
expq(v)
v.
However, i f
of
(p,p)
E
M x M.
of a l l p a i r s p
i s always uniquely defined.
i s defined f o r a l l
The manifold 9
E
M
i s geodesically complete
M
and a l l v e c t o r s
v
l e n t t o t h e following requirement: [a,bl
yo:
y :
(p,v) if
(u'
TM
with
i t should be p o s s i b l e
.
R +M
a f t e r proving some l o c a l r e s u l t s .
We w i l l r e t u r n t o a study of completeness Let
-M
t o an i n f i n i t e geodesic
To
p
be t h e tangent manifold of E
,. . . ,un)
M,
v
E
T%.
We give
M,
c o n s i s t i n g of a l l p a i r s
t h e following
TM
i s a coordinate system i n a n open s e t
tangent vector a t
q
E
a,
=
Lemma
a
says t h a t for each (q,v)
-
i s defined throughout a neighborhood
p
E
M
(P,O)
(q, exp,(v)).
i s non-singular.
U x U C M x M by
Then we have proven the following.
LEMMA
10.3. For each p E M t h e r e e x i s t s a neighborhood and a number E > 0 s o that: (1) Any two p o i n t s of W a r e joined by a unique geodesic i n M of l e n g t h < E . (2) T h i s geodesic depends smoothly upon t h e two p o i n t s . ( I . e . , if t - * e x p q l ( t v ) , 0 5 t 5 1 , i s t h e geodesic j o i n i n g q 1 and q,, then the p a i r ( q l , v ) E TM depends d i f f e r e n t i a b l y on ( q l ,q2) . ) each q E W t h e map expq maps t h e open diffeomorphically onto an open s e t
us 1 w. REMARK.
With more c a r e i t would be p o s s i b l e t o choose
W
the geodesic j o i n i n g any two o f i t s p o i n t s l i e s completely w i t h i n pare J . H . C . Whitehead,
Convex regions i n t h e geometry of paths,
s o that W.
Com-
Quarter-
l y J o u r n a l of Mathematics (Oxford) Vol. 3 , ( 1 9 3 2 ) , pp. 3 3 - 4 2 .
the map
Now l e t u s study t h e r e l a t i o n s h i p between geodesics and a r c - l e n g t h . of t h e p o i n t
V
Now consider the smooth f u n c t i o n =
so that
p
expq(v)
V.
more this map i s d i f f e r e n t i a b l e throughout
F(q,v)
of
of
then every
t ' a , +. . .+ tnan, ,..., tn c o n s t i t u t e
U can be expressed uniquely a s
s 10.2
structure:
C"
U C M
. Then the functions u1 ,..., u n , t ' '9 a coordinate system on t h e open set TU C TM.
where
W
U'
W
For every geodesic segment t o extend
belongs t o a given neighborhood
q
Choose a smaller neighborhood
E.
consists
V'
expq ( v )
T h i s i s c l e a r l y equiva-
T%.
E
if
such t h a t
/ / v / /<
and such that
59
We may assume that t h e f i r s t neighborhood
(q,v)
F(V1) 1 W x W . DEFINITION.
GEODESICS AND COMPLETENESS
§lo.
F:
V
-
(p,O)
E
TM.
Further-
THEOREM 1 0 . 4 .
Let
y :
M x M
We claim t h a t the Jacobian of
defined by F
a t the point
o:
be a s i n Lemma 1 0 . 3 .
E
[0,11
<
be t h e geodesic of l e n g t h and l e t
I n f a c t , denoting t h e induced coordinates on
(d ,.. . , u ~ , u1 2 ..., , u : ) ,
and
W
-*
[0,11
M
j o i n i n g two p o i n t s of
E
-*
Let
W,
M
be any other piecewise smooth p a t h j o i n i n g t h e same two p o i n t s . Then
we have
s 1
Il$lldt
F*(
- a- ) atJ
Thus t h e Jacobian matrix of
F
at
=
0
a a$
(p,O)
of
Thus
(p,o)
E
TM
,
0
c~([o,il)
has t h e form
It follows f r o m t h e i m p l i c i t f u n c t i o n theorem that V'
Il$lldt
where e q u a l i t y can hold only i f t h e p o i n t s e t coincides with y ( [ 0 , 1 1 ) .
hence i s non-singular.
neighborhood
5
F
m P S Some
diffeomorphically onto some neighborhood
y
i s t h e s h o r t e s t p a t h j o i n i n g i t s end p o i n t s .
The proof w i l l be based on two lermnas. be a s i n 1 0 . 3 .
Let
q = y ( o ) and l e t
Us
§TO.
11. RIEMANNIAN GEOMETRY
60 LEMMA 1 0 . 5 .
I n Uq, t h e geodesics t w o u g h q t h e orthogonal t r a j e c t o r i e s of hypersurfaces
{ expq(v)
:
61
Thus t h e s h o r t e s t p a t h j o i n i n g two concentric s p h e r i c a l s h e l l s
are around
q
1.
11v11= constant
v E TMq,
GEODESICS AND COMPLETENESS
i s a r a d i a l geodesic.
PROOF:
Let
f(r,t)
exp ( r v ( t ) ) , q
=
s o that
w(t)
=
f(r(t),t)
Then PROOF:
t - v(t)
Let
denote any curve i n
-
with
TMq
Ilv(t) I(
=
1,
We must show t h a t t h e corresponding curves
t in
us,
where
<
0
ro
<
a r e orthogonal t o t h e r a d i a l geodesics
E,
r
exp9( r v ( t O ) )
I n terms of t h e parametrized surface f(r,t)
Since the two v e c t o r s on t h e r i g h t a r e mutually orthogonal, and since llaFll ar = I , this gives
exPq(rov(t ))
where e q u a l i t y holds only i f
given by
f
o5r <
exp,(rv(t)),
=
. E
<
Zlat;
>
a =
0
Now
a
ar
ar ar
D
af
r ( t ) i s monotone and
r
<waFzE>
+
piecewise smooth path
from
w
q'
-, f ( r , t ) where
The second expression i s equal t o
0
<
r
<
llvll
E ,
af
Ilv(t) 11
=
r.
dent of
But for r
Therefore t h e q u a n t i t y
1.
= = 0
af
=(o,t)
Therefore
= 0.
ar ar <=,=.) i s indepen-
we have f(o,t)
hence
Then for any
6
>
0
=
exp ( 0 ) = q 9 ar i s i d e n t i c a l l y zero, which com-
m:
segment w i l l be
2r
- 6; hence l e t t i n g
<
r(t)
<
w(t) E,
and
Ilv(t)II
LEMMA 1 0 . 6 .
equal t o function
= 1,
The l e n g t h
-
v(t)
6
s", 11$11
does not coincide with
m i s t con-
t o the spherical The l e n g t h of this t h e l e n g t h of
0
7([O,ll),
m
then w e
T h i s completes the proof of 1 0 . 4 .
i s the following.
.
exp,(r(t)v(t))
with
PROOF:
Consider any segment of
above, and having l e n g t h i s @ e a t e r than
Or
where e q u a l i t y holds Only if the r(a)1, r ( t ) i s monotone, and t h e f u n c t i o n v ( t ) is constant. Ir(b)
tend t o
w
COROLL'WY 1 0 . 7 . Suppose t h a t a p a t h 0: [O,ll -. M, parametrized by arc- length, has l e n g t h l e s s than or equal t o t h e l e n g t h of any other p a t h from ~ ( 0 t)o m( l ) Then m i s a geodesic.
TMq. dt
w([O,ll)
6
An important consequence of Theorem 1 0 . 4
.
can be expressed uniquely i n t h e form
2 r . If
e a s i l y obtain a s t r i c t inequality.
ar <=,=>
[ a , b l * Uq - (q)
6
and l y i n g between t h e s e two s h e l l s .
w i l l be
Consider any
the p a t h
r,
Now consider any piecewise smooth curve
0
Thus
exp ( r v ) E Us ; 9
s h e l l of r a d i u s
p l e t e s t h e proof.
Each p o i n t
0.
t o a point
=
1.
=
q
t a i n a segment j o i n i n g the s p h e r i c a l s h e l l of r a d i u s
l a Fl l
=
v ( t ) i s constant.
The proof of Theorem 1 0 . 4 i s now straightforward.
af D a f
=
The f i r s t expression on t h e r i g h t i s zero s i n c e t h e curves
since
$
hence only i f
T h i s completes the proof.
3-F
a r e geodesics.
0;
a
where e q u a l i t y holds only i f
(r,t).
f o r all
=
,
we must prove that af af
af
10.4.
<
Hence t h e e n t i r e p a t h DEFINITION.
w
l y i n g w i t h i n an open s e t
W, as
T h i s segment must be a geodesic by Theorem
E..
m
A geodesic
i s a geodesic. y:
[a,bl
-. M w i l l
be c a l l e d minimal
if
about
its l e n g t h i s l e s s than or equal t o t h e l e n g t h of any other piecewise smooth
p.
Since
p0
Theorem 1 0 . 4 asserts t h a t any s u f f i c i e n t l y small segnent of a
on
S
expp(6v),
=
f o r which t h e d i s t a n c e t o
I f such a n a r c has l e n g t h p e a t e r than
I[,
it i s c e r t a i n l y
n o t minimal. I n general, m i n i m a l geodesics a r e not unique.
=
q.
T h i s implies that the geodesic segnent
t
-,
is a c t u a l l y a minimal geodesic f r o m p
to
However, t h e following a s s e r t i o n i s t r u e .
we w i l l prove that
Define t h e d i s t a n c e
p(p,q)
between two p o i n t s
p,q
E
M
t o be t h e
T h i s c l e a r l y makes
M
i n t o a metric space.
T h i s identity, f o r
C O R O W Y 10.8. Given a compact s e t K C M t h e r e e x i s t s a number 6 > 0 s o t h a t any two p o i n t s of K w i t h d i s tance l e s s than 6 a r e joined by a unique geodesic of l e n g t h l e s s than 6 . Furthermore this geodesic i s m i n i m a l ; and depends d i f f e r e n t i a b l y on i t s endpoints.
small enough s o that any two p o i n t s i n
K
as i n
10.3,
t
=
r,
t
5
r,
t
E
[6,rl
.
r-t
w i l l complete the proof.
and l e t
with d i s t a n c e l e s s than
p
to
q
must pass through
p(p,q) = f i n
(P(P,s)
+
i s true.
(18)
Therefore
p(p0,q) = r - 6 .
to E [ 6 , r l
Let
i s true.
If
be
6 6
to < r
p(s,q))
= 6 +
p o i n t of
M i s g e o d e s i c a l l y complete i f every geo-
po
p(po,q)
6
about t h e p o i n t
with minimum d i s t a n c e from = fin
P(7(t0),ci)
Let
7(t0) ;
S'
f o r which
i s true also.
)
(1
denote a s m a l l spheri-
and l e t
p;
be a
E S'
(Compare D i a g r a m l o . )
q.
I ( ~ ( 7 ( t 0 ) , +s )p(s,q)) = 6 1 + p(p;,q)
S € S
d e s i c segment can be extended i n d i f i n i t e l y .
t
denote the supremum of those numbers
w e w i l l obtain a contradiction.
S'
.
this prqves ( I 6 ) .
= 7(6),
Then by c o n t i n u i t y t h e e q u a l i t y
c a l s h e l l of r a d i u s
lie
Since
Since every
w e have
S,
S E S
This completes t h e proof.
R e c a l l t h a t t h e manifold
5
0
I n f a c t for each
q.
F i r s t we w i l l show t h a t the e q u a l i t y
(It)
by open s e t s W,,
expp(tv),
I t follows p a t h from
K
7(t) =
q.
to
~(7(t),9)=
e a s i l y from 1 0 . 4 t h a t this metric i s compatible with t h e usual topologg of M.
Cover
must g e t c l o s e r and c l o s e r
7
(It)
g r e a t e s t lower bound f o r t h e a r c - l e n g t h s of piecewise smooth paths j o i n i n g
in a common W,.
We w i l l prove that
The proof w i l l amount t o showing that a p o i n t which moves along the
For example two a n t i geodesic
PROOF.
1,
=
i s minimized.
q
expp(rv)
podal p o i n t s on a u n i t sphere a r e joined by i n f i n i t e l y many minimal geodesics.
these points.
IIvII
On t h e other hand a long geodesic may n o t be minimal.
For example we w i l l see s h o r t l y t h h t a p e a t c i r c l e a r c on t h e u n i t sphere
i s a geodesic.
63
i s compact, t h e r e e x i s t s a p o i n t
S
p a t h j o i n i n g i t s endpoints.
geodesic i s minimal.
GEOD_ESICSAND COMPLETENESS
§ 10.
11. RIEMANNIAN GEOMETRY
62
Then
,
hence THEOREM 1 0 . 9 (Hopf and Rinow*). I f M is geodesically complete, then any two p o i n t s can be joined by a minimal geodesic. PROOF. Up
*
Given
p,q
as i n L e m 10.3. L e t
E
M
S C U
with distance P
r
>
(2)
p(p;,q)
PA
W e claim that
=
(r
-
i s equal t o
to) -
7(t0
6'
.
+ sl).
I n f a c t the t r i a n g l e
i n e q u a l i t y states t h a t 0,
choose a neighborhood
denote a s p h e r i c a l s h e l l of r a d i u s
6
<
P(P,Pd)
E
Compare p . 341 of G . de Rham, Sur l a r e d u c t i b i l i t 6 d ' u n espace de Riemann,Comentarii Math. H e l v e t i c i , Vol. 26 (1952); a s w e l l as H. Hopf and W. Rinow, Ueber den B e g r i f f d e r - v o l l s t h d l g e n differentialgeometrischen F l i c h e , C m e n t a r i i , V o l . 3 (1931), pp. 209-225.
(making use of (2)). PA
2 P(P,q) - P(P;),q)
=
to + 6'
But a p a t h of l e n g t h p r e c i s e l y
obtained bY following
7
fmm
P
to
7(t0),
to + 6' from p t o and then following
a minimal geodesic from Y ( t , ) t o Pd. Since t M s broken geodesic has minimal length, i t f o l l o w s f r o m C o r o l l a r y 1 0 . 7 that i t I s an (unbroken)
FAMILIAR EXAMPLES OF GEODESICS.
and hence coincides with 7 . Now t h e e q u a l i t y ( 2 ) becomes Thus 7 ( t 0 + 8 ' ) = p;. p(7(t0
+ s'),q)
T M c~o n t r a d i c t s t h e d e f i n i t i o n of
=
r - ( t o+
to;
GEODESICS AND COMPLEIENESS
§ 10.
11. FJ3IMANNIAN GEOMETRY
64
I n Euclidean n-space,
Rn,
with
x,, ... ,xn and the u s u a l Riemannian metric dxn we have r& = 0 and t h e equations for a geo-
t h e usual coordinate system
.
8 ' )
65
dxl
@
desic
and completes the proof.
.
dx, +. .+ dxn @ 7,
given by
t
-
. . ., x n ( t )
(x, ( t ) ,
become
2
xi = O J dt whose s o l u t i o n s a r e t h e s t r a i g h t l i n e s . TMs could a l s o have been seen as T
'I1
follows:
i t i s easy t o show that the formula for a r c l e n g t h
coincides with t h e u s u a l d e f i n i t i o n of a r c length as t h e l e a s t upper bound of t h e lengths of inscribed polygons;
from this d e f i n i t i o n i t i s c l e a r that
s t r a i g h t l i n e s have minimal length, and a r e t h e r e f o r e geodesics. The geodesics on i n t e r s e c t i o n s of
PROOF.
D i a p a m 10
Reflection through a plane
whose f i x e d point s e t i s
A s a consequence one has t h e following.
with a unique geodesic
C
between C'
.
I(x)
=
x
and
Sn n E 2 .
=
Let
x
I(C1)
I(y)
=
y.
Sn.
i s an isometry
E2
and
y
I:
C'
=
-
Then, since
i s a geodesic of the same length a s Therefore
Sn
be two p o i n t s of
of minimal length between them.
C'
i s an isometry, the curve COROLLARY 1 0 . 1 0 . If M i s geodesically complete then every bounded subset of M has compact closure. Consequently M i s complete as a metric space ( i . e . , every Cauchy sequence converges)
are p r e c i s e l y t h e g r e a t c i r c l e s , t h a t i s , t h e
Sn
Sn with the planes through t h e center of
I(C').
Sn C
I
C'
T h i s implies that
c c. F i n a l l y , since t h e r e i s a g r e a t c i r c l e through any point of
Sn
in
any given d i r e c t i o n , these a r e a l l t h e geodesics. PROOF.
emp:
5 - M
If
X C M
has diameter
maps t h e d i s k of r a d i u s
d d
then for any in
TMp
X
E
X
t h e map
onto a compact subset
of M which (making use of Theorem 1 0 . 9 ) contains X. of
p
Hence t h e closure
i s compact. Conversely, i f
Antipodal p o i n t s on t h e sphere have a continium of geodesics of minimal length between them.
All other p a i r s of p o i n t s have a unique geo-
d e s i c of minimal l e n g t h between them, but an i n f i n i t e family of non-minimal geodesics, depending on how many times t h e geodesic goes around t h e sphere
M
i s complete as a metric space, then i t i s not
d i f f i c u l t , using Lemma 1 0 . 3 , t o prove that
M
and i n which d i r e c t i o n i t starts.
i s geodesically complete.
For d e t a i l s t h e reader i s r e f e r r e d t o Hopf and Rinow.
Henceforth we w i l l
not d i s t i n g u i s h between geodesic completeness and metric completeness, b u t w i l l r e f e r simply t o a complete Riemannian manifold.
By t h e same reasoning every meridian l i n e on a surface of revolution i s a geodesic.
The geodesics on a r i g h t c i r c u l a r cylinder
Z a r e t h e generating l i n e s , t h e c i r c l e s cut by planes perpendicular t o t h e generating l i n e s , and
11. RIEMANNIAN GEOMETRY
66 the h e l i c e s on
67
Z.
PART I11
/----
THE CALCULUS OF VARIATIONS APPLIED TO GEODESICS
511.
Let
The Path Space of a Smooth Manifold.
be a smooth manifold and l e t
M
-
s a r i l y d i s t i n c t ) points of M. If
PROOF:
isometry
I:
Z
-
is a generating l i n e of Z t h e n w e can set up an 2 by r o l l i n g Z onto R2: L *R L
w i l l be m e a n t a map 1')
[0,11
2)
M
01
m(0)
=
0 =
to < t,
by I
-.
,-
I
I
L
R(M;p,q),
o r b r i e f l y by
Later ( i n §16)
n
in R
2
.
Z
a r e j u s t the images under
Two p o i n t s on
Z
I-'
of the straight l i n e s
have i n f i n i t e l y many geodesics between them.
<
... <
tk
=
1
of
Cm;
or
R(M)
to
q
in M
w i l l be denoted
R.
w i l l be given the s t r u c t u r e of a topological
space, but for the moment this w i l l not be necessary. as being something l i k e an
The geodesics on
from p t o q
q.
~ ( l )=
The s e t of a l l piecewise smooth paths from p L
be two ( n o t neces-
i s d i f f e r e n t i a b l e of c l a s s
[ti-,,ti] and
p
q
such that
there e x i s t s a subdivision
s o that each
and
By a piecewise smooth p a t h
[o,ll
u):
p
We w i l l think of
" i n f i n i t e dimensional manifold."
R
To start the
analogy we make the following d e f i n i t i o n .
By the tangent space
of
0
a t a path
w i l l be m e a n t t h e vector
u)
space consisting of a l l piecewise smooth vector f i e l d s W which W(0) =
and
0
W ( 1 ) = 0.
The n o t a t i o n
along
u)
for
TRu) w i l l be used for this
vector space. If
F
is a real valued function on F*:
i t is n a t u r a l t o a s k w h a t
R
Tau) +TRRu))
,
the induced map on the tangent space, should mean.
When F
is a function
which is smooth i n the usual sense, on a smooth manifold
F,: TMp - T R R p ) u -+ a ( u ) i n M,
as follows.
Given X
which is defined for ff(0)
=
P,
c -E
TMp
M, we can define choose a smooth p a t h
m(0) dff =
E
x .
,
s o that
60
$1 1
111. CALCULUS OF VARIATIONS
Then F,(X)
i s equal t o
(-&)F(p)
mF(P)
d(Fi:(U)) l u =o, m u l t i p l i e d by the b a s i s vector
f u n c t i o n on
il,
.
THE PATH SPACE
we attempt t o d e f i n e
F,: as follows.
*
n
I n order t o c a r r y out a n analog9us c o n s t r u c t i o n for F:
Given
W
f
TOw
-
69
mq,)
Taw choose a v a r i a t i o n
R, E(0)
=
&dO5)
a,
=
E:
(-E,E)
+
n
with
w
t h e following concept i s needed.
F,(W)
and set A v a r i a t i o n of
DEFINITION.
(keeping endpoints f i x e d ) i s a
o
(
function
6: f o r some
>
E
1)
[o,il
t o have these p r o p e r t i e s .
o
defined by
i
= 1,
to < t l
0 =
cr(u,t)
x [0,11
(-E,E)
c"
6(u)(t) is
=
... <
tk =
-+
M
F:
on each s t r i p
(-E,E)
X
[ti-,,ti],
3)
p,
=
O(U,O)
belongs t o a(u,l)
W e w i l l use e i t h e r
a
=
of
t i o n of -
a.
6
Now vector''
i n Rn,
0
d6 =(o)
then
a
R(M;p,q),
for a l l
q
or
R =
u
E
note that:
(or
(-E,E.)
6)
Trio
More
i s replaced by a neighbor-
i s c a l l e d an n-parameter v a r i a n.
may be considered as a "smooth path" i n E
i s defined t o be t h e v e c t o r f i e l d
W
Its "velocity along
o
given
by Wt
Clearly
W
E
TRo.
=
&It
=
T h i s vector f i e l d
W
t o r f i e l d associated w i t h the variation Given any W
6:
(-E,E)
-+
R
E
an
,(o,t)
*
i s a l s o called the variation=a.
TRo note that t h e r e e x i s t s a v a r i a t i o n
which s a t i s f i e s t h e conditions
5(0) =
w,
d5 ~ ( 0 ) =
W.
In f a c t one can set .(u)(t)
=
expo(t)(u Wt)
.
By analogy w i t h t h e d e f i n i t i o n given above, i f
F
F
5. must s a t i s f y i n order for F,
Ere have i n d i c a t e d how
F, might be defined only
-
R
i s a r e a l valued
A path
if and only i f
o
i s a c r i t i c a l p a t h for a f u n c t i o n
l u F0
i s zero for every v a r i a t i o n
6
of
o.
derivatives
.
(-E,E)
5 t o r e f e r t o the variation.
generally i f , i n t h e above d e f i n i t i o n ,
p
R
DEFINITION.
EXAMPLE. 6(u)
t h e r e i s no guarantee t h a t
t o motivate t h e following.
1
...,k. Since each
hood
<
that t h e map
a:
lu=o m u l t i p l i e d by t h e tangent vector
W e w i l l not i n v e s t i g a t e w h a t conditions
t h e r e i s a subdivision SO
d(F(:(u)) u
equal t o
Of course without hypothesis on F
this d e r i v a t i v e w i l l e x i s t , or w i l l be independent of t h e choice of
R,
4
such that
0,
6(0) =
2)
of
(-E,E)
. )F(m)
If
F
takes on i t s minimum a t a p a t h
a r e a l l defined, then c l e a r l y
oo
mo,
and i f the
i s a c r i t i c a l path.
CALCULUS OF VARIATIONS
111.
70
THE FNERGY OF A PATH
$12.
71
takes on i t s m i n i m u m d2 p r e c i s e l y on t h e s e t o f m i n i m a l geodesics from p t o q.
Suppose now that tor
v
M
i s a Riemannian manifold.
TMP w i l l be denoted by
E
energy of
from a t o b
w
We w i l l write
E
We w i l l now see which p a t h s
The Energy of a Path.
$12.
\\v(\
(where
0
=
5a
< v,v >*. < b 5 1)
energy f u n c t i o n Let E :
The l e n g t h of a vec-
For
Eo. to
Vt
=
At
=
AtV
=
v e l o c i t y v e c t o r of
=
D dm
=
-
Vt-
Vt+
AtV
THEOREM
=
Wt
=
=a (uo , t )
Furthermore, l e t :
,
w
w
,
d i s c o n t i n u i t y i n the v e l o c i t y v e c t o r a t
=
<
o
t
<
t,
.
1
f o r a l l b u t a f i n i t e number of values of
0
12.2
and l e t
o,
a c c e l e r a t i o n v e c t o r of
where
a
( F i r s t v a r i a t i o n formula).
t.
The d e r i v a t i v e
Applying Schwarz's i n e q u a l i t y
a f(t)
be a v a r i a t i o n of
given by
b
Of course
with
dw
are c r i t i c a l paths f o r the
E 0
be the a s s o c i a t e d v a r i a t i o n v e c t o r f i e l d .
as
T h i s can be compared with the a r c - l e n g t h from a
as follows.
(-E,E)
define t h e
w E 0
1
for
-
E.
o
1
=
and
a
g(t)
=
11~11
a PROOF:
I
we see that
According t o Lemma 8 . 3 , we have
a
aa aCY <=,x >
=
2
D aa aa <Jiix,x >.
Theref ore where e q u a l i t y holds i f and only i f
t
t h e parameter
g
i s constant; t h a t i s i f and only if
q
= ~ ( 1 ) .
from p
0
= ~ ( 0 )
Then
Here t h e e q u a l i t y
L( 7 )
=
=
L ( 7 ) 2 5 L ( o ) ~5 E(w) L(w)
geodesic, possibly reparametrized. L(w)
toarc-lengthalong
a minimal geodesic.
D
=
w.
E( w)
Choose
.
each s t r i p
i s a l s o a minimal
can hold only i f
w
(Compare $ 1 0 . 7 . )
On t h e other hand
[ti-l,til,
E(7)
<
E(w)
unless
w
-
as follows.
a
i s also
implies that
M be a complete Riemannian manifold have d i s t a n c e d. Then t h e energg
E: n(M;p,q)
tl
<...<
tk =
D for Jii 1
~ C Y
so t h a t
x [ti-1, t i ] . Then we can
aa
i n t h i s l a s t formula. CY
i s d i f f e r e n t i a b l e on
" i n t e g r a t e by p a r t s " on
The i d e n t i t y
aCY
73€ c 3 i m
I n other words:
LFNM!l 1 2 . 1 . Let and l e t p,q E M function
(-E,E)
<
&
i
can hold only i f t h e parameter i s p r o p o r t i o n a l
T h i s proves that
0 = to
~ C Y~ C Y
0
By Lemma 8.5 we can s u b s t i t u t e E(Y)
the equality
7
1
d
du
Now suppose that t h e r e e x i s t s a minimal geodesic to
1
is p r o p o r t i o n a l t o a r c - l e n g t h .
R !
>
=
+
aa D aa (2ii.xx
>
Adding up t h e corresponding formulas f o r aa
that
=
for t
0
= 0
or
i
=
1,
. . . , k;
PROOF:
and using t h e f a c t
c r i t i c a l point.
t h i s gives
1,
f(t)
1
dE(6(u)) du 2 1
aa
=
i=l
u
= 0,
There i s a v a r i a t i o n of
1 ~m (
= 0 )
I n t u i t i v e l y , t h e f i r s t term i n t h e expression POP i n the d i r e c t i o n of decreasing
dE.6
T(o)
/
\
k'
\
path E(E) I smaller em
Diagram 1 1 .
1
The second term shows that varying the curve i n t h e d i r e c t i o n cf i t s a c c e l e r a t i o n vector D tends t o reduce E .
(2)
a,
of
is a,
Cm
(o
A(t)
E
on t h e whole i n t e r v a l
i s i d e n t i c a l l y zero along
n
i s c a l l e d a geodesic
[O, 11,
i f and only if
and t h e a c c e l e r a t i o n v e c t o r
(o.
COROLLARY 1 2 . 3 . The p a t h (o f u n c t i o n E i f and only i f
i s a c r i t i c a l point for the i s a geodesic.
0)
D
dm
m(x)
i s d i f f e r e n t i a b l e of c l a s s
everywhere:
"kink," tends t o
.
R e c a l l that t h e p a t h
f (t ) A ( t)
where
Then
> dt.
A(t),A(t)
=
c
for a l l
t.
Hence each
W(ti)
=
at2V.
(ol[ti,ti+ll
Then
C',
even a t the p o i n t s
ti.
from the uniqueness theorem f o r d i f f e r e n t i a l equations that
shows
path
/
-!f ( t ) <
=
be a
0 a,
E ; see D i a g r a m 1 1
ti.
1
Now p i c k a v a r i a t i o n such that
This completes t h e proof.
decrease
W( t )
0)
i s a geodesic.
1
(o
with
Let
0
we now o b t a i n t h e required formula
that varying the p a t h
(o
i s p o s i t i v e except t h a t i t vanishes a t the
D aa
0
t
73
Clearly a geodesic i s a c r i t i c a l p o i n t .
This i s zero i f and only i f Setting
THE ENERGY OF A PATH
§12.
CAlLXLUS OF VARIATIONS
111.
72
thus
(o
i s a n unbroken geodesic.
Now i t follows (o
is
Cm
-
Continuing with t h e analogy developed i n t h e preceding s e c t i o n , we now wish t o d e f i n e a b i l i n e a r f u n c t i o n a l
-
E**: TnY x TO Y when
i s a c r i t i c a l p o i n t of t h e f u n c t i o n
y
point
i . e . , a geodesic.
E,
can be defined as follows. +
values i n
x TiV$
E
at
y .
M
where
with c r i t i c a l
V
=
Given
X,,X,
c
T%
(0,O)
of the energy
in
(0,O)
DWl
=
f i e l d and where
Dwl
E ( t + )- x(t-)
denotes t h e jump i n DW1 a t one of i t s f i n i t e l y many p o i n t s of d i s c o n t i n u i t y i n the open u n i t i n t e r v a l .
choose a smooth map
a ( u l , u 2 ) defined on a neighborhood of M,
a2E
$& denotes t h e v e l o c i t y vector At
R
-r
1
.
This
DW,
f,,:
(u1,u2)
Then t h e second d e r i v a t i v e
then t h e Hessian T%
6: U-r n with
f u n c t i o n i s equal t o
f i s a r e a l valued f u n c t i o n on a manifold
p,
75
R
b i l i n e a r f u n c t i o n a l w i l l be c a l l e d t h e Hessian of If
E
THEOREM 1 3 . 1 (Second v a r i a t i o n formula). L e t be a 2-parameter v a r i a t i o n of t h e geodesic y v a r i a t i o n vector f i e l d s Wi = xa6 i(O,O) E TRY, i = 1,2
The Hessian of t h e Energy Function a t a C r i t i c a l Path.
$ 1 3.
THE HESSIAN OF
S13.
CALCULUS OF VARIATIONS
111.
74
2
R , with
s o that
PROOF:
According t o 1 2 . 2 we have
Then Theref ore
T h i s suggests d e f i n i n g
as follows.
E,,
Given v e c t o r f i e l d s
W1,W2
E
TnY
-f<
choose a 2-parameter v a r i a t i o n
a: where
U
U x
i s a neighborhood of
4O,O,t)
(Compare 5 1 1 . )
Y(t),
=
[O,ll
(0,o)
aa
x 1(O,O,t)
Then the Hessian
-r
M
i n R2, =
,
0
L e t us evaluate this expression
s o that
Wl(t),
aa
an unbroken geodesic, n e have
(O,O,t)
=
.
W2(t)
=2
E*+(W1,W2) w i l l be defined t o be t h e
second p a r t i a l d e r i v a t i v e
Rearranging t h e second term, we obtain a2E(E(ul,u2) ) au, au2
where
n
1 ; (0,O)
6(ul,u2)(t) = a2E -(O,O) U1 u2 The following theorem i s needed t o prove that
E(u1,u2)
E
so that t h e f i r s t and t h i r d terms a r e zero.
denotes t h e p a t h
a(u1,U2,t)
.
This
I n order t o interchange t h e two o p e r a t o r s
.
second d e r i v a t i v e w i l l be w r i t t e n b r i e f l y as
E,,
b r i n g i n the curvature formula, 1s
W e l l
defined.
dU1
and
$ ,
we need t o
CALCULUS OF VARIATIONS
111.
76
Together with t h e i d e n t i t y
D =,V
=
D aa J - - F ~ ,
=
JACOB1 FIELDS
$14.
zDw l ,
77
this y i e l d s
(13.3)
Jacobi F i e l d s :
514.
S u b s t i t u t i n g this expression i n t o ( 1 3 . 2 ) this completes t h e proof of 1 3 . 1 a ‘E = .-(o,o) COROILARY 1 3 . 4 . The expression E,,(W1,W2)
.
A vector f i e l d
i s w e l l defined.
E,,(W1,W2)
and
W,,
T h i s formula a l s o shows that
D2J + R ( V , J ) V
where
so that
i s bilinear.
E,,
along a geodesic
J
V
=
i s c a l l e d a Jacobi f i e l d
2.
=
dtP
0
T h i s i s a l i n e a r , second order d i f f e r e n t i a l equation.
[ I t can be p u t i n a more f a m i l i a r form by choosing orthonormal p a r a l l e l vector fields
The symmetry property
y
E,,
i f i t s a t i s f i e s the Jacobi d i f f e r e n t i a l equation
i s a well defined symmetric and b i l i n e a r f u n c t i o n of W1 and W,. a2E PROOF: The second v a r i a t i o n formula shows that -(O,O) depends only on the v a r i a t i o n vector f i e l d s W1
The Null Space of
Pl,
..., Pn
along
Then s e t t i n g J ( t )
y .
c
=
fi(t)Pi(t),
the
equation becomes E*,(W1 ,W2)
E**(W2, Wl )
=
i s not a t a l l obvious from the second v a r i a t i o n f o r n u l a ; b u t does follow a2E a2E immediately from the symmetry property dU1dU2 - 7 q - q . REMARK 1 3 . 5 . E,,
The diagonal terms
of t h e b i l i n e a r p a i r i n g
E,,(W,W)
can be described i n terms of a 1-parameter v a r i a t i o n of
7.
In fact
<
a? = R(V,P.)V,Pi> . I Thus the Jacobi equation has 2n l i n e a r l y J J independent s o l u t i o n s , each of which can be defined throughout 7 . The
where
solutions are a l l
(?-differentiable.
given Jacobi f i e l d
A
J
i s com-
p l e t e l y determined by i t s i n i t i a l conditions: J(O),
where field
6: d5 =(o)
(-E,E)
+
n
denotes any v a r i a t i o n of
equal t o W .
7
with v a r i a t i o n vector
To prove t h i s i d e n t i t y it i s only necessary t o
Let with
p
and
= y(a)
q
=
DJ
~ ( 0 E )TMr(o)
y(b)
*
be two p o i n t s on t h e geodesic
7,
a f b.
introduce t h e two parameter v a r i a t i o n B(u1,u2)
=
DEFINITION.
6(Ul + u,)
non-zero Jacobi f i e l d
and t o note t h a t
The m u l t i p l i c i t y of
and
p J
p
q
along and
q
a r e conjugate, along y
i f there e x i s t s a
y
which vanishes f o r
t
=
a
and
t
=
b.
a s conjugate p o i n t s i s equal t o t h e dimen-
sion of t h e v e c t o r space c o n s i s t i n g of a l l such Jacobi f i e l d s . A s an a p p l i c a t i o n of t h i s remark, we have t h e following.
Now l e t LEMMA 1 3 . 6 . I f 7 i s a m i n i m a l geodesic from p t o q i s p o s i t i v e semi-definite. then t h e b i l i n e a r p a i r i n g E,, Hence t h e index h of E,, i s zero.
y
be a geodesic i n
R =
The i n e q u a l i t y
E(G(u))
2 E(7)
=
E(E(o))
R e c a l l that t h e
s-
space of the Hessian E,,:
Tny x TO - R
i s t h e vector space c o n s i s t i n g of those PROOF:
R(M;p,q).
7
W1 E Tn
implies that
*
If
7
such that
Y
E,,(W1,W2)
has s e l f - i n t e r s e c t i o n s then this d e f i n i t i o n becomes ambiguous.
One should r a t h e r say that t h e parameter values with respect t o y .
a
and
b
are conjugate
= 0
111. CALCULUS OF VARIATIONS
78
f o r a l l W,.
The n u l l i t y
n u l l space.
of
v
i s degenerate
E,,
i s equal t o t h e dimension of this
E,,
if
514.
>
v
entire unit interval.
JACOB1 FIELDS
T h i s completes t h e proof of 1 4 . 1 .
It follows that t h e n u l l i t y
0.
79
of
v
i s always f i n i t e .
E,,
THEORFM 1 4 . 1 . A v e c t o r f i e l d W1 E Tn belongs t o t h e 7 n u l l space of E,, i f and only i f W1 i s a Jacobi f i e l d . Hence E,, i s degenerate i f and only i f t h e end p o i n t s p and q a r e conjugate along 7 . The n u l l i t y of E,, i s equal t o t h e m u l t i p l i c i t y of p and q a s conjugate p o i n t s .
REMARK 1 4 . 2 .
Actually t h e n u l l i t y
satisfies
v
t
the space of Jacobi f i e l d s which vanish f o r precisely
n,
i t i s c l e a r that
v
5
n.
vanishes a t
(Compare the proof of 1 2 . 3 . )
p
and
q,
then
1
Hence
c e r t a i n l y belongs t o
1
+
<W2(t),0>
=
Conversely, suppose t h a t Choose a subdivision
WII [ti-,,til
Tn7
.
The second
0 =
to
<
tl
i s smooth f o r each
W1
f
= 1.
T h i s w i l l imply that
V
=
denotes t h e v e l o c i t y v e c t o r f i e l d .
a T -dt
<W2,0>
.
0
=
(Since DV iTt; = o ) ,
0
<...<
tk =
Let
i.
f:
[o,ll
of
1
[0,11
-
of
J
E,,.
[O,ll
t
1.
u n i t sphere Since this i s zero, i t follows that
i s a Jacobi f i e l d f o r
WII[ti-,,til
= 1,2
,..., k-1.
E
Tn
7
be a f i e l d such t h a t
n-1.
DW1
W;(ti)
Suppose that
-
=
Ati
Sn,
for
Then
p
and l e t
and
has no jumps.
k
R(V,J)V
=
tR(V,V)V
Since
Jo
= 0,
J,
#
Thus this
0,
Then the Jacobi equation becomes
p
a r e a n t i p o d a l p o i n t s on t h e
q
and
q
p to
q.
a r e conjugate w i t h m u l t i p l i c i t y
Thus i n this example t h e n u l l i t y
v
of
E,,
takes i t s
The proof w i l l depend on the following
discussion.
But a s o l u t i o n W1
completely determined by t h e v e c t o r s lows that t h e
v
be a p e a t c i r c l e a r c from
y
a
be a 1-parameter v a r i a t i o n of
t h e endpoints f i x e d , such t h a t each Hence
Then
i s " f l a t " i n t h e sense that t h e curva-
M
Suppose that
l a r g e s t p o s s i b l e value.
Let
DW1
where
i s skew symmetric i n the f i r s t two v a r i a b l e s .
R
Then w e w i l l see that
i. I
Jt = tVt
non-degenerate.
1
EXAMF'LF 2 .
W2
Since
b u t not f o r
= 0,
Furthermore
0.
n.
D ~ J= 0 . S e t t i n g J ( t ) = C f i ( t ) P i ( t ) where Pi a r e p a r a l l e l , dt this becomes d2fi = 0 . Evidently a Jacobi f i e l d along 7 can have dt a t most one zero. Thus t h e r e a r e no conjugate p o i n t s , and E,, is
-
Now l e t
=
ture tensor i s i d e n t i c a l l y zero.
and i s
Dositive otherwise; and l e t
each
2
D2J
<
has dimension
= 0
In fact l e t
14J+tg
z
hence
n.
s a t i s f i e s the Jacobi equation.
EXAMPLE
be a smooth
,,...,t k
v
v
completes the proof.
s o that
tO,t
f u n c t i o n which vanishes f o r t h e parameter values
Then
since
= 0
belongs t o the n u l l space
<
t
=J
t belongs t o t h e n u l l space.
J
i s a Jacobi f i e l d which
J
states that
v a r i a t i o n formula ( $ 1 3 . 1 )
- 7E,,(J,W2)
J
If
5
0
7.
We w i l l c o n s t r u c t one
example of a Jacobi f i e l d which vanishes f o r
PROOF:
For
t h e r e a r e only f i n i t e l y many l i n e a r l y independent Jacobi f i e l d s along
W1 (ti) and
Jacobi f i e l d s W I I [ t i - l , t i l ,
t o give a Jacobi f i e l d W 1 which i s
z(ti).
of t h e Jacobi equation i s
i = 1,
C"-differentiable
Thus i t fol-
...,k,
f i t together
throughout t h e
a:
be a by
Cm map such that
O(u) ( t )
=
a(u,t)l
(-E,E)
a(0,t)
=
7,
not n e c e s s a r i l y keeping
z(u) i s a geodesic. x [O,lI y(t),
i s a geodesic.
That i s , l e t
+M and such that each
6(u)
[given
111. CALCULUS OF VARIATIONS
80
LEMMA
14.3. If a i s such a v a r i a t i o n of geodesics, then t h e v a r i a t i o n vector f i e l d i s a Jacobi f i e l d along y .
PROOF:
If
i s i d e n t i c a l l y zero.
a
i s a v a r i a t i o n of
7
y
through
W( t )
=
%(
LFXVA 1 4 . 4 .
Every Jacobi f i e l d along a geodesic y : [O,ll - M may be obtained by a v a r i a t i o n of y through geodesics.
0 , t)
D through geodesics, then x
aa x
Hence
PROOF:
of
D D
aa
Suppose that
c o n s t r u c t a Jacobi f i e l d
a(o) Therefore t h e v a r i a t i o n vector f i e l d
t
=
~ ( 0 )and
t
and
= 0
b:
y(t)
along
W
yl
for
U
E
[o,sl
-
(-E,E)
a:
to
b(u).
a(u):
5
t
5
We w i l l f i r s t
6.
a:
-U
(-E,E)
so that
g(0)
Thus one way of obtaining Jacobi f i e l d s i s t o move geodesics around. by l e t t i n g
0
with a r b i t r a r i l y prescribed
i s any prescribed v e c t o r i n TMr(o). db U with b ( 0 ) = y ( 6 ) and )O(, arbitrary.
Now d e f i n e the v a r i a t i o n
field.
s o that any two p o i n t s
y(0)
Choose a curve
= 6.
so that
S i m i l a r l y choose
i s a Jacobi
of
U
U a r e joined by a unique minimal geodesic which depends d i f f e r e n t i a b l y
values a t
(Compare 9 1 3 . 3 . )
Choose a neighborhood
on t h e endpoints.
D D aa
81
JACOB1 FIEU)S
$14.
-
[O,61
M
-
M
be t h e unique minimal geodesic from
t
Then the formula
[0,61
(-€,El
+
%(o,t)
a(u)
d e f i n e s a Jacobi f i e l d with the
given end conditions. Any Jacobi f i e l d along
formula
W
-
vector f i e l d along t h e geodesic and
q.
p
and
g
P ( Y ) +TMy(o)
w i l l be a Jacobi f i e l d vanishing a t
y
We have j u s t shown that P
fixed, the variation p
Rotating i n n-1 d i f f e r e n t d i r e c t i o n s one obtains n-1 l i n e a r l y
dimension
2n
along
7,
If
then the
d e f i n e s a l i n e a r map
(W(O), w ( 6 ) )
Now l e t u s r e t u r n t o the example of two a n t i p o d a l p o i n t s on a u n i t Rotating the sphere, keeping
can be obtained i n this way:
[O,sl
denotes t h e vector space of a l l Jacobi f i e l d s W
,y(y)
n-sphere.
y (
i s onto.
i t follows that
i s determined by i t s values a t
P
x TMy(6)
Since both v e c t o r spaces have t h e same
i s a n isomorphism.
y(0)
and
~ ( 6 ) .
I . e . , a Jacobi f i e l d
(More g e n e r a l l y a Jacobi
f i e l d i s determined by i t s values a t any two non-conjugate p o i n t s . )
There-
f o r e t h e above c o n s t r u c t i o n y i e l d s a l l p o s s i b l e Jacobi f i e l d s along 71 [O,6l.
The r e s t r i c t i o n of If
u
Xu)
t o the interval
[0,6l
i s not e s s e n t i a l .
i s s u f f i c i e n t l y small then, u s i n g t h e compactness of
[o,ll,
can be extended t o a geodesic defined over t h e e n t i r e u n i t i n t e r v a l
$u) [0,11.
This y i e l d s a v a r i a t i o n through geodesics: a':
(-€',El)
X
[O,ll
-*
M
w i t h any given Jacobi f i e l d as v a r i a t i o n v e c t o r . independent Jacobi f i e l d s . multiplicity
n-1
.
Thus
p
and
q
a r e conjugate along
7
with RFJURK 14.5.
This argument shows that i n any such neighborhood U U a r e uniquely determined
t h e Jacobi f i e l d s along a geodesic Segment i n
1
111.
a2
CALCULUS OF VARIATIONS
515.
THE INDEX THEOREM
515.
The Index Theorem.
a3
by t h e i r values a t t h e endpoints of t h e geodesic.
REMARK 1 4 . 6 . (-6,b)
of
7 ( o ) along 7(o)
o
The proof shows a l s o , that t h e r e i s a neighborhood
s o that if t c 7.
(-6,6)
then
7(t)
i s not conjugate t o
The index
of t h e Hessian
X
We w i l l see i n 515.2 that t h e s e t of conjugate p o i n t s t o
along the e n t i r e geodesic
7
T O 7 x TO 7 + R
E,,:
has no c l u s t e r p o i n t s .
i s defined t o be the maximum dimension of a subspace of
i s negative d e f i n i t e .
TO Y
on which
E,,
We w i l l prove the following.
THEOREM 1 5 . 1 (Morse). The index A of E,, i s equal t o the number of p o i n t s 7 ( t ) , with 0 < t < 1 , such t h a t 7 ( t ) i s conjugate t o y ( 0 ) along 7 ; each such conjugate p o i n t being counted w i t h i t s m u l t i p l i c i t y . T h i s index A i s always f i n i t e * . A s an immediate consequence one has:
-
C O R O W Y 1 5 . 2 . A geodesic segment 7 : [ 0 , 1 I M can contain only f i n i t e l y many p o i n t s which a r e conjugate t o 7 ( 0 ) along 7 .
I n order t o prove 15.1 we w i l l f i r s t make an estimate f o r s p l i t t i n g the vector space one of which
7(t)
i s contained i n a n open s e t
to < t l
<...< tk =
so that each s e p e n t that each Let
1
510.)
Choose a subdivision
of the u n i t i n t e r v a l which i s s u f f i c i e n t l y f i n e
y[ti-,,til
l i e s w i t h i n such an open s e t
...,tk) C
Tn7(t0,tl,t2,
1)
W \[ t i - , , t i l
2)
W
along
7
U;
and s o
TQ7 be the vector space c o n s i s t i n g of
such that
vanishes a t t h e endpoints
Tn7(to,tl,
..., tk)
71 [ti-l,til
i s a Jacobi f i e l d along
t
= 0,
t
f o r each
i;
= I .
1s a f i n i t e dimensional v e c t o r space c o n s i s t i n g of
broken Jacobi f i e l d s along
*
such that any two
[ti-l,ti3 i s minimal.
71
a l l vector f i e l d s W
Thus
U
U a r e joined by a unique minimal geodesic which depends d i f f e r -
e n t i a b l y on t h e endpoints. (See 0 =
by
i s positive definite.
E,,
Each p o i n t p o i n t s of
A
TQ7 i n t o two mutually orthogonal subspaces, on
7.
For g e n e r a l i z a t i o n of this r e s u l t see: W. Ambrose, The index theorem i n Riemannian geometrp, Annals Of Mathematics, V o l . 7 3 (1961), pp. 49-86.
Let
T I C TR
be the v e c t o r space c o n s i s t i n g of a l l vector f i e l d s
Y
for which W ( t o )
W E TRY
W(t,)
= 0,
= 0,
The vector space Tny sum T n y ( t o , t l , . . . , t k )@ T I . THese mutually perpendicular w i t h r e s p e c t E,,. Furthermore, E,, restricted definite. LF,MMA 1 5 . 3 .
PROOF:
Given any vector f i e l d
= O,l,
..., k. W - W,
and
generate
TI
If
n u l l space.
W1 ( t i ) = W( ti)
second d e r i v a t i v e
~
Y
d2E
O
'
belongs t o
W,
y
du2 with v a r i a t i o n vector f i e l d
W
belongs t o
Points
y(tO),y(tl),
c(u)(ti)
=
..., y ( t k ) for i
y(ti)
Proof that
each
0,l
=
2
E,,(W,W)
wise smooth p a t h from
fixed.
~ ( 0 t) o
E,,.
-
(-E,E)
(Compare 13.5.)
W.
Each
6(u) E R
...
to
y(t2)
than any other p a t h between i t s endpoints.
E,,(W,W)
were equal t o
I n f a c t for any Ex,(Wl,W)
E,,(W,W)
=
0.
0.
>
0
for W
Then W
If
For if
T
<
to
( E: ),
Hessian
i s a piece-
~ ( 1 ) .
E(C(0))
=
E
TI,
u
between
But
Of
y (T)
yT
and
X(T)
5
W # 0.
2 0.
hence
Suppose that
would l i e i n t h e n u l l space of
W2 E T I
the i n e q u a l i t y
denote
X(T)
First
which vanish a t
dimensional space ?' of
X(T)
y(0)
T,
and
y ( ~ )
such that t h e
7 ( ~ ' ) . Thus
yT,
we obtain a
( E:' ),
on which
yTl A( T)
which vanishes i d e n t i c a l l y dimensional v e c t o r space
i s negative d e f i n i t e .
A(T)
= 0
T
A(T)
=
o
for small values of
note that
i s s u f f i c i e n t l y small then
yT
is a minimal geodesic,
by Lemma 1 3 . 6 .
X(T)
,
=
A(T).
X(T).
1s Continuous from t h e l e f t :
ASSERTION ( 3 ) . A(T-E)
Hence
T.
Now l e t us examine the d i s c o n t i n u i t i e s of t h e f u n c t i o n E,,.
Each vector
A ( T ~ ) .
For i f must be
Let
Y(T).
i s negative d e f i n i t e on this vector space.
f i e l d s along
.
= 0,
to
[O,T~.
which i s a s s o c i a t e d with this geodesic.
then t h e r e e x i s t s a
T I
v e c t o r f i e l d s along
W1 E TR ( t o , t l..., , tk) w e have a l r e a d y seen that
For any
y(0)
i s a monotone f u n c t i o n of
This proves that
Therefore t h e second d e r i v a t i v e , evaluated a t Proof that
( E: ),
X(T)
ASSERTION (2).
E(&(u))2 E(Y)
This
i s the index which we a r e a c t u a l l y t r y i n g t o compute.
ASSERTION ( 1 ) .
15 i s chosen s o a s t o leave the
E TI.
to
W = 0.
TI.
t o the i n t e r v a l
y
f i e l d i n 9Y extends t o a v e c t o r f i e l d along
for W
Since
note t h a t :
i s a minimal geodesic, and t h e r e f o r e has smaller energy
yl[ti-l,til
A(1)
i s any v a r i a t i o n of
R
I n other words we may assume that
y(tl)
i s p o s i t i v e d e f i n i t e on
i s a geodesic from
~ O , T ]- M
yT:
Thus
,..., k.
0
c o n s i s t s of Jacobi f i e l d s .
The proof is s t r a i g h t f o r w a r d . Let y T denote t h e r e s t r i c t i o n of
can be i n t e r p r e t e d a s the
g(0) equal t o
then we may assume that
TI
l i e s i n the
T I ,
the index of the Hessian
E,,(W,W)
6:
Thus W
0.
...,
Thus
where
(0);
Exx(W2,W2)
LEMMA 1 5 . 4 . The index (or t h e n u l l i t y ) of E,, i s equal t o the index (or n u l l i t y ) of E,, r e s t r i c t e d t o t h e space T R y ( t O , t , , ..., tk) of broken Jacobi f i e l d s . I n p a r t i c u l a r (since T n Y ( t 0 , t l , tk) i s a f i n i t e dimensional vector space) t h e index A i s always f i n i t e .
0
the Hessian
E,,
=
2
for
T Q y ( t O , t l , ..., tk)
Thus t h e two subspaces a r e mutually perpendicular with r e s p e c t t o E Tn
E,,(W,,W)
+ c
An immediate consequence i s t h e following:
denote the unique
W1
<W2(t)
t
For any W
But t h e n u l l space of
Thus t h e q u a d r a t i c form E,,
1
-
implies that
c
2c E,,(W2,W)
=
contains no Jacobi f i e l d s other than zero, this implies that
TI
13.1) takes the form
then t h e second v a r i a t i o n formula
+ c W2, W + c Wp)
E,,(W
for a l l values of
e x i s t s and i s unique.
W1
T R y ( t o t l , .. . , t k ) and
belongs t o
=
= 0.
and have only the zero v e c t o r f i e l d i n common.
TRY,
$E,,(W1,W2)
let
Y
Thus the two subspaces,
TI.
W(tk)
THE INDEX THEOREM
completes t h e proof of 1 5 . 3 .
Y
belongs t o
W1
= 0,.
5
0
s p l i t s as t h e d i r e c t two subspaces a r e t o t h e i n n e r product t o T I is p o s i t i v e
W E TR
It follows from 914.5 that
Clearly
. .,
W(t2)
TR ( t o , t l ,... ,tk) such t h a t
"broken Jacobi f i e l d " i n i
5 15.
CALCULUS OF VARIATIONS
111.
84
For a l l s u f f i c i e n t l y s m a l l
E
> o
we have
First
CALCULUS OF VARIATIONS
111.
86
PROOF. According t o 15.3 t h e number
S15. i s negative d e f i n i t e .
can be i n t e r p r e t e d as
X(1)
f i e l d s along
t h e index of a quadratic form on a f i n i t e dimensional v e c t o r space
~n,(t,,t,, say
ti
<
...,t k ) . T
<
We may assume t h a t t h e subdivision i s chosen
ti+,. Then t h e index
of a corresponding q u a d r a t i c form broken Jacobi f i e l d s along u s i n g t h e subdivision
Y,.
<
0
<
tl
...,J v
be
l i n e a r l y independent Jacobi
v
a l s o vanishing a t the endpoints.
y,,
Note that t h e
v
can be i n t e r p r e t e d as t h e index
X(T)
on a corresponding v e c t o r space of H, T h i s vector space i s t o be constructed
. . < ti
t2 <.
J1,
a7
vectors
that
SO
Let
THE INDEX THEOREM
<
of
T
a r e l i n e a r l y independent. X1,
...,Xy
along
7T+E
Hence i t i s p o s s i b l e t o choose
vector f i e l d s
v
s o that t h e matrix
Since a
[O,T].
broken Jacobi f i e l d i s uniquely determined by i t s values a t t h e break p o i n t s 7(ti),
this v e c t o r space i s isomorphic t o the d i r e c t sum
i s equal t o t h e v x v Jh
over
7T+E
i d e n t i t y matrix.
Extend t h e vector f i e l d s Wi
by s e t t i n g these f i e l d s equal t o
for
0
T
5 t 5
T
+
and E.
Using t h e second v a r i a t i o n formula we see e a s i l y that
c
Note that this vector space r a t i c form
H,
Now
A(,).
on
v a r i e s continuously with
Evidently t h e quad-
T. T.
c
i s negative d e f i n i t e on a subspace '?'C
H,
For a l l
c
i s independent of
sufficiently close t o
T I
negative d e f i n i t e on 97.
Therefore
then we also have
5
A(,)
Let
v
A(,-E)
ASSWTION ( 4 ) .
2
by Assertion 1 .
Hence
=
X(T+E)
E
>
A(,-E)
a conjugate p o i n t of m u l t i p l i c i t y
v
<
T
Now l e t
w,,
( E: )**.
. t
when t h e v a r i a b l e
along
of Assertion 3 .
Since
some subspace 9J1C l y close t o
T,
A(,+€)
c
5
dim ,Z
=
Let
H,
w e see that
ni
of dimension
i t follows that
.
+ v
A(,)
n i - A(,)
HT1
-
c
and
=
(Kronecker d e l t a )
2~~
..., W A ( , ) ,
c - l J 1 - c X,
,...,
A(T)
+
vector f i e l d s
c - l J V - c Xv
+
dimension
X(T)
definite.
I n f a c t the matrix of
v
on which t h e q u a d r a t i c form
( Ei+E),,
( E:+E),
i s negative
w i t h r e s p e c t t o this b a s i s i s
passes Clearly
I
H,
i s p o s i t i v e d e f i n i t e on
V .
For all
sufficient-
i s p o s i t i v e d e f i n i t e on P . Hence
- 4 I + c2 B
c At
be as i n t h e proof
T I
v
We claim that these vector f i e l d s span a v e c t o r space of
y,+€.
this a s s e r t i o n w i l l complete t h e proof of t h e index theorem.
PROOF that
Jh, Xk)
be a small number, and consider the
c
= A(,).
and i s continuous otherwise.
V ;
E
we have
0
A(?-)+ v
jumps by
h(t)
-
= T
T I
be t h e n u l l i t y of t h e Hessian
Then f o r a l l s u f f i c i e n t l y s m a l l
Thus the f u n c t i o n
But i f
A(,).
( E:+€)*,(
HTI i s
i t follows that
T
A(,!)
of dimension
where
and
A
B
a r e f i x e d matrices.
If
c
i
i s s u f f i c i e n t l y small, this
compound matrix i s c e r t a i n l y negative d e f i n i t e .
T h i s proves Assertion (4).
The index thearem 15.1 c l e a r l y follows from t h e Assertions ( 2 ) , ( 3 ) , and (4). PROOF that
f i e l d s along
y,,
a(,+€)
2
A(,)
+
V .
Let
W,,...,WA(,)
be
X(T)
vanishing at t h e endpoints, such that t h e matrix
vector
CAEULUS OF VARIATIONS
111.
aa
AN APPROXIMATION TO
Sl6.
PROOF:
Let
denote t h e b a l l
S
(X E M : p(x,p)
816. Let
M
A F i n i t e Dimensional Approximation t o
be a connected Riemannian manifold and l e t
two ( n o t n e c e s s a r i l y d i s t i n c t ) p o i n t s of piecewise
Rc
Note t h a t every p a t h and
q
n(M;p,q)
R =
to
q
can be topologized as follows.
denote t h e topological metric on
M
coming from i t s Riemann m e t r i c .
a,
mf
p t h s from p
The s e t
M.
p
.
Cm
s(t), s'(t)
with arc-lengths
E 0
be
Since
Let
p
Given
r e s p e c t i v e l y , d e f i n e the d i s t a n c e
exists
E
>
0
M
S
x
to
y
x
E
and
S
<
of l e n g t h and
T h i s follows
Hence by 10.8
p(x,y)
<
there
there i s a
E
and so that this geodesic
E;
y.
...,tk)
(tO,tl,
Choose t h e subdivision
ti - ti-l
i s a compact s e t .
x, y
s o that whenever
depends d i f f e r e n t i a b l y on
difference
S C M.
<E5 c .
i s complete,
unique geodesic from
.
561
l i e s within this subset
Rc
LU E
from the i n e q u a l i t y L2
of
89
Rc
i s l e s s than
of
[o,ll
s o that each
Then for each broken geodesic
E2/c.
C
n(tO,tlj".,tk) (The l a s t term i s added on s o that t h e energy f u n c t i o n , we have a
w i l l be a continuous function from induces t h e required topology on Given
c
I n t nc
and l e t
>
0
let
Rc
T h i s metric
<
subset
E-' ( [O,c) )
E-'([O,cl) C
(where E
We w i l l study the topology of
Rc
=
R
1 EO: R
-
Thus t h e geodesic
R
Choose some subdivision
Let
n(to,tl,
..., tk)
by construct-
0 =
to
cn([ti-l,til
[0,11
-M
The broken geodesic
p
~ ( 0 = )
2)
m([ti-l,til
and
u(t,), m(t2),...,cu(tk-l)
<...< n
be the subspace of
tk =
1
of t h e u n i t i n t e r -
i = 1,
.. . , k .
(LU(tl),...>LU(tk-l))
. . . ,tk)
I n t R ( to,tl ,
subset of the ( k - l ) - f o l d product
M x...x M.
and a c e r t a i n open
Taking over t h e d i f f e r e n t i a b l e
s t r u c t u r e from this product, this completes the proof of 1 6 . 1 .
F i n a l l y we d e f i n e t h e subspaces
ci(tO,tl ,...,t k ) c
M x M x...x M.
Evidently this correspondence
d e f i n e s a homeomorphism between
i s a geodesic f o r each
E
c o n s i s t i n g of paths
,
( ~ ( 1 )= q
i s uniquely determined by t h e ( k - l ) - t U p l e
LU
such that
1)
.
is uniquely and d i f f e r e n t i a b l y determined by
-L
0:
E2
t h e two end p o i n t s .
i n g a f i n i t e dimensional approximation t o i t .
val.
<
(ti - ti-l)c
R.
denote the closed subset
denote the open
is t h e energy f u n c t i o n ) .
t o the r e a l numbers.)
R
=
nc n
R(tO,tl
Int n ( t o , t l , . ..,tk)c =
(Int
0')
n
,..., tk) n ( t o , ...,tk) .
To shorten t h e n o t a t i o n , l e t us denote this manifold Int n ( t o , t l , .
..,t k ) c of
broken geodesics by
.
Let
E': B - R denote t h e r e s t r i c t i o n t o
LEMMA 1.6.1. Let M be a complete Riemannian manifold; and l e t c be a f i x e d p o s i t i v e number such t h a t n c # 0 . Then f o r a l l s u f f i c i e n t l y f i n e subdivisions ( t O , t l..., , tk) of [0,1 1 t h e s e t I n t n ( t o , t l , .. ,tk) can be given t h e s t r u c t u r e of a smooth f i n i t e dimensional manifold i n a natural way.
B.
B
of t h e energy f u n c t i o n
E : R
+R.
111.
CALCULUS OF VARIATIONS
$1
THEOREM 1 6 . 2 . T h i s f u n c t i o n E l : B - R i s smooth. Furthermore, f o r each a < c t h e set Ba = ( E l ) - ' [ o , a ] i s compact, and i s a deformation r e t r a c t " of t h e corresponding s e t R a . The c r i t i c a l p o i n t s of E ' a r e p r e c i s e l y t h e same as t h e c r i t i c a l p o i n t s of E i n I n t nc: namely the unbroken geodesics from p t o q of l e n g t h l e s s than 6.The index [or t h e n u l l i t y ] of t h e Hessian El,, a t each such c r i t i c a l p o i n t y i s equal t o t h e index [ or t h e n u l l i t y 1 of E,, a t y . Thus t h e f i n i t e dimensional manifold
f o r t h e i n f i n i t e dimensional p a t h space
B
For
c
AN APPROXIMATION TO
the s e t
,..., p k b l ) E
S x S x.. .x S
po
A retraction
A s a n immediate conse-
Int nc
r:
THEOREM 1 6 . 3 . L e t M be a complete Riemannian manifold and l e t p,q E M be two p o i n t s which a r e not conjugate along any geodesic of l e n g t h 5 &. Then Ra has t h e homotopy type of a f i n i t e CW-complex, w i t h one c e l l of dimension X f o r each geodesic i n R a a t which E,, has index X .
denote the unique broken geodesic i n
<
a geodesic of l e n g t h
from
E
implies that
m[O,ll C S.
implies t h a t
r(m)
Clearly
geodesics.)
on t h e
Since t h e broken geodesic
1
53.5. E
B
depends smoothly
( k-1 ) - t u p l e
,...,m ( t k - l )
m(tl),o(t2)
i t i s c l e a r that t h e energy tuple.
u)
E
M
x . . . ~M
E 1 ( m ) a l s o depends smoothly on this
I n f a c t we have t h e e x p l i c i t formula
9.) A s a closed subset
Then (k-1)-
B
such t h a t each to
5
E(r(m))
I [O,ti-l
<
E(m)
c.
5u 5
ti
-
r(m)I[ti-l,til
is
The i n e q u a l i t y
c
=
m i n i m a l geodesic from
ru(m) I[u,il
ml[u,il
=
f i t s into a
"(ti-,) t o
m(u)
I n t nC,
and
rl
=
r. It i s e a s i l y v e r i -
i s continuous as a f u n c t i o n of both v a r i a b l e s .
i s a deformation r e t r a c t o f E(rU(m))
5 E(m)
,
.
ro i s the i d e n t i t y map of
mation r e t r a c t of
1-
I [O,ti-l I ,
r u ( o ) I [ ti-l ul
,
r
let
r(m)
Since
r(m)
Int nC
=
B
<
This r e t r a c t i o n
I
ru(m)
.(ti).
E
Let
can be s o defined.
For ti-l
ru(w)
fied that that
i s defined as follows.
5 (L
ru: I n t n c as follows.
PROOF of 1 6 . 2 .
=
Hence t h e i n e q u a l i t y
contains only f i n i t e l y many
T h i s follows from 1 6 . 2 together with
pk
p,
m(ti-l)
parameter family of maps
PROOF.
such that
=
B
.-L
p(p,m(t)) 2
na
91
of a compact s e t , this i s c e r t a i n l y compact.
quence we have t h e following b a s i c r e s u l t .
( I n particular it i s asserted that
Qc
i s homeomorphic t o t h e s e t of a l l ( k - 1 ) -
Ba
(Here i t i s t o be understood that
provides a f a i t h f u l model
I n t Rc.
(p,
tuples
<
a
6.
T h i s proves
I n t a'.
i t i s c l e a r t h a t each Ba
i s a l s o a defor-
n".
Every geodesic i s a l s o a broken geodesic, s o i t i s c l e a r t h a t every ' l c r i t i c a i point" of
E
in
Int nc
automatically l i e s i n t h e submanifold
Using t h e i'irst v a r i a t i o n formula ( $ 1 2 . 2 ) Points of
* Similarly
B
i t s e l f i s a deformation r e t r a c t of
Int nc.
Y .
El
i t i s c l e a r t h a t the c r i t i c a l a r e p r e c i s e l y t h e unbroken geodesics.
Consider the tangent space TBy t o the manifold B T h i s w i l l be i d e n t i f i e d with t h e space T R y ( t o , t l , . . . , t k )
a t a geodesic
of broken
B.
96
111.
geodesics
CAXULUS OF VARIATIONS
T ~ , Y ~ , Y ~ , . . from .
p
s h o r t g r e a t c i r c l e a r c from p circle arc
pq'p'q;
let
r2
to
as follows.
q,
to
q;
$17.
let
denote t h e a r c
y1
Let
yo
denote t h e
denote t h e long great
pqp'q'pq;
and s o on.
The
REMARK.
THE FULL PATH SPACE
More g e n e r a l l y i f
M
not c o n t r a c t i b l e then any two non-conjugate p o i n t s of
denotes t h e number of times that
occurs i n the
or p '
p
rk.
i n t e r i o r of
The index
h(yk)
or p '
=
v 1 +...+
i s equal t o
k(n-1),
i n t h e i n t e r i o r i s conjugate t o
of the p o i n t s
p
plicity
Therefore we have:
n-1.
pk
p
since each
with multi-
COROLTARY 17.4. The loop space n(Sn) has t h e homotopy type of a CW-complex w i t h one c e l l each i n t h e dimensions 0 , n-1, z ( n - 1 ) , 3(n-1) ,...
.
For
n
>
2
from this information.
t h e homology of Since
n(Sn)
a(?)
can be computed immediately
has n o n - t r i v i a l homology i n i n f i n i t e -
l y many dimensions, we can conclude: COROLLARY 1 7 . 5 . Let M have t h e homotopy type of Sn, for n > 2 . Then any two non-conjugate p o i n t s of M a r e joined by i n f i n i t e l y many geodesics. T h i s follows s i n c e t h e homotopy type of
n(M))
depends only on t h e homotopy type of
geodesic i n 3(n-1),
n(M)
and s o on.
with index
0,
M.
n*(M)
(and hence of
There must be a t l e a s t one
a t l e a s t one w i t h index
n-1, 2(n-1),
a r e joined by
Compare p. 484 of J . P . S e r r e , Homologie singulitSre des espaces f i b r 6 s , Annals of Math. 54 ( 1 9 5 1 ) , pp. 425-507. A s another a p p l i c a t i o n of 1 7 . 4 ,
k
M
i n f i n i t e l y many geodesics.
thal suspension theorem.
subscript
97
i s any complete manifold which 1s
(Compare
one can give a proof of t h e Freuden-
522.3.)
918.
CALCULUS OF VARIATIONS
111.
Now suppose t h a t exp,
NON-CONJUGATE POINTS
i s non-singular a t
-
99
v.
Choose
n
-
independ-
e n t v e c t o r s X1, ...,% i n T ( T ) v . Then exp,(X1), re l i n e a r l y independent. I n T% Mp choose p a t h s u v l ( u ) , exp,(%) u v a(u) 5.1 8.
Existence of Non-Conjugate P o i n t s .
with
Theorem 1 7 . 3 gives a good d e s c r i p t i o n of t h e space v i d i n g that t h e p o i n t s geodesic.
p
and
pro-
n(M;p,q)
Then
are n o t conjugate t o each other along any
q
T h i s s e c t i o n w i l l j u s t i f y this r e s u l t by s h a r i n g that such non-
conjugate p o i n t s always e x i s t .
f: N - r M between manifolds of t h e same x
dimension i s c r i t i c a l a t a p o i n t f,:
E
TNx
i f t h e induced map
N
...,Wn
-dvi a + O (u) )
and
. . .,an,
al ,
along
=
p.
Since t h e W i ( l ) independent, no n o n - t r i v i a l l i n e a r combination of t h e Wi Since
which vanish a t
n
n
xi .
constructed as above, provide
vanishing a t
yV,
n
Jacobi f i e l d s =
exp,(Xi)
are
can vanish a t i s t h e dimension of the space of Jacobi f i e l d s along y V ,
p,
c l e a r l y no n o n - t r i v i a l Jacobi f i e l d along
a t b o t h p and exp v.
yv
vanishes
T h i s completes t h e proof.
TMf(x)
+
COROLLARY 18.2. L e t p E M. Then f o r almost a l l p i s not conjugate t o q along any geodesic.
We w i l l apply this d e f i n i t i o n t o the ex-
1- 1.
of tangent spaces i s not
W1,
exp v.
R e c a l l that a smooth map
~ ~ ( =0 v )
..., ...,
q c M,
p o n e n t i a l map exp ( W e w i l l assume that
M
=
expp:
.
TMp+M
PROOF.
i s complete, s o that exp
theorem ( 9 6 . 1 ) .
is everywhere defined;
although this assumption could e a s i l y be eliminated.) THEOREM 18.1. The p o i n t exp v i s conjugate t o p along t h e geodesic 7,, from p t o exp v i f and only i f t h e mapping exp i s c r i t i c a l a t v . PROOF: = 0
Suppose that
f o r some non-zero
X
E
considered as a manifold. dv = v and ,(O) = X.
v(0)
is critical at
exp T(TMp),,
Let
f i e l d along
yV.
W
Then
defined by
a
t
given by
Obviously W ( 0 ) = 0 .
-+
7v
a =(exp
to
V
u - v ( u ) be a p a t h i n T%
Then t h e map
i s a v a r i a t i o n through geodesics of t h e geodesic Therefore t h e v e c t o r f i e l d
v E TMp.
t h e tangent space a t
T,
such that
a(u,t) given by
tv(u))
exp,(X)
IuE0
=
exp t v ( u )
t - exp t v . i s a Jacobi
W e a l s o have
But t h i s f i e l d i s not i d e n t i c a l l y zero s i n c e
So there i s a n o n - t r i v i a l Jacobi f i e l d along
vanishing a t these p o i n t s ; hence
p
and
yV
from p
to
exp v,
exp v a r e conjugate along
7v
a
T h i s follows immediately from 18.1 together with S a r d ' s
T
111. CALCULUS OF VARIATIONS
100
TOPOMGY AND CURVATURE
61 9.
101
[ I n t u i t i v e l y t h e curvature of a manifold can be described i n terms of " o p t i c s " w i t h i n t h e manifold a s follows. 619.
some R e l a t i o p s Between TopoloEY and Curvature*
~u~s e c t i o n with
w i l l describe t h e behavior of geodesics i n a manifold
"negative cumaturelf or w i t h " p o s i t i v e curvature. 'I
geodesics a s being the paths of l i g h t r a y s .
Consider an observer a t
looking i n t h e d i r e c t i o n of t h e unit v e c t o r
U
A s m a l l l i n e segment a t
pointed i n a d i r e c t i o n corre-
with length
q
sponding t o the u n i t v e c t o r suppose that < R ( A , B ) A , B > 0 for every p a i r of v e c t o r s A,B i n t h e tangent 3 P c e T% and f o r every p E M. Then no two P o i n t s of
2
~ ( 1+
a r e conjugate along any geodesic.
PROOF.
Let
7
W
5
>
J
V;
and
than i t r e a l l y i s .
Then
7.
where
9
+ R(V,J)V
=
p exp(rU).
r))
.
Thus i f s e c t i o n a l curvatures a r e negative then any o b j e c t appears s h o r t e r
be a geodesic w i t h v e l o c i t y vector f i e l d
be a Jacobi f i z l d along
=
+ (terms involving higher powers of
A s m a l l sphere of r a d i u s
an e l l i p s o i d with p r i n c i p a l r a d i i let
q
would appear t o t h e observer a s a
TS,
E
L,
towards a p o i n t
l i n e segment of length
19.1.
M
Suppose that we t h i n k of the
0
W
-
K1 , K 2 , .
. .,K,
E(
1
+
+ TK1
q
would appear t o be
. . .) , . . ., E ( 1
+
2
$K, + . . . )
denote the eigenvalues of the l i n e a r transformation
Any small o b j e c t of volume
R(U,W)U.
at
E
r2
v(1 + Tr2 ( ~ l + K, +...+
+ ( h i g h e r terms))
Q)
t o t h e "Ricci curvature"
v
K(U,U),
would appear t o have volume where
K~ +...+
K,
i s equal
as defined l a t e r i n this s e c t i o n . ]
Here are some f a m i l i a r examples of complete manifolds w i t h curvaTherefore
J) Thus the f u n c t i o n DJ # 0 . so if If
J
< $, J)
DJ <= ,J>
a l s o vanishes a t
J
and
0
[O,tOl.
quantity A
and
B.
If
A
and
20
-
B
to >
0,
then t h e f u n c t i o n
t o , and hence must vanish i d e n t i c a l l y
T h i s implies that =
i s i d e n t i c a l l y zero.
REMARK.
and a t
0
J(0)
so that
1 $1 2
i s monotonically i n c r e a s i n g , and s t r i c t l y
vanishes both a t
throughout t h e i n t e r v a l
+
ture
g(0)
=
The Euclidean space with curvature
(2)
The paraboloid
(3)
The hyperboloid of r o t a t i o n
<
ture
0.
z = x 2 - y2, with curvature
x2 + y2
-
z2
<
0.
= 1,
w i t h curva-
0.
(4) The h e l i c o i d x cos z + y s i n z
= 0,
t r a r i l y close t o
w i t h curvature
<
0.
It i s n o t known whether o r n o t t h e r e e x i s t s a complete
0.
surface i n 3-space with curvature negative and bounded away from
T h i s completes t h e proof.
p
then t h e
It i s equal t o t h e Gaussian curvature of t h e s u r f a c e
* expP (ulA + u2B)
spanned by t h e geodesics through p with v e l o c i t y v e c t o r s i n t h e subspace (see f o r example, Laugwitz "Differential-Geometrie," spanned by A and B.
0.)
A famous example of a manifold w i t h everywhere negative s e c t i o n a l
curvature i s t h e pseudo-sphere
i s c a l l e d t h e s e c t i o n a l curvature determined by
(ul,u2)
p. 101.)
(1)
(REMARK. I n a l l of t h e s e examples t h e curvature t a k e s values a r b i -
0,
a r e orthogonal unit v e c t o r s a t
1. 0 :
z
=
-
Jl - x 2
-
y2' +
with the Riemann metric induced from R 3 . t h e constant value
sech-'
m, > z
o
Here t h e Gaussian curvature has
- 1.
N o geodesic on this surface has conjugate p o i n t s although two geo-
d e s i c s may i n t e r s e c t i n more than one p o i n t .
The pseudo-sphere g i v e s a
81 9 .
111. CALCULUS OF VARIATIONS
102
Theref ore, the exponential map
geometry, i n which the sum of t h e angles of any t r i a n g l e i s
<
radians.
T h i s manifold i s not complete.
TOPOMGY AND CURVATURE
I n f a c t a theorem of H i l b e r t
But i t follows from 18.1 t h a t
onto.
T%
expp:
exp
P
s o that
exp i s l o c a l l y a diffeomorphism. P see t h a t exp i s a global diffeomorphism. P
107
--L
i s one-one and
M
i s n o n - c r i t i c a l everywhere; Combining t h e s e two f a c t s , we This completes t h e proof of
19.2.
More generally, suppose t h a t
M
5
complete and has s e c t i o n a l curvature f l a t torus
S1 x S1,
negative curvature.) space
of
i s not simply connected; b u t i s 0.
2
or a compact surface of genus
with constant
2
Then Theorem 1 9 . 2 a p p l i e s t o t h e u n i v e r s a l covering
For i t i s c l e a r that ?i
M.
(For example M might be a
i n h e r i t s a Riemannian metric
from M which i s geodesically complete, and has s e c t i o n a l curvature 5 Given two p o i n t s paths from
s t a t e s t h a t no complete s u r f a c e of constant negative curvature can be imbedded i n R3.
(See e . g . Willmore, " D i f f e r e n t i a l Geometry,'' p. 1 3 7 . )
2. 2 .
topology of
Such a manifold can even be compact; f o r example,
THEOREM 1 9 . 2 (Cartan*). Suppose t h a t M i s a simply COMeCted, complete Riemannian manifold, and that t h e s e c t i o n a l curvature (R(A,B)A,B) i s everywhere < 0. Then any two p o i n t s of M a r e joined by a unique geod e s i c . Furthermore, M i s diffeomorphic t o t h e Euclidean space Rn.
index theorem t h a t every geodesic from
p
Theorem 1 7 . 3 a s s e r t s that t h e p a t h space
i s connected.
For example:
to
q
n(M;p,q)
has index
homology group of
A = 0.
that
n,(M)
=
i
>
k
Hk(G)
of =
M
Since
-
M
is
=
Hk(M)
can be i d e n t i f i e d with the co-
of t h e group
Hk(n,(M))
for
ni(M)
0
nl(M).
1.
(See for example PP.
Academic Press, 1959.)
contains a n o n - t r i v i a l f i n i t e c y c l i c subgroup
has the homotopy type
i s simply connected implies that
ni(M)
S. T . Hu "Homotopy Theory,"
s u i t a b l e covering space
Thus
CW-complex, w i t h one v e r t e x f o r each geodesic. M
Clearly
c o n t r a c t i b l e t h e cohomology group
we have
Hk(k)
=
0
x1
(c) for
=
G; k
>
200-
Now suppose G.
Then f o r a
hence n
.
But t h e cohomology groups of a f i n i t e c y c l i c group a r e n o n - t r i v i a l i n a r b i O(M;p,q)
Since a CoMeCted 0-dimensional CW-complex must c o n s i s t of
a s i n g l e p o i n t , i t follows that t h e r e i s p r e c i s e l y one geodesic from
p
to
t r a r i l y high dimensions.
See E. Cartan, "Lecons s u r l a GBometrie des Espaces de Riemann," P a r i s , 1926 and 1951.
T h i s gives a c o n t r a d i c t i o n ; and completes the
proof. Now w e w i l l consider manifolds with " p o s i t i v e curvature."
9.
*
i s c o n t r a c t i b l e p u t s s t r o n g r e s t r i c t i o n s on the
1
PROOF:
Since t h e r e a r e no conjugate p o i n t s , i t follows from the
The hypothesis t h a t
2
COROLLARY 1 9 . 3 . I f M i s complete with ( R ( A , R ) A , B > then t h e hornotopy groups ni(M) a r e zero for i > 1 ; and n l ( M ) contains no element of f i n i t e order; other than t h e i d e n t i t y .
202
of a 0-dimensional
M.
contains p r e c i s e l y one geodesic.
<0 -
(Compare H i l b e r t and Cohn-Vossen, "Anschauliche
Geometrie," p. 2 2 8 . )
PROOF:
q
0.
it follows t h a t each homotopy c l a s s of
(See f o r example Laugwitz, " D i f f e r e n t i a l -
curvature which a r e complete.
a surface of genus
to
The f a c t that
However, t h e r e do e x i s t R i e m a n n i a n manifolds of constant negative
Geometrie," pp. 1 1 4 - 1 1 7 . )
p
p,q E M,
Of
Instead
considering t h e s e c t i o n a l curvature, one can o b t a i n sharper r e s u l t s i n
this case by considering t h e R i c c i t e n s o r (sometines c a l l e d t h e "mean curva-
t u r e tensor" )
.
The R i c c i t e n s o r a t a p o i n t
DEFINITION.
fold
M
8 19.
CALCULUS OF VARIATIONS
111.
104
p
TOPOLOGY AND CURVATURE
of a Riemannian mani-
i s a bilinear pairing
U L
0
K: defined a s follows.
Let
TMp
-R
x Ti‘f$
105
1
be t h e t r a c e of t h e l i n e a r transforma-
K(U,,U,)
tion
for i
S&ng
= 1,
...,n-1
we o b t a i n
W -R(U,,W)U2
from T%
to
( I n c l a s s i c a l terminology t h e tensor
TMp.
from R
by c o n t r a c t i o n . )
K ( U , ,Up)
=
K
It follows e a s i l y from 9 9 . 3 that
i s obtained K
i s symmetric:
K(U,,U,)
Now i f Let
The R i c c i tensor i s r e l a t e d t o s e c t i o n a l curvature a s follows. U1,U2,
...,Un
be an orthonormal b a s i s f o r t h e tangent space
PROOF:
<
>
R(Un,Ui)Un,Ui
By d e f i n i t i o n
>) .
( (R(Un,Ui)Un,Uj
zero, we obtain a sum of
for
K(Un,Un)
i
= 1,2,.
0.
7
i s equal t o t h e sum of the s e c t i o n a l curva-
ASSERTION. K(Un,Un) tures
T%*
<
. . ,n-1 .
i s equal t o the t r a c e of t h e matrix
5:
(-E,E)
-
n-1 s e c t i o n a l curvatures, a s a s s e r t e d .
P1,
-c
...,Pn
be a geodesic of l e n g t h
M
along
y
u
Hence
= 0.
p o i n t s along
y,
V
Wi(t)
=
(sin
L.
y.
(n-l)/r2.
3
nt) Pi(t).
LF,,
and
=
0
.
-< IT.
S. B. Myers, Riemann manifolds with p o s i t i v e mean Curvatwe,
Math. J o u r n a l , Vol. 8 ( 1 9 4 1 ) , pp. 4 0 1 - 4 0 4 .
Duke
o,
d2E(E(u)) du2
,
for small values of u # 0.
Hence
l/r2.
<
then
Wi
K(U,U)
r
This com-
then every s e c t i o n a l
takes the constant value
>
nr
con-
a best possible r e s u l t .
If
M.
p,q y
E
M
let
must be
5
y
nr.
be a m i n i m a l geodesic from
p
to
9.
Therefore, a l l p o i n t s have d i s t a n c e
Since closed bounded s e t s i n a complete manifold a r e compact, i t
follows that
* See
=
In fact i f
COROLLARY 19.5.
Then t h e l e n g t h of
Then
contains conju-
I t follows from 19.4 t h a t every geodesic of l e n g t h
PROOF. =
y
I f M i s complete, and K(U,U) 2 ( n - l ) / r 2 > 0 f o r a l l u n i t v e c t o r s U, then M i s compact, w i t h diameter 5 n r .
Choose
s o that =
then this expression i s
nr
This implies t h a t the index of
i s a sphere of r a d i u s
t a i n s conjugate p o i n t s :
W e may assume t h a t
DPi
>
i s not a minimal geodesic.
y
E(n(u)) < E(y)
If M
EXAMPIX.
which a r e orthonormal a t one
p o i n t , and hence a r e orthonormal everywhere along
i.
dE(C(u)) du
Since the n - t h diagonal term of this matrix i s
[O,ll
for some
L
i s a variation with variation vector f i e l d
n
curvature i s equal t o
y:
I t follows a l s o that
for every u n i t v e c t o r U a t every p o i n t of M ; where r i s a p o s i t i v e c o n s t a n t . Then every geodesic on M of l e n g t h > n r contains conjugate p o i n t s ; and hence i s not minimal. Let
and
gate p o i n t s .
p l e t e s t h e proof.
p a r a l l e l vector f i e l d s
Let
E,,(Wi,Wi)
THEOREM 1 9 . 4 (Myers*). Suppose that the R i c c i curvature K satisfies K ( U , U ) 2. ( n - l ) / r ’
PROOF:
(n-i)/r2
i s p o s i t i v e , and hence, by the Index Theorem, that
for
Pn
Hence
2
K(Pn,Pn)
M i t s e l f i s compact.
This c o r o l l a r y a p p l i e s a l s o t o the u n i v e r s a l covering space 2 2 i s COmpaCt, i t follows t h a t t h e fundamental group n,(M)
Since
finite.
T h i s a s s e r t i o n Can be sharpened as follows.
of is
CALCULUS OF VARIATIONS
111.
106
$19.
THEOREM 1 9 . 6 . I f M i s a compact manifold, and i f t h e R i c c i tensor K of M i s everywhere p o s i t i v e d e f i n i t e , then the p a t h space Q(M;p,q) has t h e homotopy type of a CW-complex having only f i n i t e l y many c e l l s i n each dimension.
PROOF.
>
(n-l)/r2>
a m i n i m u m , which we can denote by E
n(M;p,q)
of l e n g t h
>
has index
nr
X
similar argument shows that i
= i,2,
...,k
has index
7
(
Clearly
0
E,,(Xi,Xj)
=
dimensional subspace of
),
,
TO7
on which
Now suppose that the p o i n t s geodesic. from
p
X
2 k.
p
t a k e s on
0
so that
>
knr.
manifold,
Then a
Xi
along
XI,
which vanishes
7
<
E,,(Xi,Xi)
...,Xk
E,,
i s negative d e f i n i t e .
and
q
q
of length
<
d e s i c s w i t h index
REM!UK.
k.
5
knr.
k-
a r e not conjugate along any
Hence t h e r e a r e o n l y . f i n i t e l y many geo-
Together with $17.3, this completes t h e proof.
I do n o t know whether or not t h i s theorem remains t r u e i f
M i s allowed t o be complete, b u t non-compact.
The p r e s e n t proof c e r t a i n l y
breaks down s i n c e , on a manifold such as t h e paraboloid curvature
z
=
x2 + y2 ,
the
K(U,U) w i l l not be bounded away from zero.
It would be i n t e r e s t i n g t o know which manifolds can c a r r y a metric
s o that a l l s e c t i o n a l curvatures a r e p o s i t i v e . provided by the product
Sm x Sk
An i n s t r u c t i v e example i s
of two spheres; w i t h
m,k
manifold t h e R i c c i t e n s o r i s everywhere p o s i t i v e d e f i n i t e .
2 2.
For this
However, t h e
s e c t i o n a l curvatures i n c e r t a i n d i r e c t i o n s (corresponding t o f l a t t o r i S' x S' C Sm x Sk)
a r e zero.
I t i s n o t known whether or n o t
be remetrized s o that a l l s e c t i o n a l curvatures a r e p o s i t i v e . p a r t i a l r e s u l t i s known:
Sm x Sk
can
The following
If such a new m e t r i c e x i s t s , then it can not be
i n v a r i a n t under t h e i n v o l u t i o n from a theorem of Synge.
(x,y)
--L
(-x,-y)
of
Sm x S k .
T h i s follows
(See J . L. Synge, On t h e c o n n e c t i v i t y
Of
On curvature and c h a r a c t e r i s t i c c l a s s e s of a Riemann
Abh. Math. Sem., Hamburg, Vol.
20
( 1 9 5 5 ) , pp. 1 1 7 - 1 2 6 .
Comptes Rendus Acad. S c i . , P a r i s , Vol. 747 (1958), pp. 1165-1168. S. I . Goldberg, "Curvature and Homology," Academic Press, 1 9 6 2 .
0.
span a
Annals
M. Berger, Sur c e r t a i n e s vari6ti.s Riemanniennes B courbure p o s i t i v e ,
Then according t o J 1 6 . 3 t h e r e a r e only f i n i t e l y many geodesics to
K. Yano and S. Bochner, "Curvature and B e t t i Numbers," Studies, No 3 2 , Princeton, 1953.
I n f a c t f o r each
and such that
for i # j ;
sources a r e u s e f u l .
S. S. Chern,
of l e n g t h
7
one can c o n s t r u c t a v e c t o r f i e l d
outside of the i n t e r v a l
For other theorems r e l a t i n g topology and curvature, the following
2 1.
More generally consider a geodesic
a u a r t e r l y J o u r n a l of Mathematics (Oxford), Vol. 7
( 1 9 3 6 ) , PP. 3 1 6- 3 2 0 .
Then every geodesic
0.
107
U on M
Since t h e space c o n s i s t i n g of a l l u n i t v e c t o r s
i s compact, it follows t h a t t h e continuous f u n c t i o n K(U,U)
of p o s i t i v e curvature,
TOPOMGY AND CURVATURE
spaces
PART IV APPLICATIONS TO LIE GROUPS
520.
AND SYMMETRIC SPACES
Symmetric Spaces.
A symmetric space i s a connected Riemannian manifold
f o r each
p
E
M
t h e r e i s an isometry
and reverses geodesics through then
Ip(7(t))
p,
:
IP i.e., i f
-
M
M
M
which leaves
such that, p
i s a geodesic and
7
fixed 7(O)
= 7(-t).
LEMMA
k0.1 L e t y be a geodesic i n M, and l e t p = y ( 0 ) and q = y ( c ) . Then I q I p ( y ( t ) ) = 7 ( t + 2c) (assuming 7 ( t ) and 7 ( t + 2c) a r e d e f i n e d ) . Moreover, IqIp preserves p a r a l l e l vector f i e l d s along 7 .
PROOF: Let ~ ' ( 0 = )
Y'(t
q.
+ c)
y'(t)
=
Therefore
I I (7(t)) q P 7(t + 2 c ) .
=
I f t h e vector f i e l d parallel (since IppuV(t )
=
7(t
=
.
Therefore
7'
I (y(-t)) 9
=
i s p a r a l l e l along
7
then
V ( 0 ) = -V(O);
P* Iq* I,*(V(t))
=
IP*(V)
=
is
therefore
V ( t + 2c).
shows that geodesics can be i n d e f i n i t e l y extended.
COROLLARY 20.3.
Ip i s unique.
Since any p o i n t i s joined t o
COROLLARY
I
I (7'(-t - c)) q
M i s complete.
COROLZSlRY 2 0 . 2 . 20.1
i s a geodesic and
Then
Ip i s an isometry) and
-V( -t)
Since
V
+ c).
20.4.
f i e l d s along f i e l d along
7
p
by a geodesic.
If U,V and W a r e p a r a l l e l vector then R(U,V)W i s a l s o a p a r a l l e l
y.
PROOF. If X denotes a f o u r t h p a r a l l e l vector f i e l d along 7 , note that the q u a n t i t y < R(U,V)W,X > i s constant along 7 . I n f a c t ,
=
p
IV. APPLICATIONS
110
@ven
P
q
= 7(0),
carries
p
to
q.
= 7 ( c)
,
<
Since
T
vector f i e l d
=
R(T,Up,T,Vp)T,Wp,T,Xp>
Thus
remains t r u e everywhere along
i s parallel.
R(U,V)W
7.
=
eiUi
Any v e c t o r f i e l d W along
may be
y
expressed uniquely as
i s constant f o r every p a r a l l e l
111
the condition R(V,Ui)V
I t c l e a r l y follows that
X.
17(c/2)Ip which
i s an isometry, this q u a n t i t y i s equal t o
.
R(Up,Vp)Wp,%>
=
Then
> <
by 20.1.
T
consider t h e isometry
SYMMETRIC SPACES
$20.
Manifolds with t h e property of 2 0 . 4 a r e c a l l e d l o c a l l y symmetric.
W(t)
w,(t)U,(t)
=
Then t h e Jacobi equation D2w + %(W)
dt2
w,(t)Un(t)
=
takes the form
0
1 dt2 ui 1
(A c l a s s i c a l theorem, due t o Cartan s t a t e s that a complete, simply connected,
d2wi
l o c a l l y symmetric manifold i s a c t u a l l y symmetric.)
+
.
+...+
eiwiUi
0.
=
I n any l o c a l l y symmetric manifold t h e Jacobi d i f f e r e n t i a l equations
Let
have simple e x p l i c i t s o l u t i o n s . Let
l y symmetric manifold.
V =
7:
R
M
-*
be a geodesic i n a l o c a l -
be t h e v e l o c i t y v e c t o r a t
%(O)
p
= y ( 0 ) .
Since t h e
Ui
are everywhere l i n e a r l y independent t h i s i s equivalent t o
the system of
n
equations d2wi
Define a l i n e a r transformation
q: T% by*
%(W)
=
R(V,W)V.
Let
2+
-+T%
e l , ...,en
W e are i n t e r e s t e d i n s o l u t i o n s t h t vanish a t
denote the eigenvalues of
%.
F i r s t observe t h a t
< q(w),w'>
%
If
wi(t)
i s self-adjoint:
T h i s follows immediately from t h e symmetry r e l a t i o n
<
R(V,W)Vl,Wl)
=
Therefore we may choose an orthonormal b a s i s %(Ui)
where
el,
...,en
along
7
by p a r a l l e l t r a n s l a t i o n .
*
=
a r e t h e eigenvalues.
...,Un
U,,
eiUi
Mp
s o that
,
Extend the
Then since
. for
M
should n o t be confused w i t h t h e R i c c i t e n s o r
Ui
t o vector f i e l d s
is l o c a l l y symmetric,
Of
$19.
=
ei
=
ci s i n
wi(t)
(5 t),
O
*
t
If
= 0.
f o r some constant
a r e a t the m u l t i p l e s of
t
<
=
wi(t)
t
T h i s completes t h e proof of 20.5.
then
ci s i n h ( J l e i l t ) = 0.
=
cit
and i f
for some constant
ei ci.
0
ei
>
0
then
ci.
.
n/Gi
-
= 0
vanishes only a t
<W,KJ(W>
=
wi(t) Then t h e zeros of
THEOREM 2 0 . 5 . The conjugate p o i n t s t o p along 7 are the p o i n t s y ( c k / G i ) where k i s any non-zero i n t e g e r , and ei i s any p o s i t i v e eigenvalue of %. The m u l t i p l i c i t y of 7 ( t ) a s a conjugate p o i n t I s equal t o t h e number of ei such t h a t t i s a mult i p l e of c / G i . PROOF:
=
then
Thus i f
ei
1. 0 ,
wi(t)
LIE GROUPS
§21.
IV. APPLICATIONS
112
113
LEMMA 2 1 . 2 . The geodesics y i n G with ~ ( 0 =) e a r e p r e c i s e l y t h e one-parameter subgroups of G .
PROOF:
Lie Groups as Symmetric Spaces.
521.
G
I n this s e c t i o n we consider a Lie group
the map
with a Riemannian metric
so
which i s i n v a r i a n t both under l e f t t r a n s l a t i o n s
Let
I y ( t ) I e takes
G
+
L,(u)
G,
T d
=
~
and right t r a n s l a t i o n , certainly exists.
as follows:
R,(u)
If
the Haar measure on new i n n e r product
If
UT.
i s commutative such a metric
G
GI and L e t
be any Riemannian metric on
G.
Then
<(,>>
<>
on
1
=
i s r i g h t and l e f t i n v a r i a n t .
I.I
G
denote
I.I
Define a
by
>
LU,RTy(W)
<<,>>
gent v e c t o r of Hence
and only if
GxG
Then
geodesic through y 7'
e
at
e.
Ie: G
-
G
L,
and
Iex: TGe
-+
-
Ie*: Gu
1,
Finally, defining I,
Q
t'
i s rational so that n'
n" then
and
y(nt)
n't
=
+ t ")
y(t'
= y(t)n
t"
and
n"t
=
= y(t)n'+n"
=
i s a homomorphism.
be a
1-parameter subgroup.
Let
at
7'
be the
7'
i s t h e tan-
e
i s a 1-parameter sub-
7'
on a Lie group
=
Xa.b [X,YI
G
i s called l e f t invariant
for every a and i s also.
b
in
If X
G.
The Lie algebra
Q
[
of
if
and G
Y
i s the
I.
i s a c t u a l l y a Lie algebra because the Jacobi I d e n t i t y + "Y,Zl,Xl
+ "Z,Xl,YI
=
0
holds f o r a l l ( n o t n e c e s s a r i l y l e f t i n v a r i a n t ) v e c t o r f i e l d s
.
-1
THEOREM 2 1 . 3 .
RU- 1 I,',,
so i s certainly
X,Y
and
Z.
=
TU-~T,
u
E
G.
Since
the identity
I,
=
G
i s a C"
i t s tangent v e c t o r a t
e.
G
W
.
A s i n 5 0 we w i l l use t h e n o t a t i o n
PROOF:
e.
d e r i v a t i v e of
R,IeRil
homomorphism of R
and
1,
i s a n isometry which r e v e r s e s geodesics through of
be a Lie group with a l e f t and
<
i t r e v e r s e s geodesics through
e,
G
<
-1
I t i s w e l l known that a 1-parameter subgroup of
Princeton, 1946.)
e;
Let
r i g h t i n v a r i a n t Riemannian metric. I f X,Y,Z a r e l e f t i n v a r i a n t vector f i e l d s on G then: a) [x,Yl,Z> = <x,[Y,zl> b) R ( X , Y ) Z = ~[X,Yl,Zl C) = [X,YI,[Z,W1>
Now t h e i d e n t i t y
=
IT(u)
A 1-parameter subgroup G.
u
i s an isometry f o r any
G -1
r e v e r s e s t h e tangent space a t
shows that each
=
r e v e r s e s t h e tangent space of
TGe
an isometry on this tangent space.
shows that
Iy(t)Ie(~ = )y ( t ) u 7 ( t )
By induction it follows that
We have j u s t seen that
X
(La)*(Xb)
the bracket
Define a map
are isometries.
By Lemma 20.1
by Ie(a)
Then
R,
e.
T h i s completes t h e proof.
= 7.
"X,Yl,Zl By hypothesis
Now
2t).
=
v e c t o r space of a l l l e f t i n v a r i a n t v e c t o r f i e l d s , made i n t o a n a l g e b r a by
LEMMA 2 1 . 1 I f G i s a Lie group with a l e f t and right i n v a r i a n t metric, then G i s a symmetric space. The r e f l e c t i o n I, i n any p o i n t T E G i s given by t h e formula ~ , ( u ) = T o - ' , . PROOF:
7(O)
such that t h e tangent v e c t o r of
a r e l e f t i n v a r i a n t then
i s l e f t and r i g h t i n v a r i a n t .
G
7
7(u +
into
t l / t "
-
R
y:
A vector f i e l d
.
dp(d) dp(T)
If
By c o n t i n u i t y
Now l e t
group.
n.
and some i n t e g e r s
7(t1)y(t").
i s compact then such a metric can be constructed
G
<,>
Let
=
t
f o r some
y(u)
y(u + 2 t ) .
y ( t ) ~ ( u ) y ( t )=
f o r any i n t e g e r
L,:
R - + G be a geodesic with
y:
T.
into
i s determined by
(Compare Chevalley, "Theory of Lie Groups,"
Y
i n the direction X.
X k Y
f o r t h e covariant
For any l e f t i n v a r i a n t
X
t h e iden-
tity
x
FX
= 0
is s a t i s f i e d , s i n c e t h e i n t e g r a l e w e s of
X
a r e l e f t t r a n s l a t e s of
Parameter subgroups, and t h e r e f o r e a r e geodesics.
1-
IV. APPLICATIONS
114
$21
COROLLWY 2 1 . 4 . [ X , Y I , [X,YI only i f [x,YI
Theref ore ( X + Y) I- ( X + Y)
=
( X I- X) + ( X l- Y)
+
( Y I- X) + ( Y I- Y)
<
the s e t of
X I - Y + Y l - X
=
0 .
x
=
[X,YI
On the other hand k Y
-
Y I-
x
X
115
The s e c t i o n a l curvature < R ( X , Y ) X , Y > = i s always 2 0 . Equality holds i f and
= 0.
such t h a t
0
E
LIE GROUPS
>
c
R e c a l l that the c e n t e r
i s zero; hence
of a Lie algebra
[X,YI
= 0
for a l l Y
i s defined t o be
0
0.
E
COROLLARY 21.5. I f G b s a l e f t and r i g h t i n v a r i a n t m e t r i c , and i f t h e Lie algebra g has t r i v i a l c e n t e r , then G i s compact, w i t h f i n i t e fundamental group.
.
Adding these two equations we obtain:
by 58.5. d) NOW
.
2X I- Y
.
[X,YI
=
PROOF:
u n i t vector i n
r e c a l l the i d e n t i t y Y
(See 5 8 . 4 . ) constant.
<x,z>
=
< Y I-
x,z>
+
I-
<X,Y
z
>.
The l e f t s i d e of this equation i s zero, since S u b s t i t u t i n g formula ( d ) 0
T h i s follows from Meyer's theorem ( 8 1 9 ) .
=
and extend t o a orthonormal b a s i s
0
+
<X,Z>
is
K(X1 , X 1 )
F i n a l l y , using t h e skew commutativity of
[Y,XI,
...,%.
X1,
be any The R i c c i
< R ( X 1 ,Xi)Xl ,Xi>
=
i=1
> .
<X,[Y,ZI
X1
curvature
i n this equation we o b t a i n
< [Y,XI,Z>
Let
must be s t r i c t l y p o s i t i v e , since
we o b t a i n the required
*
-
K(X1 ,X1)
#
[Xl,Xil
1. Furthermore
i s bounded away from zero, since the unit sphere i n
Therefore, by Corollary 19.5, the manifold
formula
for some
0
G
0
i s compact.
i s compact.
T h i s r e s u l t can be sharpened s l i g h t l y as follows. <[X,YI,Z> By d e f i n i t i o n ,
- x 1
+ Y I- ( X k Z)
I- ( Y I- 2 )
[X,YI I-
+
z.
this becomes
- F[x,[Y,zll
1
+ FIY,[x,zll
1
+ T"x,Yl,zl
Using t h e Jacobi i d e n t i t y , t h i s y i e l d s t h e required formula R(X,Y)Z
(b) The formula
COROLLARY 2 1 . 6 . A simply connected Lie group G with l e f t and right i n v a r i a n t metric s p l i t s as a Cartesian product G ' x Rk where G I i s compact and Rk denotes t h e a d d i t i v e Lie group of some Euclidean space. Furthermore, t h e Lie algebra of G I has t r i v i a l c e n t e r .
i s equal t o
R(X,Y)Z
(a),
SiJbstituting formula
.
<X,[Y,Zl>
=
=
T;"X,YI,ZI 1
( c ) follows from ( a ) and ( b )
.
PROOF. g'
.
Let
c
=
IX
be the c e n t e r of t h e Lie algebra E
0 :<X,C>
be t h e orthogonal complement of
* It follows
that t h e t r i - l i n e a r f u n c t i o n X,Y,Z .-, Q X , Y I , Z > i s skewsymmetric i n a l l t h r e e v a r i a b l e s . Thus one obtains a l e f t i n v a r i a n t d i f f e r e n t i a l 3-form on G, r e p r e s e n t i n g a n element of t h e de R h a m cohomology group H3(G). I n this way Cartan w a s a b l e t o prove that H3(G) # 0 i f G i s a non-abelian compact connected Lie group. (See E. Cartan, "La Topologie des Espaces Representatives des Groupes de Lie," P a r i s , Hermann, 1936.)
GI x Rk possesses a
Conversely it i s c l e a r that any such product l e f t and right i n v a r i a n t metric.
if
x,y
E
and
01
C
c
E
hence
[x,yI
E
0'.
Hence
o for a l l
.
Then
c
=
I t follows that G
gr
C
E
and l e t
c 1
i s a Lie sub-algebra.
For
then
< [X,YI,C > Lie a l g e b r a s .
=
B
<X,[Y,CI> g
=
0;
s p l i t s as a d i r e c t sum
S p l i t s as a C a r t e s i a n product
i s c m p a c t by 21.5 and. GI'
GI
x G";
f l l (3
C
where
of GI
1s Simply connected and a b e l i a n , hence isomorphic
$21
IV. APPLICATIONS
116
LIE GROUPS
117
by
T h i s completes the
t o some Rk. (See Chevalley, "Theory of Lie Groups.")
.
Ad V(W)
proof.
=
[V,WI
we have
%
THEOREM 2 1 . 7 ( B o t t ) . Let G be a compact, simply connected Lie group. Then t h e loop space n(G) has t h e homotopy type o f a CW-complex with no odd dimensional c e l l s , and with only f i n i t e l y many X-cells for each even value of h . Thus t h e X-th homology groups of
n(G)
i s zero f o r
A
The l i n e a r transformation
odd, and i s
+
G
(-a1O
REMARK
an example, i f
1.
G
T h i s CW-complex w i l l always be i n f i n i t e dimensional.
i s t h e group
that t h e homology group REMARK 2.
0
Ad V =
(Ad
.
V)
i s skew-symmetric; that i s
-
<
W,Ad
)
This follows immediately from t h e i d e n t i t y 21.3a.
even.
X
(Ad V)
< Ad V(W),W' > an orthonormal b a s i s f o r
f r e e a b e l i a n of f i n i t e rank for
-
=
V(W')
- : 2 3 *.
g1
.
.
Therefore we can choose
so that t h e matrix of
As
>
Ad V
t a k e s t h e form
of u n i t quaternions, then we have seen
S3
i s i n f i n i t e c y c l i c f g r a l l even values of i.
HiO(S3)
It follows that t h e composite l i n e a r transformation
T h i s theorem remains t r u e even f o r a non-compact group.
(Ad V) (Ad V) 0
has
matrix
I n f a c t any connected Lie group contains a compact subgroup as deformation (See K. Iwasawa, On some types of topological proups, A n n a l s of
retract.
Mathematics 50 ( 1949), Theorem 6 . ) PROOF of 2 1 . 7 .
Choose two p o i n t s
conjugate along any geodesic.
p
and
By Theorem 17.3,
to
each
q X.
of index
X.
G
O(G;p,q)
type of a CW-complex with one c e l l of dimension p
in
q
A
which a r e not
has t h e homotopy
Therefore t h e non-zero eigenvalues of
f o r each geodesic from
By 6 1 9 . 4 t h e r e a r e only f i n i t e l y many
Thus i t only remains t o prove that the index
X
= - &Ad
V)*
a r e p o s i t i v e , and
occur i n p a i r s .
A-cells for
I t follows from 20.5 that t h e conjugate p o i n t s of
of a geodesic is
occur i n p a i r s .
always even.
p
7
starting at
v
=
$(O)ETG
P
t h e conjugate p o i n t s o f
p
with velocity vector
p
on
y
a r e determined by the
eigenvalues of t h e l i n e a r transformation
q:
TGp - T G
P
'
defined by
%(W)
=
R(V,W)V
=
Defining the a d j o i n t homomorphism Ad V:
kl
-.,
geodesic from
' a .
$[V,Wl,Vl
.
.*
along
7
also
I n other words every conjugate p o i n t has even m u l t i p l i c i t y .
Together with the Index Theorem, this implies that the index
Consider a geodesic
According t o 6 2 0 . 5
%
p
to
q
i s even.
T h i s completes t h e proof.
X
of any
MANIFOLDS OF MINIMAL GEODESICS
$22.
IV. APPLICATIONS
118
119
Thus we obtain: COROLLARY 2 2 . 2 . With t h e same hypotheses, isomorphic t o I X ~ + ~ ( M )f o r 0 5 i 5 X o -
Whole Manifolds of Minimal Geodesics
$22.
n(M;p,q) based on two p o i n t s
So f a r we have used a p a t h space
special position. let
p,q
A s an example l e t
be a n t i p o d a l p o i n t s .
d e s i c s from
p
to
q.
M
p,q
a smooth manifold of dimension
n
Rn2
is
and contain two conjugate p o i n t s , each of m u l t i p l i c i t y
2n.
=
Xo
For any non-minimal geodesic must wind one and a h a l f times around
S"";
n, i n i t s i n t e r i o r .
T h i s proves the following.
Then t h e r e a r e i n f i n i t e l y many minimal geo-
I n f a c t t h e space
)
2.
t h e ( n + l ) - s p h e r e . Evidently t h e hypotheses a r e s a t i s f i e d with
i n some
Sn+l , and
be t h e unit sphere
d
L e t u s apply t h i s c o r o l l a r y t o the case of two a n t i p o d a l p o i n t s on
However, Bott has pointed out
p,q E K which a r e i n "general p o s i t i o n . " t h a t very u s e f u l r e s u l t s can be obtained by considering p a i r s
ni(0
of m i n i m a l geodesics forms
COROLLARY 2 2 . 3 . (The Freudenthal suspension theorem.) The homotopy group ni(Sn) i s isomorphic t o n i + l ( S n + l ) f o r i 5 2n-2.
which can be i d e n t i f i e d with the equator
Sn C S n + l . We w i l l see t h a t this space of minimal geodesics provides a
Theorem
P
space
-
n
a l s o implies that t h e homology groups of the loop
22.1
ad
a r e isomorphic t o those of
f a c t follows from
22.1
for example Hu, p .
206.
i n dimensions
5
Lo
-
2.
This
together with the r e l a t i v e Hurewicz theorem. Compare a l s o
(See
J . H. C . Whitehead, Combinatorial
homotopy I , Theorem 2 . ) The r e s t of
$22
w i l l be devoted t o the proof of Theorem 2 2 . 1 .
The
proof w i l l be based on t h e following l e m a , which a s s e r t s t h a t t h e condition " a l l c r i t i c a l p o i n t s have index
2 Xot'
remains t r u e when a f u n c t i o n i s
jiggled slightly. 4
Let
f a i r l y good approximation t o t h e e n t i r e loop space Let
M
n(Sn+').
be a complete Riemannian manifold, and l e t
points w i t h distance
p(p,q)
=
be a compact subset of the Euclidean space Rn;
K
a neighborhood of
p,c, E M
K;
be
U - R
f:
be a smooth f u n c t i o n such that a l l c r i t i c a l p o i n t s of
>
U
and l e t
be two
a.
let
f in
K
have index
lo.
THEOREM 2 2 . 1 . I f the space ad of minimal geodesics from p t o g i s a t o p o l o g i c a l manifold, and i f every non-minimal geodesic from p t o q has index 2 L o , then t h e r e l a t i v e homotopy group n i ( n , n d ) i s zero f o r o 5 i < h0.
LEMMA
If
22.4.
i s "close" t o
g: f,
-
U R i s any smooth f u n c t i o n which i n t h e sense t h a t
I t follows that t h e i n c l u s i o n homomorphism ni(O
is a n isomorphism for i group
ni(n)
5
Xo
i s isomorphic t o
-
d 2.
)
-
ni(n)
ni+l(M)
f o r a l l values of
S. T. Hu, "Homotopy Theory," Academic P r e s s , 1959, p. 817.1.)
uniformly throughout
But i t i s w e l l known that t h e homotopy
111;
i.
f o r some s u f f i c i e n t l y small constant
then a l l C r i t i c a l P o i n t s of
(Compare
together with
K,
(Note that the a p p l i c a t i o n , points.)
f
g
g
in
K
have index
2
E,
Xo.
1s allowed t o have degenerate c r i t i c a l p o i n t s .
In
w i l l be a nearby f u n c t i o n without degenerate c r i t i c a l
IV. APPLICATIONS
120
PROOF of 2 2 . 4 .
The f i r s t d e r i v a t i v e s of
g
a r e roughly described
U; which vanishes p r e c i s e l y a t t h e c r i t i c a l p o i n t s of
d e r i v a t i v e s of
g
121
To complete the proof of 2 2 . 4 , i t i s only necessary t o show t h a t
the i n e q u a l i t i e s ( * ) w i l l be s a t i s f i e d providing t h a t
by the s i n g l e r e a l valued f u n c t i o n
on
MANIFOLDS OF MINIMAL GEODESICS
§ 22.
can be roughly described by
n
g.
for s u f f i c i e n t l y small
The second
T h i s follows by a uniform c o n t i n u i t y argument
E .
which w i l l be l e f t t o t h e reader (or by the footnote above ) .
continuous f u n c t i o n s
We w i l l next prove an analogue of Theorem 2 2 . 1
for r e a l valued
functions on a manifold.
as follows.
Let
Let 1
eg(x)
of
g
eg(x)
F...
n eigenvalues of t h e matrix
denote the x
5
2
has index
2
such that each
ei(x)
a2
( 6-$) . xi x j
i f and only i f the number
A
f:
Thus a c r i t i c a l p o i n t
MC
+
R be a smooth r e a l valued f u n c t i o n with minimum i s compact.
f-'[O,cl
=
0,
LIDlMA 2 2 . 5 . If the s e t Mo of minimal p o i n t s i s a manifold, and if every c r i t i c a l p o i n t i n M - Mo has index 2 A o , then nr(M,M 0 ) = 0 f o r 0 5 r < X o .
i s negative.
ek(x)
M
e x follows from the f a c t that the 63 * A-th eigenvalue of a symmetric matrix depends continuously on t h e matrix The c o n t i n u i t y of t h e functions
.
T h i s i n t u r n follows from the f a c t that the r o o t s of a polynomial depend
continuously on the polynomial.
U C M.
(See § 1 4 of K . Knopp, "Theory of Functions,
P a r t 11," published i n the United S t a t e s by Dover, 1 9 4 7 . ) LO m ( x ) denote t h e l a r g e r of the two numbers kg ( x ) and -eg ( x ) . g Similarly l e t mf(x) denote t h e l a r g e r of t h e corresponding numbers k f ( x )
Let
and index mf(x)
The hypothesis that a l l c r i t i c a l p o i n t s of
-e)(x).
2
>
implies that
A.
0
for a l l x
Let
6
>
i s so c l o s e t o
0
f
E
AO
-ef ( x )
>
0
whenever
kf(x)
f
= 0.
in
K
PROOF:
have
F i r s t observe t h a t
Mo
I n f a c t Harmer has proved that any manifold
neighborhood r e t r a c t .
mf
on
K.
absolute neighborhood r e t r a c t s , A r k i v f o r Matematik, V o l . 389-408.)
Replacing
i s joined t o the corresponding p o i n t of Mo
U
by a unique m i n i m a l geodesic.
Let '1
Thus
h: (Ir,Tr) be any map.
can be deformed i n t o
U
denote t h e u n i t cube of dimension
g
that
(1950), pp.
1
by a smaller neighborhood i f necessary, we may
U
assume that each point of
I n other words
Now suppose that
i s an absolute
Mo
(See Theorem 3.3 of 0 . Harmer, Some theorems on
K.
denote t h e minimum of
i s a r e t r a c t of some neighborhood
We must show that
h
-
r
<
Mo
within
M.
and l e t
Xo,
(M,Mo)
i s homotopic t o a map
h'
with
h'(Ir) C Mo. (*)
for a l l x
Let E
K.
c r i t i c a l p o i n t of
Then g
m ( x ) w i l l be p o s i t i v e for x g i n K w i l l have index 2 L o .
E
K;
hence every
mum of
f
c
be t h e maximum of
on t h e s e t
since each subset
M - U.
MC - U
f
on
h(1').
(The f u n c t i o n
f
Let
36
>
0
be t h e mini-
has a minimum on M - U
i s compact.)
Now choose a smooth f u n c t i o n
* T h i s statement can be sharpened as follows. Consider two nxn symmetric matrices. If corresponding e n t r i e s of t h e two matrices d i f f e r by a t most E, then corresponding eigenvalues d i f f e r by a t most ne. T h i s can be proved using Courant's minimax d e f i n i t i o n of t h e A - t h eigenvalue. (See J l of Courant, Uber d i e Ab%n@;igkeit d e r Schwingungszahlen e i n e r Membr an..., Nachrichten, Koniglichen G e s e l l s c h a f t d e r Wissenschaften zu Gottingen, Math. Phys. Klasse 1 9 1 9 , pp. 255-264.)
Which approximates I s p o s s i b l e by J 6 . 8 .
f
Mc+26 g: +R c l o s e l y , b u t has no degenerate c r i t i c a l p o i n t s . To be more p r e c i s e t h e approximation should be s o
close t h a t : (I)
I f ( x ) - g(X) 1
<
6
for a l l
x
E
Mc+26;
and
This
ILC
IV. APPLICATIONS
122 (2)
The index of pact s e t
g
§22.
a t each c r i t i c a l p o i n t which l i e s i n the com-
f-l[6,C+26]
Then
2 x0.
is
I t follows from Lema 2 2 . 4 that any
F
g
which approximates
f
I n f a c t the compact s e t
f-’[6,C+261
each of which l i e s i n a coordiKi, Lemma 2 2 . 4 can then be applied t o each %.
The proof of 22.5 now proceeds as follows. g-l[26,c+61
smooth on the compact region
c
f-’[
p o i n t s a r e non-degenerate, with index
2 Xo.
g-’ (-m,c+61
g-’ ( - m , 2 6 1
2
has t h e homotopy type of
The f u n c t i o n
~ , C + ~ F I ,
g is
and a l l c r i t i c a l
Hence the manifold w i t h c e l l s of dimension
attached.
Xo
Now consider the map
Ir,fr
h: Since
r
<
i t follows that
Xo
C g-’(-m,C+6l,M
MC,Mo
-C
h
0
i s homotopic within
. g-’
(-m,C+61
,Mo
to
a
Ir,fr
h’:
+
But t h i s l a s t p a i r i s contained i n Mo
h”:
within
Ir,fr
M.
-
It follows t h a t
g-’(-m,26I,M and
(U,Mo);
.
0
U
T h i s completes the proof of
Mo,Mo.
The o r i g i n a l theorem,
can be deformed i n t o
(M,Mo)
i s homotopic w i t h i n
h’
t o a map
22.5.
now can be proved as follows.
22.1,
Clearly
i t i s s u f f i c i e n t t o prove t h a t
nc,nd)
ni(lnt f o r a r b i t r a r i l y l a r g e values of
a smooth manifold
c.
=
o Int nc
A s i n $ 1 6 the space
..
I n t R C ( t O , t l , . ,tk) as deformation r e t r a c t .
contains The space
ad of minimal geodesics i s contained i n this smooth manifold. The energy f u n c t i o n Int nC(tO,tl,
...,t k ) ,
difficulty i s that
when r e s t r i c t e d t o
ranges over t h e i n t e r v a l To c o r r e c t this, l e t
[O,m).
F:
be any diffeomorphism.
+R,
almost s a t i s f i e s the hypothesis of
E(u)
the required i n t e r v a l
E: R
[d,c)
+
LO,=)
d
E
22.5.
<
c,
E:
ni(Int n c ( t o
can be
covered by f i n i t e l y many compact s e t n a t e neighborhood.
0
Int nc(to,t
s a t i s f i e s the hypothesis of 22.5.
s u f f i c i e n t l y c l o s e l y , t h e f i r s t and second d e r i v a t i v e s a l s o being approximated, w i l l s a t i s f y ( 2 ) .
MANIFOLDS OF MINIMAL GEODESICS
The only i n s t e a d of
i s zero fop
i
<
Lo.
,,..., tk)
+
R
Hence
,..., t k ) , n d
E
n i ( l n t nc,nd)
T h i s completes t h e proof.
IV. APPLICATIONS
124
Since U(n) Riemannian metric.
Cn
The u n i t a r y group U(n)
of a l l l i n e a r transformations
Cn + C n
S:
i s defined t o be t h e group
n x n
complex matrices
denotes t h e conjugate transpose of For any
n x n
such t h a t
S
A
S S*
=
where
I;
t h e exponential of
A
S*
If
and
A
B
1
+ -3. A3
=
E
let
g
=
.
T(exp A ) T
-1
T h i s i n n e r product on
In particular:
0
i n t h e space of
n x n matrices diffeomorphically onto a neighborhood of I . A i s skew-Hermitian ( t h a t i s i f A + A* = O), then it fol-
If
U(n) can be defined as follows. denote the r e a l p a r t of the complex
.
I
The f u n c t i o n exp maps a neighborhood of
(4)
number
...
+
A,B
Riemannian metric on U ( n ) . (exp A)(exp -A)
(3)
U(n) (by Assertion ( 2 ) above) ; and
Clearly t h i s i n n e r product i s p o s i t i v e d e f i n i t e on
(exp A) (exp B)
=
e x p ( t A)
A s p e c i f i c Riemannian metric on
i s defined
A
commute then
exp ( A + B)
-L
d e f i n e s a 1 -parameter subgroup of
Given matrices
The following p r o p e r t i e s a r e e a s i l y v e r i f i e d : (1) exp (A*) = (exp A)*; exp (TAT- ’ ) (2)
the correspondence
A
hence defines a geodesic.
I + A + &A2
=
I n f a c t f o r each skew-Hermitian matrix
t
by the convergent power s e r i e s expansion exp
fold.
i s the
U(n)
S.
complex matrix
f i n e d (as i n 9 1 0 ) by following geodesics on the r e s u l t i n g Riemannian mani-
which preserve this i n n e r
Equivalently, u s i n g the matrix r e p r e s e n t a t i o n ,
group of a l l
TU(nII - U ( n )
defined by exponentiation of m a t r i c e s coincides w i t h the f u n c t i o n exp de-
of complex numbers, w i t h t h e u s u a l H e r -
be the space of n - t u p l e s
m i t i a n inner product.
product.
Note t h a t t h e f u n c t i o n exp:
F i r s t a review of w e l l known f a c t s concerning t h e u n i t a r y group.
125
i s compact, i t possesses a l e f t and r i g h t i n v a r i a n t
The Bott P e r i o d i c i t y Theorem f o r t h e Unitary Group.
523.
Let
THE BOTT PERIODICITY THEOREM
523.
g
,
determines a unique l e f t i n v a r i a n t
g
To v e r i f y t h a t the r e s u l t i n g metric i s a l s o
right i n v a r i a n t , we must check that i t is i n v a r i a n t under t h e a d j o i n t a c t i o n of
U(n)
on 0 .
DEFINITION of the a d j o i n t a c t i o n .
Each
S E U(n)
determines an
i n n e r automorphism lows from ( 1 ) and ( 3 ) that exp A
i s unitary.
Conversely i f
exp
A
is
belongs t o a s u f f i c i e n t l y small neighborhood of 0 , then i t follows from ( 1 ) , ( 3 ) , and ( 4 ) that A + A* = 0 . From t h e s e f a c t s one
u n i t a r y , and
x
A
of the group U ( n ) .
-s x
s-l
=
(L~R~-’)X
The induced l i n e a r mapping
e a s i l y proves t h a t :
( 5 ) U(n) (6)
i s a smooth submanifold
t h e tangent space
TU(n)I
of t h e space of
n x n matrices;
can be i d e n t i f i e d w i t h t h e space of
n x n skew-Hermitian m a t r i c e s . Therefore t h e Lie a l g e b r a
g
of
U(n)
i s called U(n).
for
t h e bracket product of l e f t i n v a r i a n t vector f i e l d s corresponds t o the
Ad(S).
=
AB - BA
of matrices.
Ad(S)
i s a n automorphism of the Lie algebra of
Using Assertion ( 1 ) above we o b t a i n t h e e x p l i c i t formula Ad(S)A
For any tangent vector a t I extends uniquely t o a l e f t i n v a r i a n t v e c t o r f i e l d on U ( n ) . Computation shows that
[A,B1
Thus
can a l s o be i d e n t i f i e d with
the space of skew-Hermitian matrices.
product
Ad(S).
A E
Q,
s
E
=
3M-l
,
U(n).
The i n n e r Product I n f a c t if A1
=
Ad(S)A,
i s i n v a r i a n t under each such automorphLsm B1 = Ad(S)B then t h e i d e n t i t y
THE BOTT PERIODICITY THEOREM
$23.
IV. APPLICATIONS
126
SO
implies that
t h a t we may a s w e l l assume that
A
127
i s already i n diagonal form
t r a c e (AB") ;
t r a c e ( A ~ B ~ *= ) t r a c e (sAB*s-') =
A =
and hence t h a t (A1,Bl
>
.
=
I t follows that the corresponding l e f t i n v a r i a n t metric on U ( n )
I n t h i s case,
i s also
right i n v a r i a n t .
Given T
E
A
E
so that
U(n)
g
we know by ordinary matrix theory
TAT-'
that t h e r e e x i s t s
i s i n diagonal form
so that
where t h e
a i l s a r e real.
Also, given any
there i s
S E U(n),
a
T
E
u(n)
exp A
-I
=
if and only if
f o r some odd i n t e g e r s
k,,
...
has the form
A
7%.
Since the l e n g t h of the geodesic
is n
where again the
are real.
exp:
Thus we see d i r e c t l y that
B
-+U(n)
i s onto.
.
c.
i s defined as t h e subgroup of
terminant
1.
i n t h e same way.
v
c o n s i s t i n g of matrices of de-
U(n)
I f exp i s regarded as the ordinary exponential map of
d e t (exp A) 3 s i n g this equation, one may show that the s e t of a l l matrices
such that
A
etrace
=
81
,
A + A*
0
=
and
SU(n)
is
I n other words, we look for a l l
E
A
i s such a matrix;
U(n) be such that exp TAT-l
TAT-l =
E
i s i n diagonal form.
T(exp A)T-l
=
T(-I)T-'
is
A
and
Since Cn
&I.
observe t h a t
Now, regarding A
i s completely
the v e c t o r space c o n s i s t i n g of a l l
Eigen( - i n ) ,
the space of a l l
s p l i t s a s the orthogonal sum
i s then completely determined by
A
from I
Cn
x
Cn.
v
E
Cn
E i g e n ( i n ) 6?
Eigen(in),
Thus the space of a l l minimal geo-
-I may be i d e n t i f i e d with t h e space of a l l
to
and S U ( n ) ,
has Components of varying dimensions.
I
to
-I.
by SU(n)
T h i s d i f f i c u l t y may be removed by and s e t t i n g n = 2m. I n this case, a l l the
above considerations remain v a l i d .
a, +...+
a
-
2m
-
o w i t h ai
=
2
n
But t h e a d d i t i o n a l condition t h a t r e s t r i c t s Eigen(in) t o b e i n g an a r b i -
trarg m dimensional sub-vector-space of C2m. This proves t h e following:
Then =
=
= 0.
TU(n)I = g such that exp A = -I. i f i t i s n o t already i n diagonal form, l e t A
t
Unfortunately, this space i s r a t h e r inconvenient t o use s i n c e i t
trace A
from
to
= 0
sub-vector - spaces of Cn.
r e p l a c i n g U(n) we begin by considering the s e t o f a l l geodesics i n U(n)
-inv.
to
Eigen(in), inv;
=
the matrix
d e s i c s i n U(n)
the Lie algebra of =
Av
Av
which i s an a r b i t r a r y subspace of
I n order t o apply Morse theory t o t h e topology of U ( n )
T
and i n t h a t case, t h e l e n g t h i s
such that
Eigen(-in),
.
A
c Cn
t
determines a minimal geodesic i f and only i f each
A
as a l i n e a r map of Cn
A
such that
matrices, i t i s easy t o show, u s i n g t h e diagonal form, that
Suppose
Thus
5 I,
equals
such an
from
m,the l e n g t h of the geodesic determined by
determined by specifying One may t r e a t the s p e c i a l unitary group SU(n)
SU(n)
=
Jk? +. .+
$ ails
[A[
t - exp t A
-1
1
LEMMA
called the
The space of m i n i m a l geodesics from I t o -I i n t h e s p e c i a l u n i t a r y group SU( 2m) is homeomorphic t o t h e complex Grassmann manifold Gm(Cm), c o n s i s t i n g of a l l m dimensional v e c t o r subspaces of c2m. 23.1.
( i - i l - s t s t a b l e homotopy group of t h e u n i t a r y group.
fli-l U. The same exact sequence shows that, f o r
nrn
-
U(m)
i n sU(2m)
to
I
Combining t h e s e two lenunas w i t h 822 w e obtain:
.
an
for
5
i
r
x2m
u(m+i)
n2m U
s
i = 2m+i,
t h e homomorphism
is onto.
From the e x a c t sequence of the f i b r a t i o n
-I
we see that
-
u(m) u(m))
ni(U(2m)/
-c
U(2m)
the c o s e t space
o for i <
=
U ( 2 m ) / U(m) x U(m).
sequence of t h e f i b r a t i o n
-
u(m)
ni+,SU(2m)
~ ( 2 m /)
for
i
5
2m.
Gm(Cem) can be i d e n t i f i e d w i t h (Compare Steenrod 8 7 . ) From t h e e x a c t
- cm(c2”)
U(m)
N Z ~ G ~ ( C ’ ~)
On t h e other hand u s i n g standard methods of homotopy theory one o b t a i n s somewhat d i f f e r e n t isomorphisms.
u(m)
-
we see now t h a t
2111.
U(a)/
-c
The complex Grassmann manifold
THEOREM 2 3 . 3 ( B o t t ) The i n c l u s i o n map Gm(Can) fi(SU(2m); 1,-I) induces isomorphisms of homotopy groups i n dimensions 5 2m. Hence ni Gm(C )
They w i l l
The complex S t i e f e l m a n i f o l d is defined t o be t h e c o s e t space U ( m ) /V(m).
Every non-minimal geodesic from has index >_ an+2.
129
be denoted b r i e f l y by
We w i l l prove t h e following r e s u l t a t the end of this s e c t i o n . 23.2.
THE BOTT PERIODICITY THEOREM
823.
IV. APPLICATIONS
128
u(m)
ni-l
2m.
-
F i n a l l y , f r m t h e e x a c t sequence of t h e f i b r a t i o n SU(m)
-u(m)
S’
w e see t h a t
n
SU(m)
I
U(m)
n
for
j
j
completes the proof of Lemma 2 3 . 4 .
+
I.
his
j
Combining L e m 23.4 w i t h Theorem 23.3 w e see t h a t for
1.
for PROOF.
-
-
F i r s t note that f o r each U(m)
-
-
U(m+ 1 )
-c
ni
u(m)
s ~ ~ + ni-’ ’
-
ni-l
Sm+’
- ...
of this f i b r a t i o n we see that ni-l
U(m) s ni-’
U(m+l)
for i
5 an.
(Compare Steenrod, “The Topology of F i b r e Bundles,” Princeton, 1951, p. 35 and p . 90.) It follows that the i n c l u s i o n homomorphisms ni-’
U(m)
a r e a l l ismorphisms f o r
-
i
u(m+l)
5 an.
-
5
i
5
2m.
ni-l
U(m)
r
niGm(C 2m )
Y
ni+l SU(2m)
ni-, U(m+2)
- ...
These mutually iSOmOrPhiC Goups are
s
“i+l
Thus we obtain:
U
PERIODICITY THEOREM.
S2m+l
niV1 U\rn+l)
1
=
there e x i s t s a fibration
m
From the homotopy e x a c t sequence
...
U
fli-l
#
j
r
ni+l
U
for
12 I.
To evaluate t h e s e groups i t is now s u f f i c i e n t t o observe that U ( i ) I
is a c i r c l e ;
SO
that no
u
=
no
x1
U
=
n1 U(1)
As a check, s i n c e SU(2)
U(1)
=
0
Z
is a 3-sphere, w e have:
n2u
=
u
=
n*SU(2)
=
0
SU(2) Y we have proved the following r e s u l t .
z
n3
(infinite cyclic).
n3
.
IV. APPLICATIONS
130
THEOREM 2 3 . 5 ( B o t t ) . The s t a b l e homotopy groups ni u of t h e u n i t a r y groups a r e p e r i o d i c with period 2 . I n f a c t t h e groups no
u
2
...
U r n 5 U r
...
r n,U
s
n4u
a r e zero, and t h e groups n1
u
g n3
2
KA(W
Now we f i n d a b a s i s f o r For each place,
must compute t h e index of any non-minimal geodesic from
i s even.
n
matrix
A
E
n x n
A
t h e matrix
with
Eja
+I
to
-I
2
on
have t h e form
inkl
<e
j
2
where
kl
A given
,...
7%
are
%(kj
with
k
j
>
-
I
along t h e geodesic
.
exp( t A )
values of t h e l i n e a r transformation KA:
g’
-
Ejn
=
R(A,W)A
2
e
=
% ( k j - ka)
2
>
-
ke)2
.
r(t)
exp t A .
=
Each eigenvalue
gives r i s e t o a s e r i e s of conjugate p o i n t s along
0
t (See $20.5.)
g’
=
2n/v5,
n/&,
4 “A,WI,AI .
e,
t h i s gives
6
4
=
- ’1 ’ k j - ‘P ’ k j - P’ ’ The number of such values of t i n t h e open i n t e r v a l k - k equal t o -J-.-& - I . Now l e t u s apply t h e Index Theorem.
(Compare $ 2 1 . T . ) A
2(
t o t h e index.
W
[A,WI
=
=
For each
... . (0,l)
is e v i d e n t l y
j , Q with
k
>
kp
we o b t a i n two copies of t h e eigenvalue %2( k j - k Q ) 2 , and hence a jc o n t r i bution of
i s the diagonal matrix
If
y
... .
3.1-6,
S u b s t i t u t i n g i n t h e formula f o r
t
=
+i i n the
corresponding t o the values
where KA(W)
with
( Q j ) - t hplace i s an eigenvector,
’j
... 2 %.
g’
Each such eigenvalue i s t o be counted twice.
kp.
According t o Theorem 20.5 t h e s e w i l l be determined by t h e p o s i t i v e eigen-
2 k2 2
(jP)-th
%(
Now consider t h e geodesic
t
k,
i n the
Each diagonal matrix i n 8 ’ i s an eigenvector with eigenvalue 0. Thus the non-zero eigenvalues of KA a r e the numbers 2 kj - kQ)
-I i f and only i f
to
I
,..., i n %
We must f i n d t h e conjugate p o i n t s t o
the matrix
+i i n t h e
a l s o with eigenvalue
odd i n t e g e r s with sum zero.
with
a s follows:
2
( j Q ) - t hplace and
skew-Hermitian matrices w i t h t r a c e zero.
We may assume that
KA,
( P j ) - t h place and zeros elsewhere, i s i n
i n the
G ( k . - kp) . J S i m i l a r l y f o r each
T(SU(n))I
=
corresponds t o a geodesic from
the eigenvalues of
I
We
R e c a l l that t h e Lie a l g e b r a
a’ c o n s i s t s of a l l
-1
.
- k p ) 2 wjg)
c o n s i s t i n g of eigenvectors of
g‘
j
( %(kj
=
and i s a n eigenvector corresponding t o the eigenvalue
The r e s t of §23 w i l l be concerned with t h e proof of Lemma 2 3 . 2 .
where
121
and
are i n f i n i t e cyclic.
sU(n),
THE BOTT PEiiIODICITY THEOREM
$23.
Adding over a l l
k. - k +A
-
1)
j , Q this gives the formula
then a s h o r t computation shows
( w J. P )
that (in(kj
-
kQ)wjn)
,
hence
for t h e index of t h e geodesic [A,[A,WII
=
(-n2(kj
-
kp)2 Wjp)
,
A s a n example, i f
y
7.
is a IIIinimal geodesic, then a l l of t h e kj
P
$24.
IV. APPLICATIONS
132
are equal t o +.
1
.
Hence
as was t o be expected.
A = 0,
Let
CASE 1 .
a r e ( s a y ) negative.
n
133
The P e r i o d i c i t y Theorem for t h e Orthogonal Group.
$24.
Now consider a non-minimal geodesic.
THE ORTHOGONAL GROW
= 2m.
T h i s s e c t i o n w i l l c a r r y out a n analogous study of t h e i t e r a t e d loop
A t least
of t h e
m+l
case a t l e a s t one of t h e p o s i t i v e
A2
kits
ki
2
must be
( 3 - (-1)
-
2)
=
3,
I n this
and we have
2(m+1)
space of t h e orthogonal group.
However t h e treatment i s r a t h e r sketchy, and
The p o i n t of view i n this s e c t i o n was suggested
many d e t a i l s a r e l e f t out.
by t h e paper On C l i f f o r d modules ( t o a p p e a r ) , by R . Bott and A . Shapiro,
.
which r e l a t e s t h e p e r i o d i c i t y theorem w i t h t h e s t r u c t u r e of c e r t a i n C l i f f o r d
1
algebras.
CASE a l l are
+-
2.
1.
m
of the
ki
Then one i s
2
m are negative b u t n o t
are p o s i t i v e and 3
and one is
Consider t h e v e c t o r space Rn
s o that
-3
mf
(3 1
-
(- 1)
-
2)
+ m$
(1
-
(- 3)
-
2)
+ (3
-
(- 3)
-
T : Rn *Rn
2)
1
which preserve this i n n e r product. real
n x n
matrices
T
Alternatively
such that
T T*
=
I.
O(n)
c o n s i s t s of a l l
T h i s group O(n)
considered as a smooth subgroup of t h e u n i t a r y group U ( n ) ; Thus i n e i t h e r case w e have
A
2 2m+2.
T h i s proves Lemma 2 3 . 2 ,
The
c o n s i s t s of a l l l i n e a r maps
orthogonal group O(n)
A 2
w i t h t h e u s u a l i n n e r product.
can be
and t h e r e f o r e
Inherits a r i g h t and l e f t i n v a r i a n t Riemannian metric. Now suppose t h a t
and t h e r e f o r e completes t h e proof of t h e Theorem 2 3 . 3 .
J : Rn -Rn,
the i d e n t i t y on Rn
J2
i s even.
A complex s t r u c t u r e
DEFINITION.
mation
n
J
on R~
i s a linear transfor-
belonging t o t h e orthogonal group, which s a t i s f i e s
-I. The space c o n s i s t i n g of a l l such complex s t r u c t u r e s
=
w i l l be denoted by
n1( n ) .
We w i l l see p r e s e n t l y (Lemma 2 4 . 4 ) that O l ( n )
is a smooth sub-
manifold of t h e orthogonal group O ( n ) .
REMARK. Given some f i x e d 9f O(n) J 1t
J,
E
n,(n)
let
U(n/2)
be t h e subgroup
c o n s i s t i n g of a l l orthogonal transformations which commute with
Then n , ( n )
can be i d e n t i f i e d with t h e q u o t i e n t space O ( n ) m n / e ) .
LEMMA
24.1. The space of m i n i m a l geodesics from I t o -I on O ( n ) is h m e m o r p h i c t o t h e space n l ( n ) of complex s t r u c t u r e s on R".
PROOF:
The space O(n)
Wthogonai matrices.
the space of
n x n
can be i d e n t i f i e d w i t h t h e group of
I t s tangent space
s
= TO(nII
skew-sgmmetric matrices.
n x n
can be i d e n t i f i e d with
Any geodesic
7
with
where 7(0) =
can be w r i t t e n uniquely as
I
for some
A
n
Let T
E
O(n)
J J
O:y
9.
E
2m.
=
Since
TAT-’
=
a 1, a 2 , .
. . ,am2 0.
A s h o r t computation shows that
sin na,
0
0
- sin n a l
cos “al
0
0
Thus exp(nA)
0
cos n a 2
sin na2
0
0
- sin n a 2
cos < a 2
...
Therefore the geodesic
If
7
=
a2
...
...
=
r(t)
...
=
Thus
-J.
i s skew-symmetric.
J
=
e x p ( n t A)
a1,a2,
am =
from
and some
0
Now the i d e n t i t y
T.
and hence that
exp n J
Hence
I
\
...,am
to
J2
implies
-I
=
-I. T h i s completes the
=
A)T-’
t
the form
is
--L
e x p ( n t A)
I
where
a r e odd i n t e g e r s . 2
-I
with
A =
\
2
2(a1
to
Suppose that the geodesic has
a,
2 a 2 2 . . . 2 am >
Computation shows t h a t the
a r e odd i n t e g e r s .
0
non-zero eigenvectors of the l i n e a r transformation
i s e a s i l y seen t o be
n
...
-I i f and only i f
i s equal t o
and only if a 1
...
0
The i n n e r product
=
LEMMA 2 4 . 2 . Any non-minimal geodesic from i n O(2m) has index 2 2 m - 2 .
T(exp
*.. ...
...
J*
The proof i s similar t o that of 2 3 . 2 .
c o s na,
\
-I t h i s implies that
Together with t h e i d e n t i t y
J.
proof.
0
equal t o
/
denotes t h e transpose of
f o r some & , , a 2 ,... am -> that a, = . .. = = 1;
-?2* ..
-a with
=
J*
135
i s skew-symmetric, t h e r e e x i s t s an element
A
(
so that
THE ORTHOGONAL G R O W
$24.
IV. APPLICATIONS
134
I)
for each i
2)
f o r each
i
< <
j
the number
j
with
ai
KA
=
(ai + a j ) 2 / 4 ,
# a j the number
-
& (Ad A ) 2
and
-
(ai
2
+ a2 +...+a,).
Each of these eigenvalues i s t o be counted twice.
-I i s minimal i f
1.
C
=
(ai + a j -
2)
+ ai
i < j
i s minimal then
For a minimal geodesic we have = 0,
as expected.
a1
1>
a
=
1
(3+1-2)
+
0
(ai - a j -
2)
.
j
a2
=
aj)2/ 4.
This l e a d s t o the formula
=
... -- & m = ’
For a non-minimal geodesic we have
2
are
2m
-
a,
s o that
2 3;
so t h a t
2.
2
hence
A
i s a complex s t r u c t u r e .
Conversely, l e t
J
This completes t h e proof.
be any complex s t r u c t u r e .
nal we have
JJ*
=
I
Since
J
i s orthogo-
NOW l e t u s apply Theorem 2 2 . 1 .
The two lennnas above, together w i t h
IV. APPLICATIONS
136
the statement t h a t a , ( n )
0 24.
i s a manifold imply the following.
a,
PROOF of 2 4 . 4 .
-
0,( n )
=
fli+,
Any p o i n t i n O(n)
expressed uniquely i n t h e form
THEOREM 2 4 . 3 ( B o t t ) . The i n c l u s i o n map (n) R O(n) induces isomorphisms of homotopy groups i n dimensions < n-4. Hence fii
THE ORTHOGONAL GROW
sgmmetric matrix. J
O(n)
exp A,
c l o s e t o the i d e n t i t y can be
where
Hence any p o i n t i n O ( n )
can be expressed uniquely as
137
J exp A,;
A
is a "small,"
skew-
c l o s e t o t h e complex s t r u c t u r e
where again
i s s m a l l and
A
skew.
for i .( n-4. ASSERTION Now we w i l l i t e r a t e t h i s procedure, studying t h e space of geodesics from
J
to
high power of
Let commute
*,
i n n l ( n ) ; and s o on.
-J
J
r # s.
anti-commutes w i t h
J,,
. . ., J k - l
be f i x e d complex s t r u c t u r e s on Rn
anti-commutes with
If A I
which anti-
i n the sense that JsJr
=
exp(J-'A J ) exp A
( J exp A ) 2
Therefore
+
=
J,
then
J-lA J
=
J - ' ( e x p A ) J exp A
.
hence
-A
Since
denote t h e s e t of a l l complex s t r u c t u r e s J
which anti-commute with t h e fixed s t r u c t u r e s
J,, . . . , J k - l .
A
J- A J
s o that
C l e a r l y each n , ( n )
C
... C n l ( n ) C
i s a compact s e t .
n a t u r a l t o define n 0 ( n )
t o be
O(n)
.
anti-commutes w i t h
J1,...,Jk-l
To complete t h e d e f i n i t i o n i t i s
=
-I
then the
=
I
.
-A
=
J.
J exp A
anti-commutes w i t h t h e complex s t r u c t u r e s
i f and only i f
commutes with
A
J1,...,Jk-l.
The proof i s similar and straightforward.
O(n)
Note that Assertions 1 and 2 b o t h put l i n e a r conditions on
**
Thus a neighborhood of
LEMMA 2 4 . 4 . Each n k ( n ) i s a smooth, t o t a l l y geodesic submanifold of O ( n ) . The space of minimal geodesics from Jf t o -Jf i n O,(n) i s homeomorphic t o n f + l ( n ) , f o r
A
J
i n %(n)
c o n s i s t s of a l l p o i n t s
A.
J exp A where
ranges over a l l small m a t r i c e s i n a l i n e a r subspace of the Lie algebra
T h i s c l e a r l y implies t h a t n k ( n )
O < f < k .
g.
i s a t o t a l l y geodesic submanifold of
O(n).
It follows that each component of
For t h e isometric r e f l e c t i o n of O ( n ) cally carry n k ( n )
A
ASSERTION 2 . C
( J exp A ) 2
i s s m a l l , this implies that 1
.
Thus we have n,(n)
Conversely i f
exp(J-lA J ) exp A
Jl,...,Jk-l.
Let n,(n)
-I.
=
=
above computation shows that
0
Suppose that t h e r e e x i s t s a t l e a s t one other complex s t r u c t u r e
DEFINITION.
n,(n)
i s a s y m e t r i c space.
i n a p o i n t of n k ( n ) w i l l automati-
to itself.
Now choose a s p e c i f i c p o i n t
2
exists a complex s t r u c t u r e J J = J@
w e see e a s i l y t h a t
cmutes with
*
These s t r u c t u r e s make Rn i n t o a module over a s u i t a b l e C l i f f o r d algebra. However, t h e C l i f f o r d a l g e b r a s w i l l be suppressed i n t h e following presentation.
**
J.
i s d i v i s i b l e by a PROOF:
which anti-commutes w i t h
on R n
n
A
2.
JrJs
for
Assume that
i s a complex s t r u c t u r e i f and only i f
J exp A
1.
A submanifold of a Riemannian manifold i s c a l l e d t o t a l l y geodesic if each geodesic i n t h e submanifold i s a l s o a geodesic i n l a r g e r manifold.
Jk.
Jk
E
nk(n),
and assume that t h e r e
which anti-commutes w i t h
...,Jk.
Setting
i s a l s o a complex s t r u c t u r e which anti-
A
However,
J1,
A
comutes w i t h
J1
,..., J k - l .
Hence t h e
formula
t d e f i n e s a geodesic f r m Jk
+
to
J k e x p ( n t A)
-Jk
i n llk(n).
Since t h i s geodesic i s
minimal i n o ( n ) , it i s c e r t a i n l y m i n i m a l i n n k ( n ) .
$24. THE ORTHOGONAL GROUP
IV. APPLICATIONS
136
a,(n).
Conversely, let 7 be any minimal geodesic from Jk to -Jk in
Cn/, as being a vector space gl4 over the quaternions H .
Setting y(t)
be the group of isometries of this vector space onto itself. Then n,(n)
=
Jk exp(nt A),
it follows from
a complex structure, and from Assertions
... ,Jk-l
Jl,
REMMRK.
geodesic
y
1,2
and anti-commutes with Jk.
that A
that A is
24.1
Before going further it will be convenient to set n
It follows easily that Jip
The point Jip
E
y($)
of the geodesic.
PROOF: Any complex structure J3 e n 3 ( 1 6 r )
In order to pass to a stable situation, note that nk(n)
of
can be
. . .,Ji
structure J f3 21; =
1,
.
on Rn’
Then each J
E
nk(n)
@
=
R’6r into two mutually orthogonal subspaces V, and V,
as fol-
are
1.
b t V, C R16r be the subspace on which JlJ2J3 equals + I; and let V,
JA for
...,k - 1 .
be the orthogonal subspace on which it equals -I. Then clearly
DEFINITION. Let SFaces a,(n),
H4r
determines a splitting
J,J2J3J,J2J3 equal to + I. Hence the eigenvalues of J,J,J3
determines a complex
on Rn @Rn’ which anti-commutes with J,
16r.
lows. Note that JlJ2J3 is an orthogonal transformation with square
imbedded in nk(n+nt) as follows. Choose fixed anti-commuting complex structures J;,
=
LEMMA 24.6 - (3). The space a,( 1 6 r ) can be identified with the quaternionic Grassmam manifold consisting of quaternionic subspaces of
flk+l(n) which corresponds to a given
has a very simple interpretation: it is the midpoint
Let Sp(n/4)
can be identified with the quotient space U(n/2)/ Sp(n/‘c).
commutes with
This completes the proof of 24.4.
belongs to Qk+l(n).
CY
139
nk
denote the direct limit as n-r
m
=
of the
with the direct limit topology.
(I.e., the fine topology.) The space 0 =nois called the infinite orthogonal group.
V, f3 V,.
Since JlJ,J3
that both V, and V,
.
commutes with J, and J,
it is clear
are closed under the action of J, and J,.
Conversely, given the splitting H4k
=
V,
@
orthogonal quaternionic subspaces, we can define J3
V, E
into mutually
n3(l6r) by the
identities It is not difficult to see that the inclusions give rise, in the limit, to inclusions
nk+l ank .
-r
0
I
n,(n)
-r
-
THEOREM 24.5. For each k 2 0 this limit map fik+l 0 ak is a homotopy equivalence. Thus we have isomorphisms nh 0
E
nhw1
n, z
nh-,
n,
g
...
=
J31V2
=
-J1J,IV1 JlJ21V2
.
T h i s proves Lemma ?4.6 - ( 3 ) .
The space “lnh-l
J3P1=
*
3
is awkward in that it contains components of
(16r)
varying dimension. It is convenient to restrict attention to the component
The proof will be given presently.
of largest dimension: namely the space of 2r-dimensional quaternionic sub-
Next we will give individual descriptions of the manifolds for
k
= 0,1,2,
ak(n)
...,8. n,(n)
this way, so that dimHV1
is the orthogonal group.
nl(n) is the set of all complex structures on Rn
.
Given a fixed complex structure J, we may think of Rn as being a vector space Cn/2 over the complex numbers. n,(n)
spaces of H4r. Henceforth, we will assume that J a
can be described as the set of “quaternionic structures” on
the complex vector space Cn’2.
Given a fixed J2 en2(n)
We m y think of
=
dimHV2
3
has been chosen in
= 217.
LEMMA 24.6 - ( 4 ) . The space n4(16r) can be identified with the set of all quaternionic isometries from Vl to V,. Thus n4(16r) is diffeomorphic to the symplectic group S p ( 2 ~ ) .
PROOF: Given J,, E n4( 16r) Commutes with JlJ,J3
.
Hence J3J4
note that the product J3J4 antimaps V,
to V2 (and V,
to V 1 )
$24. THE ORTHOGONAL GROUP
IV. APPLICATIONS
140
W i n t o two mutually orthogonal subspaces.
s p l i t t i n g of Since
J3J4 commutes w i t h
and
J,
we see that
J,
i t i s e a s i l y seen that
-V,
Conversely, given
identities:
his proves 24.6 - ( 4 )
-
REMARK.
Jil T
=
J41V2
= -T-’
-
phisms of
PROOF:
Given
commutes with
J,J,J3
Let
Since
J2
W C V,
( 5)
=
X C W, i t i s not hard t o see that
J 6 i s unique-
If O(2r) C U(2r)
keeping
W
X
denotes t h e p o u p of complex automor-
f i x e d , then t h e q u o t i e n t space
U( 2 r ) / O ( 2r)
can
a,(
J5
E
a,( 1 6 r )
note that t h e transformation
J1J4J5
+ I . Thus J,J4J5 maps V,
and has square
V,
J,J4J5 coincides with
J,J4J5, i t follows that
J2W
J,J4Jg equals
c
V,
Given
J,J6J7
commutes with
square
+ I . Thus
J1J6J7
+ I.
equals
J7,
anti-commuting with
J,J2J3, w i t h
J,,
...,J6
J,J4J5, and with
J , J 6 J 7 determines a s p l i t t i n g of
orthogonal subspaces:
into
i n t o two mutually orthogonal sub-
be the subspace on which
anti-commutes w i t h
PROOF:
X,
(mere
J,J6J7 equals
- I ) . Conversely, given
X, C X
X
note that
J2J4J6; and has
i n t o two mutually
+I) and
X,
(where
it can be sharn that
J7
I s uniquely determined.
is
T h i s space a7(16r), l i k e n3(16r),has components of varying dimen-
-I. Clearly
sion.
W.
Again we w i l l r e s t r i c t a t t e n t i o n t o t h e component of l a r g e s t dimen-
sion, by assuming that Conversely, given t h e subspace
dim X ,
W, i t i s n o t d i f f i c u l t t o show t h a t
REMARK. morphisms of
V,
If
U ( 2 r ) C Sp(2r)
keeping W
ASSERTION.
denotes t h e group of quaternionic auto-
f i x e d , then t h e q u o t i e n t s w c e
can be i d e n t i f i e d with n 5 ( 1 6 r )
Sp( 2 r ) / U( 2 r )
commutes b o t h with Since
J6
.
E
n6(16r) note that the transformation
J,J2J3 and with
(J,J4J6)2
=
r.
=
I,
J 1 J 4 J g . Hence
it follows that
J,J4J6 maps
J2J4Jg
The l a r g e s t component of
n7(1 6 r )
i s diffeomorphic t o R2’.
a
UMNA 2 4 . 6 - ( 8 ) . The space a , ( 1 6 r ) can be i d e n t i f i e d w i t h t h e s e t of a l l r e a l i s o m e t r i e s from X, t o X,.
PROOF. c-utes
Given
dim X p
t h e Grassmann manifold c o n s i s t i n g of r- dimensional subspaces of
can be i d e n t i f i e d LEMMA 2 4 . 6 - (6). The space a6( 161’) with the set of a l l real subspaces X C W such that W s p l i t s as t h e orthogonal sum X fB J , X . PROOF.
=
Thus we obtain:
J 5 i s uniquely determined.
itself.
-I.
L F M 24.6 - ( 7 ) . The space i 6 r ) can be i d e n t i f i e d with the r e a l Grassrnann manifold c o n s i s t i n g of real subspaces of X s Rzr.
.
p r e c i s e l y the orthogonal subspace, on which JIW
w i l l be the orthogonal
+ I . Then J , X
I
i t s e l f ; and determines a s p l i t t i n g of spaces.
be t h e
be i d e n t i f i e d with a6(16r).
J3
The space n5( 1 6 r ) can be i d e n t i f i e d with t h e s e t of a l l v e c t o r spaces W C V, such that ( I ) W i s closed under J , ( i . e . , W i s a complex vector space) and (2) V1 s p l i t s a s t h e orthogonal sum W 8 J, W .
LEMMA 24.6
X C W
Let
l y determined.
J 4 i s uniquely determined by the
J41V,
J2J4J6 equals
subspace on which i t equals
Conversely, given any such isomorphism
is a quaternionic isomorphism. T : V,
subspace on which
-+v,
J 3 J 4 1 V , : V,
141
J,J4J6 W
determines a
into
&nce
with
JrJ8
determines
If
J8
E
a8(16r) then t h e orthogonal transformation
J,J2Jj, J,J4J5,
maps
X,
and
J2J4J6; b u t anti-commutes w i t h
isomorphically onto
X,.
J7J8 J,J6J7.
Clearly this isomorphism
J8 uniquely.
Thus w e see that
n8( 16r)
i s diffeomorphic t o t h e orthogonal
IV. APPLICATIONS
142
$24. THE ORTHOGONAL GROUP
group* O ( r ) .
DEFINITION.
Let us consider this diffeomorphism
r
the l i m i t a s
-+O(r),
and pass t o
( J , , . . . , Jk) -space i f no proper, non-
i s a minimal
V
J l ,...,
t r i v i a l subspace i s closed under t h e a c t i o n of
It follows that O8 i s homeomorphic t o t h e i n f i n i t e
-c m .
orthogonal group 0 .
a8( 16r)
143
Jk. Two such
and
minimal v e c t o r spaces a r e isomorphic i f t h e r e i s a n isometry between them
Combining t h i s f a c t with Theorem 24.5, we obtain the
J,,. . . , Jk.
which commutes with the a c t i o n of
following. LEMMA 2 4 . 8 ( B o t t and Shapiro) . F o r k f 3 (mod 4), any two minimal ( J 1 , . , . , J k ) vector spaces a r e isomorphic.
THEOREM 2 4 . 7 ( B o t t ) . The i n f i n i t e orthogonal group 0 has t h e same homotopy type as i t s own 8 - t h loop space. Hence the homotopy group n i 0 i s isomorphic t o ni+8 0 f o r i
Sp
If
= a 4denotes
argcunent a l s o shows t h a t 0 and t h a t
n R n R Sp,
Rnnn 0 .
Sp
%
20.
L
The proof of 24.8 follows t h a t of 2 4 . 6 .
complex numbers o r the quaternions.
has t h e homotopy type of t h e & - f o l d loop space
For
has t h e homotopy type of the & - f o l d loop space J3
1
0 1
"i SP
1
z2 0
3
Z
Z
4 5
0
Z2 Z2
6 7
T h i s g i v e s t w o non-isomorphic minimal
-J,J2.
H
to
=
5,6
complex s t r u c t u r e s spaces.
k
For
The dimension i s equal t o
HI.
and H
HI. @
HI, with
8.
we o b t a i n t h e same minimal vector space
H
@
HI. The
we again o b t a i n t h e same space, b u t t h e r e a r e two J 7 i s equal t o
p o s s i b i l i t i e s , according a s
Z
H
J5,J6 merely determine p r e f e r r e d complex or r e a l sub-
7
=
Call these
4.
a minimal space must be isomorphic t o
4
=
k
For
0
Z
However, t h e r e a r e two p o s s i b i l i t i e s , according as
+J1J2 or
k
J3J4 mapping
0
0 0
a
2
a minimal space i s s t i l l a 1-dimensional v e c t o r space
3
=
i s equal t o
For
0
2
or
= 0,1,
Clearly any two such a r e isomorphic.
spaces, both with dimension equal t o
0
z2
k
over t h e quaternions.
The a c t u a l homotopy groups can be tabulated as follows. n i 0
k
m i n i m a l space i s j u s t a 1-dimensional v e c t o r space over the reals, the
t h e i n f i n i t e symplectic group, then t h e above
i modulo 8
For
+J1J2J3J4JgJ6o r t o
-J1J2J3J4J5J6.Thus i n this case t h e r e a r e two non-isomorphic minimal
vector spaces;
The v e r i f i c a t i o n t h a t t h e s e groupsare c o r r e c t w i l l be l e f t t o the reader.
SP(1) i s a ?,-sphere, and that
(Note that
SO(3)
For
i s a p r o j e c t i v e 3-space.) with
The remainder of this s e c t i o n w i l l be concerned with the proof of
Consider a Euclidean v e c t o r space
*
For
k
Jl,
>
8
V
with anti-commuting complex
>
k
Periodically.
8
L
L'.
and
a minimal v e c t o r space must be isomorphic t o L
onto
L
L',
L'. The dimension i s equal t o 1 6 .
i t can be shown that the s i t u a t i o n r e p e a t s more o r l e s s
However, t h e cases
mk
Let
...,J k' i t can be shown that
i
8
=
mapping
For
It i s f i r s t necessary t o prove an a l g e b r a i c lemma.
Theorem 24.5.
structures
J7J8
c a l l these
k
k
5 8 w i l l s u f f i c e f o r our purposes.
denote t h e dimension of a m i n i m a l
...,J k ) - v e c t o r
(Jl,
space.
From t h e above d i s c u s s i o n we see t h a t : ak(l 6r)
I n f a c t any a d d i t i o n a l complex s t r u c t u r e s
i s diffeomorphic t o
ak-8(r).
Jg,Jlo,.. . ,Jk on R16r
give J8Jk on
...,
r i s e t o anti-commuting complex s t r u c t u r e s J8Jg, J 8 J l 0 , J 8 J l l , and hence t o an element of ak-8(r).However, f o r our purposes i t Xl; w i l l be s u f f i c i e n t t o s t o p w i t h k = 8.
For
k
>
8
mo
= 1,
m4
=
ml
mg
=
= 2,
m6
REMARK.
These numbers
=
mk
i t can be shown that
mk
m2 = m3 m7 =
=
=
8, m8
4, =
16.
16m~-~.
a r e c l o s e l y connected with t h e problem
of c o n s t r u c t i n g l i n e a r l y independent v e c t o r f i e l d s on spheres. example t h a t
J,,
...,Jk
Suppose f o r a r e anti-commuting complex s t r u c t u r e s on a vector
F IV. APPLICATIONS
144
space
of dimension
V
u
f o r each u n i t vector
d i c u l a r t o each other and vector f i e l d s on an
the
V
E
r can be any p o s i t i v e i n t e g e r .
Here
rmk.
to
Thus w e obtain
u.
linearly
k
For example we o b t a i n
(rmk-l)-sphere.
f i e l d s on a
(4r-l)-sphere;
f i e l d s on a
( 1 6 r - I ) - s p h e r e ; and s o on.
and Radon.
..., Jku
J l u , J2u,
vectors
k
7 v e c t o r f i e l d s on an
Then
eigenspaces of
A;
are perpen-
eigenvalues of
AIMh;
independent 3
(8r-l)-sphere;
...,
we see that
Commentarii Math. Helv. V o l . 1 5 ( 1 g 4 3 ) , pp. 358-366.) J .
.'
F . Adam has r e c e n t l y proved that t h e s e estimates a r e b e s t p o s s i b l e . k f 2 (mod 4 ) .
PROOF of Theorem 24.5 for minimal geodesics from
of n k ( n )
at
T
c o n s i s t s of a l l matrices
J
i s skew
2)
A
anti-commutes with commutes with
t
corresponds t o a geodesic
J,,
-
T
-
T.
can compute
E
T
Since n k ( n ) KA
e x i s t s by
j u s t as before.
from J
(nL4)
A.
to
A given
A
E
T
=
-+
24.8.
-J i f and only i f
mutes w i t h
O(n),
we
(-A2B + PABA
-
BA2)/4
-,J
S p l i t t h e v e c t o r space Rn
eigenvalue
so as
J,
,...,J k - l , J
and
A.
a l l equal, except f o r s i p . *
*
for a
e x p ( n t A)
.
Then t h e eigenvalues of For otherwise
Mh
BIMj
B
Mh
as follows.
T
Let
to M
j
. . . Q M,
of
must be
would s p l i t as a sum of
BIMj
We are d e a l i n g with t h e complex eigenvalues of a r e a l , skew-symmetric transformation. Hence these eigenvalues a r e pure imagiinars; and occur i n conjugate p a i r s .
Bw)
J
Mh
B
B
on
T.
Since
w
M
E
I.
J.
BIMh
com-
It follows e a s i l y that
J1,...,Jk-l
and anti-
i s an eigenvector of
+ 1
v
E
KA Mh
corresponding t o the then
( - A ~ B + ~ A B A- m 2 ) v 2
2
( a Bv + 2a Ba v + Bah)v j j h
Ti 1 (aj
+ ah)2 BV v
E
M
;
j*
The number of minimal spaces
Mh C Rn
i s given
&n must be 2 3 .
T h i s proves t h e following
k f 2 (mod 4) ) :
ASSERTION-
-> ( 3 + 1 1 2 / 4
B(Mh.
T.
E
For otherwise we would have a minimal geodesic. (always f o r
Such an isomorphism
Mj.
For a t least one of t h e s e t h e i n t e g e r
= n/mk+l.
where the bar i n -
It i s a l s o c l e a r that
(ah + a j ) 2 / 4 . For example i f
Now l e t u s count.
Mj;
to
for v c Mh,
and a similar computation a p p l i e s f o r
s
=
a l s o commutes with
=
by
...,k-1; +JIB .
and anti-commutes w i t h
=
on
-
be the negative a d j o i n t of
i s skew-symmetric.
Thus
J.
=
a.
A
k + 1 f 3 (mod 4)
belongs t o t h e v e c t o r space
r
as a d i r e c t sum M, 8 M2 Q
Since
T
= 1,
i s an isomorphism from
( K ~ B ) =~
mutually orthogonal subspaces which are closed and minimal under t h e a c t i o n
of
-JB and BJ'
B
t o o b t a i n a lower bound f o r t h e index of t h e corresponding geodesic
t
(J1,...,J k - , , J , J ' ) -
be a n isometry from Mh
BIMh
=
W e claim that KA
KA:
E?J
the negative a d j o i n t
,
We must c o n s t r u c t some non-zero eigenvalues of
which
we can c o n s t r u c t an eigenvector
JJ!
J1,...,Jk-l
cammutes w i t h =
Let
Finally l e t
it i s c l e a r that
determines a s e l f - a d j o i n t transformation
[A,[A,BII
h # j
with
=
i.
i s a t o t a l l y geodesic submanifold of
Mh
is
Mh
mkCl.
is
Mh
BJ,
B(Mh
Proof that
?-
by t h e formula
KAB
Thus
J.
d i c a t e s that we have changed t h e s i g n of
...,J k - , .
J exp
i s a complex s t r u c t u r e on
are mutually isomorphic.
h,j
P # h,j.
where
J
i t s eigenvalues a r e a l l odd m u l t i p l e s of
KA:
J A
a r e odd, p o s i t i v e i n t e g e r s .
of t h e l i n e a r transformation
be zero f o r
+_ i a h be the two
Let
which s a t i s f i e s t h e conditions
We must study non-
denote t h e vector space of a l l such matrices
Each such A
E: Rn -Rn
I n o t h e r words
A
A
...,Ms
Ml,M2,
-J i n n k ( n ) . R e c a l l that t h e tangent space
1)
3)
Let
to
J
J1
For each p a i r BIMp
ah JAIMh;
=
Jl
...,as
1
Hence t h e dimension of
minimal.
These r e s u l t s a r e due t o Hurwitz
al,
where
,..., J k - l , and
anti- comutes with
8 vector
145
and hence would not be minimal.
Now note that
vector
(Compare B. E c h m , Gruppentheoretischer Beweis des Satzes von
Hurwitz-Radon
THE ORTHOGONAL GROW
§ 24.
=
4.
KA
a t least
he i n t e g e r
9-1
s = n/mk+l
eigenvalues which a r e tends t o i n f i n i t y w i t h
n.
146
t
Now consider t h e geodesic
of KA t = e-l
-+
J exp(nt A ) .
Each eigenvalue
, 2e-l,
.. .
3e-’,
by 2 0 . 5 .
Thus i f
e2
2
t
from J
then one obtains a t
4
The index of a non-minimal geodesic from
to
-J
t r a c e of
in
I.
Since A
+
Thus
n %(n)
L
i t follows t h a t the i n c l u s i o n map
i : Qk+,
12
%
-
-+
n
%
have the homotopy type of a CW-complex.
f o r e , by Whitehead’s theorem, i t follows t h a t
Thus for each
=
e
n t trace A
which i s com-
S’
I t follows t h a t t h i s t r a c e i s i n n ( n k ( n ) ;J,-J) .
-
of this aeodesic can be estimated a s follows.
X
i n t o an orthogonal sum
h,
M, @
. ..
7
Thus
AIMh
For otherwise
iah.
coincides with
the a c t i o n of
...
ahJ1J2
J l , . . . , J k - l , and
Mh
and i s minimal.
A;
can have only
AIMh
would s p l i t i n t o eigenspaces.
Mh
Since
Jk-lIMh.
i s minimal under
Mh
i t s complex dimension i s
J;
As
where each
G3
the complex l i n e a r transformation
one eigenvalue, say
k $ 2 (mod 4 ) .
hence t h e
Now
is closed under the a c t i o n of J l , . . . , J k - l , J , and
There-
i s a homotopy equivalence.
i
T h i s completes the proof of 2 4 . 5 providing that
before s p l i t Rn
induces
Jk-’
A a l s o as a
i s skew-Hermitian;
A
maps t h e given geodesic i n t o a closed loop on
The index
. ..
JIJ,
=
we may t h i n k of
determinant (exp(ntA))
=
i
v a r i a n t under homotopy o f t h e geodesic within t h e p a t h space This
3.
But i t can be shown
isomorphisms of homotopy groups i n a l l dimensions.
nk+’ and
n/mk+l
Therefore, passing t o t h e d i r e c t l i m i t
n.
f
In fact
p l e t e l y determined by t h e t r a c e of A .
induces isomorphisms of homotopy groups i n dimensions number tends t o i n f i n i t y with
commutes with
i s a pure imaginary number.
f ( J exp(ntA))
i
fik+l(n)
that both
A
b
2 n/mk+’-
is
J
It follows t h a t the i n c l u s i o n map
m,
in nk(n).
-J
complex l i n e a r transformation.
ASSERTION.
n-
to
J exp(ntA)
(compare Assertion 2 i n t h e proof of 24:b)
Applying the index theorem, t h i s proves
the following.
nk(n)
-
147
NOW consider a geodesic
e2
gives r i s e t o conjugate p o i n t s along this geodesic for
l e a s t one i n t e r i o r conjugate p o i n t .
as
T.JE ORTHOGONAL GROUP
§ 24.
IV. APPLICATIONS
mk/2.
~~
PROOF of 24.5 for k
(mod 4).
2
n fik(n)
Thus
B
as follows.
Let
t u r e on Rn.
nk(n)
:
J , , ..., J k - ,
S’
(n/2)-dimensional complex vector space by
that
v
E
Rn;
where
i 2= - 1 ,
i
= c
and t h a t
l E
J-’J’
C.
...
J
E
fi,(n).
commutes w i t h
i.
k
I
commute with
For any Thus
KAB
=
f(J‘).
(-A2B + PABA
BIMh t
which commutes with
Since
Mh
-
BA’) /4
and
M. J
are
(J1,...,Jk-’,J)-
the negative a d j o i n t of
BIMh;
M h - + M. J
:
and anti-commutes with
Jl,...,Jk-l
J‘
J-lJ’
E
2
(mod 4)
guarantees
i.
nk(n)
KAB
Thus f o r each note that t h e
i s a u n i t a r y complex
Since
ah
>
aj
and l e t
BIMp
Let
J.
be zero for
=
BIMj
e # h,j.
(ah - aj)‘/4
each such eigenvalue makes a c o n t r i b u t i o n of
towards t h e index
X,
(ah - a . ) / 2 J
we o b t a i n t h e i n e q u a l i t y 2X
1.
1 a h > &j
(ah - a j -
2)
for
.
KA. 1
be
Then
.
(ah - aj)2B/4
we o b t a i n a n eigenvalue
l i n e a r transformation, and has a w e l l defined complex determinant which w i l l be denoted by
of t h e l i n e a r transforma-
an easy computation shows that
Jk-iv
The condition
J , , J , ,..., J k - l
Choose a base p o i n t composition
J,J2
=
B
minimal it follows from 24.8 t h a t t h e r e e x i s t s an isornet-y
c c
defining
for
-+
Can be constructed much as b e f o r e .
rT
iv
an eigenvector
be t h e f i x e d anti-commuting complex s t r u c -
i n t o an
Make Rn
-
we c o n s t r u c t a map
n&(n)
i ( a l +...+a r ) m k / 2 .
tion
i n g subspace n k + , ( n ) has only f i n i t e l y many.
f
h # j
Now f o r each
has i n f i n i t e l y many components, while the approximat-
To describe t h e fundamental group
i s equal t o
Therefore t h e t r a c e of A
has an i n f i n i t e c y c l i c fundamental
be ascribed t o the f a c t . t h a t n,(n) group.
The d i f f i c u l t y i n t h i s case may
IV. APPLICATIONS
148
149
Now l e t u s r e s t r i c t a t t e n t i o n t o some f i x e d component of T h a t i s l e t us l o o k only a t matrices c
such that
A
trace A
=
nClk(n).
icmk/2
where
i s some constant i n t e g e r . Thus t h e i n t e g e r s 1) 2)
3)
al,
= a2 I ... a1 +...+ a = Max Iahl 2 3 a1
h
ar c,
satisfy E 1
(mod 2 ) , ( s i n c e
=
The o b j e c t of this appendix w i l l be t o give a n a l t e r n a t i v e v e r s i o n
-I),
and
f o r t h e f i n a l s t e p i n t h e proof of Theorem 1 7 . 3 ( t h e fundamental theorem
ah i s equal t o
Let
- 3.
-9 the sum of t h e negative
ah and
exp(nA)
( f o r a non-minimal g e o d e s i c ) .
Suppose f o r example that some the positive
APPENDIX. THE HOMOTOPY TYPE OF A MONOTONE UNION
...ar
ah.
p
of Morse t h e o r y ) .
a
space
be t h e sum of
n(M;pJq),
=
a
Given t h e s u b s e t s
nal
C
R
C
na2
C
...
and given t h e information that each
of t h e p a t h
nai
has t h e
homotopy type of a c e r t a i n CW-complex, we wish t o prove that t h e union
Thus
R
a l s o has t h e homotopy type of a c e r t a i n CW-complex. P - 9
hence
2r
2p 2~
2
+ c.
=
c,
4X
1 %>
2 2p 2 r
+
q
>
r,
More g e n e r a l l y consider a t o p o l o g i c a l space
Now
(ah
X, C X1 C X2 C
-
aj
-
2)
+ c;
1
>
aj
hence
~
ah where
follows that t h e component of
r
=
>
n/mk
n %(n)
X (ah - ( - 3 )
- 3)
P
,
with
n.
=
( n ) , as
n
x,
It
-
m
=
xo
XIOJll
" xlx
[1,21
Xi?
x2 x [ 2 , 3 1
u
u
... .
X x R.
T h i s i s t o be topologized as a subset of
. DEFINITION.
Passing t o t h e d i r e c t l i m i t , w e o b t a i n a homotopy equivalence on each component.
To w h a t e x t e n t i s t h e homotopy type of
I t i s convenient t o consider t h e i n f i n i t e union
i s approximated up t o higher and
higher dimensions by t h e corresponding component of
of subspaces.
determined by t h e homotopy types of t h e
0
tends t o i n f l n i t y
...
and a sequence
X
the sequence
T h i s completes t h e proof of 24.5.
p(x,.r)
=
[Xi)
X
i s t h e homotopy d i r e c t l i m i t of
i f t h e p r o j e c t i o n map
p : X,
+
X,
defined by
i s a homotopy equivalence.
x,
EXAMPLE some Xi,
We w i l l say that
1.
and that
Suppose that each p o i n t of X
i s paracompact.
X
l i e s i n t h e i n t e r i o r of
Then using a p a r t i t i o n of u n i t y one
can c o n s t r u c t a map f :X+R tr
6*
SO that
pondence
f(x)
2
i+l
for
x q! XiJ
x + ( x , f ( x ) ) maps
and
X
20
for a l l
x.
Now t h e corres-
homeomorphically onto a subset of
X
i s c l e a r l y a deformation r e t r a c t . and
f(x)
Therefore
p
X,
which
i s a homotopy equivalence;
i s a homotopy d i r e c t l i m i t . EXAMPLE 2.
w i t h union
X.
Let
Since
X
be a CW-complex, and l e t t h e Xi be subcomplexes
P : Xz
-
X
induces isomorphisms of homotopy groups
i n a l l dimensions, i t f o l l o w s f r m W t e h e a d ' s theorem that X direct l i m i t .
i s a homotopy
150
1,
APPENDIX
151
HOMOTOPY CF A MONOTONE U N I O N
I
The u n i t i n t e r v a l
EXAMPL3 3.
[0,11
l i m i t of the sequence of closed subsets
is
LO1
not
the homotopy d i r e c t
a s follows (where it i s always t o be understood that
[l/i,ll.
n
0
and
5 t 5 1,
,...) .
= 0,1,2
The main r e s u l t of this appendix i s the following. THEOREM A . Suppose that X i s t h e homotopy d i r e c t l i m i t of (Xi] and Y i s t h e homotopy d i r e c t l i m i t of ( Y i ) . Let f : X + Y be a map which c a r r i e s each Xi i n t o Yi by a homotopy equivalence. Then f i t s e l f i s a homotopy equivalence.
f
Taking u
to Assuming Theorem A, t h e a l t e r n a t i v e proof of Theorem 17.3 can be given as follows.
of homotopy equivalences.
C K,
Since
n
C K2 C
K
U nai
=
Define
Suppose that
(obtained by r e s t r i c t i n g f ) f
and
=
Yi
fz
-
+
K
=
U Ki
a r e homotopy
homotopy inverse
g : X,
t
Y,
by
f,(x,t)
=
fi : Xi+
i s homotopic t o t h e i d e n t i t y .
(x,n+2t)
hl(x,n+t)
E
X,+l
x [n+ll
=
fn,
h:
=
identity.
o < t < $
for
35t 5
1
.
In fact a
$ can be defined by the formula o < t < +
(x,n+2t)
hLt"
=
hl(x,n+l)
Proof that the composition
Yi Of
X,.
.
hlg
=
(x,n+l)
.
i s homotopic t o the i d e n t i t y map
Note that
o < t < t
t < t < h
c P
h l ( x , n +3 ? - t) Define a homotopy
$ : %'Xn
for
i s a homotopy equivalence.
hl
=
However counter-
let
%:%-%
on t h e other hand has the following
=
hl(x,n++)
We must prove
examples can be given.
Define the hornotopy
which i s c l e a r l y homotopic
(x,n+4t)
f c must a c t u a l l y be homotopic t o the i d e n t i t y .
be a one-parameter family of mappings, w i t h
X,
+
This i s w e l l defined since
(f(x),t).
i s a homotopy equivalence.
and that each map
-
f
g(x,n+t)
Under these conditions i t would be n a t u r a l t o conjecture
For each n
X,
%
:
hl( x , n + t )
c i s a homotopy equivalence. FEWARK.
that
Xi
:
f,
It i s c l e a r l y s u f f i c i e n t t o prove t h a t
tht
hl : X,
We w i l l show that any such map
i s a l s o a homotopy equivalence.
PROOF of Theorem A.
CASE 1 .
The mapping
hO
proper t i e s :
...
d i r e c t l i m i t s (compare Examples 1 and 2 above), it follows t h a t t h e l i m i t
-
t h i s d e f i n e s a map
R e c a l l that we had constructed a commutative diagram
KO
mapping
f,.
= 0
Q : Xc
+
%
as follows.
.
h < t < 1
For
0
5u
<3
let
($1,
This i s w e l l defined since hlg(x,n+(l-u)/2)
hlg(x,n+$+u)
=
hl(x,n+l-u).
=
and conclude that
Gf,
T h i s proves t h a t
fz
argument shows t h a t
Ho i s equal t o
Now
HA i s given by
h l g and
fzG : Y,
I
'
f,
Thus
h l g i s homotopic t o t h e i d e n t i t y ; and a completely analogous argument shows that ghl i s homotopic t o t h e i d e n t i t y . T h i s completes the
Now l e t
X
and
Y
For each n
be a r b i t r a r y .
be a homotopy i n v e r s e t o
fn.
let
Note that t h e diagram
$I n'
xn
] in and
$I+1
jn
'n+ (where
lin
' %+l
1
denote i n c l u s i o n maps)
jn
i s homotopy commutative.
In
fact in% * %+lfn+iin% Choose a s p e c i f i c homotopy and
G : Yc
define
-%
h+iJnfn&
=
$
with
: Yn XnC1 by t h e formula +
G(y,n+t)
=
{
%+iJn h:
(%(Y) ,n+2t)
o < t < 4
(h:t-l
f5t5
( y ), n + l )
3
Gf,
Let
X:
c o n s i s t i n g of a l l p a i r s
5
n.
sition
denote t h e subset of (Thus Gf,
identity. mapping
$=
Xo x [ O , l l
carries I n fact
Gf,
X, u
.
= in%, hy
We w i l l show that t h e composition
T
Yc
:
.. . s-lx
+
X,
1
gn,,j n ;
=
.
i s a homotopy equivalence.
[n-1 ,nl
u
(x,r)
Xn x [ n l . )
1
with The compo-
i n t o i t s e l f by a mapping which i s homotopic t o t h e contains
restricted t o
Xnx [ n l
Xnx [nl
hence i s homotopic t o t h e i d e n t i t y .
as deformation r e t r a c t ; and the
can be i d e n t i f i e d w i t h
%fn,
and
Thus w e can apply Case 1 t o t h e sequence
i s a homotopy equivalence, s o that T h i s proves that
f,
i s a homotopy
and completes t h e proof of Theorem A.
The proof i s not d i f f i c u l t .
proof i n Case 1 .
: Yn-
-
A similar
COROLLAFZ. Suppose t h a t X i s t h e homotopy d i r e c t limit of (Xi). I f each Xi has t h e homotopy type of a CW-complex, then X i t s e l f has the homotopy type of a CW-complex.
Clearly this i s homotopic t o t h e i d e n t i t y .
$I
i s a homotopy equivalence. has a l e f t homotopy i n v e r s e .
has a right homotopy i n v e r s e .
equivalence (compare page 2 2 )
CASE 2 .
153
HOMOTOF'Y OF A MONOTONE UNION
APPENDIX
152
, t i
1
