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M, having images in a domain of local chart U c We say that p and a have a "contact of order 1" or the "same tangent line" in the point x0 £ [/if: p(Q) = CT(0) = x0, (0 € / ) , and for any function 1 is symmetric if and only if HT*M and WT*M are both Lagrangian (Corollary 8.4.2.). Therefore, we must have: Tv{u){T*2M), in the natural basis as follows d T*2M, (5.1), transforms the semispray (2)i - y iiW— — ++ 2v V
The Geometry of Hamilton & Lagrange Spaces
2
f e nv) •
iu
(ID
t=o
t=0
The relation "contact of order 1" is an equivalence on the set of smooth curves in M, which pass through the point x0- Let [p]xo be a class of equivalence. It will be called a "1-osculator space" in the point x0 e M. The set of 1-osculator spaces in the point x0 € M will be denoted by Osc^.o, and we put
OsclM = U Osci0. xo€M
One considers the mapping ir : Osc'M —>• M defined by 7r([p]I0) = x0. Clearly, ?r is a surjection. The set Osc'M is endowed with a natural differentiable structure, induced by that of the manifold M, so that it is a differentiable mapping. It will be described below. The curve p : I -> M, (Im p c U) is analyticaly represented in the local chart (U,ip) by xl = x*(t), t e I, x0 = {xi = a;'(0)), taking the function / from (1.1), succesively equal to the coordinate functions xl, then a representative of the class [p]xa is given by x*i^x'(Q)
+ t—(0), at
te(-£,e)cl.
The previous polynomials are determined by the coefficients dr1
xi=xt(0),yi = —{0).
(1.2)
Hence, the pair (TT"1 ([/),>), with 4>({p}Xo) = (ij.j/j) € R2n, V[p]xo £ ^(U) is a local chart on M . Thus a differentiable atlas AM of the differentiable structure on the manifold M determines a differentiable atlas ADSC'M o n OscxM and therefore the triple ( M ) is a differentiable bundle. Based on the equations (1.2) we can identify the point [p]xo € Osc^M with the tangent vector y'Q e TXoM. Consequently, we can indeed identify the 1-osculator bundle with the tangent bundle (TM, M). By (1.2) a transformation of local coordinates {x\yl) ->• (x\yl) on the manifold TM is given by
(
3)
^
dx>
Ch.1. The geometry of tangent bundle
3
One can see that TM is of dimension 2n and is orientable. Moreover, if M is a paracompact manifold, then TM is paracompact, too. Let us present here some notations. A point u e TM, whose projection by 7r is x, i.e = . will be denoted by (x,y), its local coordinates being (x\yl). We put TM — TM \ {0}, where {0} means the null section of TT. The coordinate transformation (1.3) determines the transformation of the natural basis I TT-'-^— I. (* = l7")i of the tangent space Tu TM at the point u € TM the \dx* dy'J following: d_ _ <9F _d_ dxi dxi dxi
dy>_ _d_ _d_ _ dy>_ _d_ dxl dyi' dyi dyl dyi
By means of (1.3) we obtain (1.4)'
— = —; — =
dyi
dx^ dx^
Looking at the formula (1.4) we remark the existence of some natural object fields on E. First of all, the tangent space Vu to the fibre 7r~: (X) in the point u € TM is locally spanned by I — , • • • ,^— \ . Therefore, the mapping V: u € TM -> Vu c T J M provides a regular distribution which is generated by the adapted basis < -^~ (i = 1,..., n). Consequently, V is an integrable distribution on TM. V is called the vertical distribution on TM. Taking into account (1.3), (1.4), it follows that (1-5)
C = y>±
is a vertical vector field on TM, which does not vanish on the manifold TM. It is called the Liouville vector field. The existence of the Liouville vector field is very important in the study of the geometry of the manifold TM. Let us consider the T(TM) -linear mapping J : X{TM) -> X(TM),
Theorem 1.1.1. The following properties hold: 1° J is globally defined on TM. 2° J o J = 0, lmJ = KeiJ = V, rank|| J\\ = n.
The Geometry of Hamilton & Lagrange Spaces
4
3° J is an integrable structure on E. 4° J C = 0. The proof can be found in [113]. We say that J is the tangent structure on E. The previous geometrical notions are useful in the next sections of this book.
1.2
Homogeneity
The notion of homogeneity of functions / G F{TM) with respect to the variables y% is necessary in our considerations because some fundamental object fields on E have the homogeneous components. In the osculator manifold OscxM = TM, a point has a geometrical meaning, i.e. changing of parametrization of the curve p : I —> M does not change the space [p]x0- Taking into account the affine transformations of parameter (2.1)
i=at + b, te I, ae R*
we obtain the transformation of coordinates of [p]xo in the form (2.2)
x' = x\ yl = ay\
Therefore, the transformations of coordinates (1.3) on the manifold E preserve the transformations (2.2). Let us consider H - {ha : R-* R | a G R+}, the group of homoteties of real numbers field R. H acts as an uniparameter group of transformations on E as follows
HxTM -> TM, (ha,u) -> ha(u), where u = ha{u), a G R+, is the point (x, y) = (x,ay), a G R+. Consequently, H acts as a group of transformations on TM, with the preserving of the fibres. The orbit of a point u0 = (xQ, y0) G E is given by ™1 — rpl
y ' = ay'o, a G R + • The tangent vector to orbit in the point u0 = hi(u0) is given by
Ch.1. The geometry of tangent bundle
5
This is the Liouville vector field C in the point u0. Now we can formulate: Definition 1.2.1. A function / : TM —> R differentiable on TM and continuous on the null section 0 : M —• TM is called homogeneous of degree r, r , Z) on the fibres of TM, (briefly: r-homogeneous with respect to ) if: (2.3)
foha
= aTf, V o £ R + .
The following Euler theorem holds [90], [106]: Theorem 1.2.1. A function f € F{TM) differentiable on TM and continuous on the null sections is homogeneous of degree r on the fibres of TM if and only if we have Ccf = rf,
(2.4)
£ c being the Lie derivative with respect to the Liouville vector field C. Remark. If we preserve Definition 1.2.1 and ask for / : TM —> R to be differentiable on TM (inclusive on the null section), then the property of 1-homogeneity o f f implies that f is a linear function in variables y\ The equality (2.4) is equivalent to
„• J£ = rf.
(2.4)' The following properties hold:
1° /ij /2-r-homogeneous=>Ai/ 1 + A2/2, Ai, A2 € R is r-homogeneous, 2° /i-r-homogeneous, / 2 s-homogeneous^=^/i • fa is (r + s)-homogeneous, 3° /1 r-homogeneous, fa / 0 s-homogeneous=>— (r — s) homogeneous.
fa
Definition 1.2.2. A vector field X £ X{E) is r-homogeneous if Xoha = ar-1h*aoX, VaER+. It follows: Theorem 1.2.2. A vector field X € X(TM) is r-homogeneous if and only if we have (2.5)
CCX = (r - \)X.
Of course, CCX — [C,X] is the Lie derivative ofX with respect to Consequently, we can prove:
The Geometry of Hamilton & Lagrange Spaces
6
d d 1° The vector fields -r—>-^—are 1 and 0-homogeneous, respectively. ox1 ay' 2° If i s F(TM) s -homogeneous and X e X(TM) is r-homogeneous then fX is s + r-homogeneous. 3° A vector field on TM: dx* 1
is r-homogeneous if and only if X^ X^1 are functions r-homogeneous.
dy'
are functions ( r - l)-homogeneous and
4° X G X(TM) is r-homogeneous and i s T(TM) s -homogeneous, then Xf G P(TM) is a (r + s - 1)-homogeneous function. 5° The Liouville vector field C is 1-homogeneous. 6° If / G T{TM) is an arbitrary s-homogeneous function, then —- is a (s - 1)<92/ homogeneous function and . . . . is (s - 2)-homogeneous function. dy'oyi In the case of q-form we can give: Definition 1.2.3. A q-form i s A'(TM) s -homogeneous if It follows [106]: Theorem 1.2.3. A q-form u> € A'(TM) is s-homogeneous if and only if £ c w = su.
(2.6) Corollary 1.2.1.
We have, [106]:
1° u) G tV(TM) s-homogeneous and u' G Av' (TM) is s' -homogeneous (s + s')-homogeneous. 2° ui G A'(TM) s-homogeneous, X, •••, X I
?
-r-homogeneous=>
w(X, --^X) is r -I- (s — I)-homogeneous. i
?
3° da; (i = l,...,n) are 0-homogeneous 1-forms. dy' (i — 1, ...,n) are 1-homogeneous 1-forms. 1
The applications of those properties in the geometry of Finsler space are numberless.
Ch.1. The geometry of tangent bundle
1.3
Semisprays on the manifold TM
One of the most important notions in the geometry of tangent bundle is given in the following definition: Definition 1.3.1. A semispray S on TM is a vector field S G X(TM) with the property: JS = C.
(3.1)
If S is homogeneous, then S will be called a spray. Of course, the notion of a local semispray can be formulated taking S e X(U), U being an open set in the manifold TM. Theorem 1.3.1. 1° A semispray S can be uniquely written in the form
(3-2) 2° The set of functions follows:
(3-3)
s = S±-X?{xtV)JL. (
i = 1,..., n) are changed with respect to (1.3) as
Fir*
2Gi = 2 |L
Bii*
tf-g^.
3° Iftheset of functions G' are a priori given on every domain of a local chart in TM, so that (3.3) holds, then S from (3.2) is a semispray. Proof.
1° If a vector field S = a'(x,y)-—- + b'(x,y) - - r is a semispray S, then ox1 oy1 { JS = C implies d = y and b\x,y) = -ZG^x^). So that Gi are uniquely determined and (3.2) holds. 2° The formula (3.3) followsfrom (1.3), (1.4) and the fact that S is a vector field onfM, i.e. 5 = y ^ - 2G>(x,y)^- = f ^ 2&(x,y)±. 3° Using the rule of transformation (3.3) of the set of functions G1 it follows that S = y'-^-r1 - 2Gi(x,y)—— 1 = f-^- 1 - 2Gl{x,y)—- 1 is a vector field which satisfies ox oy ox ay JS = C. q.e.d. From the previous theorem, it results that S is uniquely determined by G'(x,y) and conversely. Because of this reason, G1 are called the coefficients of the semispray S.
The Geometry of Hamilton
& Lagrange
Spaces
T h e o r e m 1.3.2. A semispmy S is a spray if and only if its coefficients Gl are 2-homogeneous functions with respect to y ' . Proof. By means of 1° and 3° from the consequences of Theorem 2.2 it follows d d that y*-z-^% is 2-homogeneous and —-. is 0-homogeneous vector fields. Hence, S is dx dy' 2-homogeneous if and only if Gl are 2-homogeneous functions with respect to yl. The integral curves of the semispray S from (3.2) are given by
(3.4)
fir1
dii*
* _ , < , £_-*,.<,,„,.
It follows that, on M, these curves are expressed as solutions of the following differential equations
< « '
^
+
2
G
<
(
*
•
!
)
=
»
•
The curves c : t e I -*• (xl{t)) c U C M, solutions of (3.5), are called the paths of the semispray S. The differential equation (3.5) has geometrical meaning. Conversely, if the differential equation (3.5) is given on a domain of a local chart U of the manifold M, and this equation is preserved by the transformations of local / dxi \ coordinates on M, then coefficients G*(x,y), ly' = — obey the transformations dt r (3.3). Hence Gl(x,y) are the coefficients of a semispray. Consequently: Theorem 1.3.3. A semispray S on TM, with the coefficients Gl(x,y) is characterized by a system of differential equations (3.5), which has a geometrical meaning. Now, we are able to prove Theorem 1.3.4. If the base manifold M is paracompact, then on TM there exist semispray s. Proof. M being paracompact, there is a Riemannian metric g on M. Consider Jljk(x) the Christoffel symbols of g. Then the set of functions
is transformed, by means of a transformation (1.3), like in formula (3.3). Theorem 1.3.1 may be applied. It follows that the set of functions Gl are the coefficients of a semispray S. q.e.d.
Ch.1. The geometry of tangent bundle
9
Remarks. 1° S = y%-~-[ — 2Gl-^—isaspray, where Gl(x, y) = - I1 jki^y1'yk, whose differential equations (3.5) are d2xi
l
, .dxi dxk
n
So the paths of S in the canonical parametrization are the geodesies of the Riemann space (M, g). QQi
2° —-j = jljk(x)yk
is a remarkable geometrical object field on TM (called non-
linear connection). Finally, in this section, taking into account the previous remark, we consider the functions determined by a semispray S: N
'
Using the rule of transformation (3.3) of the coefficients G* we can prove, without difficulties: Theorem 1.3.5. If are the coefficients of a semispray S, then the set of l functions N j(x,y) from (3.6) has the following rule of transformationwith respect to (1.3): — - Nm} — - — • J - " dX™ dxi In the next section we shall prove that Nlj are the coefficients of a nonlinear connection on the manifold E = TM.
1.4
Nonlinear connections
The notion of nonlinear connection on the manifold E = TM is essentially for study the geometry of TM. It is fundamental in the geometry of Finsler and Lagrange spaces [113]. Our approach will be two folded: 1° As a splitting in the exact sequence (4.1). 2° As a derivate notion from that of semispray.
The Geometry of Hamilton & Lagrange Spaces
10
Let us consider as previous the tangent bundle (TM, M) of the manifold M. It will be written in the form (E,M) with E = TM. The tangent bundle of the manifold E is (TE,E), where TXT is the tangent mapping of the projection n. As we know the kernel of 7rT is the vertical subbundle (VE,E). Its fibres are the linear vertical spaces VUE, u e E. A tangent vector vector field on E can be represented in the local natural frame
si - / v
v
dl
8x'
It can be written in the form X = (x\y\X\Yl) mapping irT : TE —> E has the local form
or, shorter, X - (x,y,X,Y).
TTTIT v X Y) = (r
The
V)
The points of submanifold VE are of the form (x, y, 0, Y). Hence, the fibres VUE of the vertical bundle are isomorphic to the real vector space Rn. Let us consider the pullback bundle •n*E = E xn E = {{u,v) e ExE
| ir(u) = 7T(V)}.
The fibres of i.e., -K*UE are isomorphic to T^M. We can define the following morphism of vector bundles 7r! : TE —> ir*E, 7r!(Xu) — (u, 7rJ(X u ).It follows that KerTr! =Ker7r T = VE. By means of these considerations one proves without difficulties that the following sequence is exact: (4.1)
0—>
VE
- ^ TE
-^t
TI'E—^0
Now, we can give: Definition 1.4.1. A nonlinear connection on the manifold E = TM is a left splitting of the exact sequence (4.1). Therefore, a nonlinear connection on E is a vector bundle morphism C : TE —t VE, with the property C o i = lVE. The kernel of the morphism C is a vector subbundle of the tangent bundle (TE,E), denoted by ( H E , E ) and called the horizontal subbundle. Its fibres HUE determine a distribution u € E —> HUE c TUE, supplementary to the vertical distribution u e E —> VUE C TUE. Therefore, a nonlinear connection N induces the following Whitney sum: (4.2)
TE = HE®VE.
Ch.1. The geometry of tangent bundle
11
The reciprocal property holds [112]. So we can formulate: Theorem 1.4.1. A nonlinear connection N on E = TM is characterized by the existence of a subbundle (HE, E) of tangent bundle ofE such that the Whitney sum (4.2) holds.
Consequences. 1° A nonlinear connection N on E is a distribution H : u £ E -* HUE C TUE with the property TUE = H»E $ VUE, V« 6 E,
(4.2)' and conversely.
2° The restriction of the morphism TT! : TE —> n*E to the HE is an isomorphism of vector bundles. 3° The component TTT : HE -» E of the mapping TT! is a morphism of vector bundles whose restrictions to fibres are isomorphisms. Hence for any vector field X on M there exists an horizontal vector field XH on E such that 7rT(XH) = X is called the horizontal lift of the vector field X on M. Using the inverse of the isomorphism n\HE we can define the morphism of vector bundles D : n*E —> TE, such that TT! O D = id| ff . £ . In other words, D is a right splitting of the exact sequence (4.1). One can easy see that the bundle Im D coincides with the horizontal subbundle HE. The tangent bundle TE will decompose as Whitney sum of horizontal and vertical subbundle. We can define now the morphism C : TE —> VE on fibres as being the identity on vertical vectors and zero on the horizontal vectors. It follows that C is a left splitting of the exact sequence (4.1). Moreover, the mapping C and D satisfy the relation: i o C + D o TT! = idrBSo, we have Theorem 1.4.2. A nonlinear connection on the tangent bundle E = {TM, i M) s characterized by a right splitting of the exact sequence (4.1), D : TT*E -> TE, such that TT! O D - \d\n.E. The set of isomorphisms ru : VUE -* TUE, u € E defines a canonical isomorphism r between the vertical subbundle and the vector bundle -n*E. Definition 1.4.2. The map K : TE —> E, given by K — p 2 ° r o C is called the connection map associated to the nonlinear connection C, where p2 is the projection on the second factor of -n*E.
The Geometry of Hamilton & Lagrange Spaces
12
It follows that the connection map K is a morphism of vector bundles, whose kernel is the horizontal bundle HE. In general, the map K is not linear on the fibres of (E, M ) . The local representation of the mapping K is (4.3)
K(x,y,X,Y) = (x,Y* + Nl{x,y)X>).
Let us consider a nonlinear connection determined by C and K the connection map associated to C, with the local expression given by (4.3). Taking into account (4.3) and the definition of C, we get the local expression of the nonlinear connection: (4.3)'
C(x, y, X, Y) = (x, y, o, Y* + N>(x, y)Xl).
The differential functions (NJ(x,y)), i,j e { 1 , 2 , . . . , « } defined on the domain of local charts on E are called the coefficients of the nonlinear connection. These functions characterize a nonlinear connection in the tangent bundle. Proposition 1.4.1. To give a nonlinear connection in the tangent bundle (TM, M) is equivalent to give a set of real functions (Nj(x, y)), i, j £ {1,2,.., n}, on every coordinate neighbourhood ofTM, which on the intersection of coordinate neighbourhoods satisfies the following transformation rule: (4 4) 1 '
~N3
Fjfi
l
Proof. The formulae (4.4) are equivalent with the second components Yj+N^(x, y)X1 of the connection map K from (4.3) under the overlap charts are changed as follows Y> + N({x, y)X' = | ^ ( y * + N?[ Applying Theorem 1.3.5 we get: Theorem 1.4.3. A semispray S on TM, with the coefficients Gl(x,y) dG* a nonlinear connection N with the coefficients N* = 3 dyi
determines
Conversely, if Nj are the coefficients of a nonlinear connection N, then (4.5) G<(x,y) = N;(x,y)yi are the coefficients of a semispray on TM. The nonlinear connection N, determined by the morphism C is called homogeneousand linear if the connection map K associated to C has this property, respectively.
Ch.1. The geometry of tangent bundle
13
Taking into account (4.3) and the local expression of the mapping nT it follows that N is homogeneous iff its coefficients Nj(x,y) are homogeneous. Exactly as in Theorem 1.3.4, we can prove: Theorem 1.4.4. If the manifold M is pracompact, connections on TM.
1.5
then there exists nonlinear
The structures P, F
Now, let ix : TXM —> TM be the inclusion and for u G TXM consider the usual identification ku : u G TXM ->• u G TU(TXM). We obtain a natural isomorphism tVu = C(u) ° ku • Tn(u)M -+ VUTM called the vertical lift
d In local coordinates, for any z = z1-^— G Tntu)M it follows
The canonical isomorphism n : VTM -> TT*TM is the inverse of the isomorphism j : TT*TM -» VTM defined by j(zi,z 2 ) = ^ ( ^ 2 ) , zi,z 2 G T M , 7r(zi) = TT(Z2)Explicitely, we have r(X u ) = (u, ( r j - l ( X u ) ) , u G T M , X u G VUTM. Consequently, we can define (5.1)
-linear mapping J : X{TM)
-> X(TM)
by
Ju = iu o ju OTT!(U).
Proposition 1.5.1. 1° The mapping (5.1) w the tangent structure J investigated in §i. 2° In the natural basis J is given by
In the same manner we can introduce the notion of almost product structure IP on TM. Based on the fact that direct decomposition (4.2)' holds when a nonlinear connection Nis given, we consider the vertical projector v : X(TM) —> X(T,M) defined by v{X) = C(X), VX G X{VTM), V{X) = 0, VX G X(HTM).
14
The Geometry of Hamilton & Lagrange Spaces
Of course, we have v2 — v. The projector v coincides with the mapping C considered as morphism between modules of sections. So, N is characterized by a vertical projector v. On the same way, a nonlinear connection on TM is characterized by a T{TM)linear mapping h : X{TM -> A'(TM) for which: ft2 = /i, Ker/i = #(VTM). The mapping h is called the horizontal projector determined by a nonlinear connection N. We have h + v = I. Finally, any vector field X G X{TM) can be uniquely written as follows X = hX + vX. In the following we adopt the notations
hX = XH, vX = Xv and we say XH is a horizontal component of vector field X, but X v is the vertical component. So, any X e ^ ( T M ) can be uniquely written in the form (5.2)
X = XH + XV.
Theorem 1.5.1. A nonlinear connection N in the vector bundle (TM,M) is characterized by an almost product structure F on the manifold TM whose distribution of eigenspaces corresponding to the eigenvalue -1 coincides to the vertical distribution on TM. Proof. Given a nonlinear connection N, we consider the vertical projector v determined by N and set P = / — 2v. It follows F 2 = /, Hence F is an almost product structure on TM. We have (*)
F(X) - -X,
VX g X{VTM).
Conversely, if an almost product structure F on TM is given, and F has the property (*), we set v := - ( / — P ) . It results that v is a vertical projector and therefore it determines a nonlinear connection N. The following relations hold: (5.3)
F = 2h - / , P = h-v,
P =
I-2v.
Taking into account the properties of the tangent structure J and almost product structure P we obtain (5.4)
JP = J, F J = - J.
Ch.1. The geometry of tangent bundle
15
Let us consider the horizontal lift determined by a nonlinear connection N with the local coefficients Nj(x,y). Denote the horizontal lift of vector fields -—, (i = l,..., n), by _6_ _ (d_
5xi ~ \dxi Remark that n\ : HTM -> n'TM is an isomorphism of vector bundle. Then the horizontal lift induced by N is just the inverse map of 7r! restricted to HTM. According to (4.3)' we have
where Nlj are the coefficients of the nonlinear connection N. Locally, if X = X{{x) —l , then XH = X{ ( — l- Ni{x,y)-. Moreover, dx \dx oyi) f S\ \ -r- ) , (i = 1,..., n), is a local basis in the horizontal distribution HTM.
\Sx'J
Consequently, it follows that ( 7— >-^— i 1 (1 = 1, ..., n), is a local basis adapted \oxl oy%J to the horizontal distribution HTM and vertical distribution VTM. - ^ T , - — is It follows ox1 oy1 J i l j (5.6) 8y = dy + N}(x,y)dx . Proposition 1.5.2. The local adapted basis -^—'-^— I and its dual (dxl,5yl) trans\oxl oy1 j form, under a transformation of coordinate (1.3) on TM, by (5 7)
-
±_&±.
d__dV_d_,
5a« ~ dx* 5xi' dy* ~ dx' dx*'
dsi-?tdxj.
sv-^Syi
~ dxi ** ' dy ~ dxi *V '
Indeed, the second formula is known from (1.4). The first one is a consequence of the formula .7—l-. = f 7-^1 1 . 5x \ dx J For the operators h, v, P we get: —\-A--h(—\(5.8)
(—\-o
(—\- —
The Geometry of Hamilton & Lagrange Spaces
16
Now, let us consider the T(TM)-inear
by
mapping F : X(TM)
-> X(TM), defined
Theorem 1.5.2. The mapping F has the properties: 1° F is globally defined on the manifold TM. 2° F is a tensor field of (1,1) type on TM. Locally it is given by (5.10)
d
•
S
F = — —-^ & dx + -z-^ ® Sy\ ay' dx*
3° F is an almost complex structure on TM: (5.11)
FoF=-/.
Proof. Since (5.9) and (5.10) are equivalent, it follows from (5.10) that defined on TM. From (5.9) we deduce (5.11).
is globally q.e.d.
By a straightforward calculation we deduce: Lemma 1.5.1. Lie brackets of the vector fields from adapted basis given by
where (5.13)
Rljk = " "k Sx
8x
Let us consider the quantities (5.14)
«>
*»••k dy
Also, by a direct calculation, we obtain:
6N
dyi
(8 d\ —l ^ 7 - ^ are \8x dy')
Ch.1. The geometry of tangent bundle
17
Lemma 1.5.2. Under a transformation of coordinates (1.3) on TM, we obtain
Consequently, the tensorial equations Wjk = 0, h ing. Therefore, we get: Lemma 1.5.3. have on TM:
= 0
ave
geometrical mean-
The horizontal distribution HTM is integrable if and only if we
Indeed, from (5.12), the Lie brackets U— ,7-r
give an horizontal vector fields
\_dx3 ox"J
if and only if &ik = 0. The previous property allows to say that Rljk is the curvature tensor field of the nonlinear connection N. We will say that t'jk from (5.14) is the torsion of the nonlinear connection N. Now, we can prove: Theorem 1.5.3. have
The almost complex structure W is integrable if and only if we # V = 0, fjk = 0.
(5.16)
Proof. Applying Lemma 1.5.1, and taking into account the Nijenhuis tensor field of the structure F [113]: Afw{X,Y)
= -[X,Y]
+ [TFX,FY] -
putting X = -r-> X — —— etc., we deduce dxl oyJ
5
M
d \ _
,
t
= Rl
6
{
d
—~ + t-
\ dyi dy J oyl Now it follows that Afw = 0 «=* {R}^ = fjk = 0}.
ox1
q.e.d.
The Geometry of Hamilton & Lagrange Spaces
18
1.6
d-tensor Algebra
Let N be a nonlinear connection on the manifold E = TM. We have the direct decomposition (4.2)'. We can write, uniquely a vector field X € X{TM) in the form X = XH + XV
(6.1)
where XH belongs to the horizontal distribution HTM. b a%s i s - l I to the direct decomposition (4.2)' we can 5x
dy }
write: X" = X^x,y)-^, Xv =
(6.1)'
With respect to (1.3) the components Xl(x,y) tively obey the rules of transformation
(6-1)"
X\x,y)j-.
and Xl(x, y) of XH and Xv respec-
X' = f^X>, # = f^X'".
Also, a 1-form field u> e ,¥*(.£) can be always set as follows (6.2)
U = LJH
V
where w"(X) = u(XH), UV{X) = to(Xv), VX e Therefore in the adapted cobasis (d:z\ wehave : w H = ujj{x,y)dx:>,
(6.2)'
The changes of local coordinate on TM transform the components (x,y), (x,y) of the 1-form w as the components of 1-forms on the base manifold M, i.e.: (6.2)"
dx' dx* ~ Uj{x,y) = — u>i(x,y); Uj{x,y) = ^—• Wj^jf).
A curve c : t £ I —> (xl(t),y'(t))
rfc £ E, has the tangent vector — = c given in the etc
form (6.1), hence:
-v dxl 6 Syi d c=c + c = + ^ c n dt 6xl dt dyl Sy' This is a horizontal curve if — = 0, Vt € / . So, ifthe functions a:J = x'(t), t € / (6.3)
are given, then the curves y1 = yl(t), t £ / , solutions of the system of differential Sy1 • dx-' equations ——h N j(x,y)—— = 0, determine a horizontal curve c in E = TM. dt dt
Ch.1. The geometry of tangent bundle
19
dxi A horizontal curve c with the property y% = —r- is said to be an autoparallel at curve of the nonlinear connection N. Proposition 1.6.1. An autoparallel curve of the nonlinear connection N, with the coefficients is characterized by the system of differential equations
Now we study shortly the algebra of the distinguished tensor fields on the manifold TM = E. Definition 1.6.1. A tensor field T of type (r, s) on the manifold E is called distinguished tensor field (briefly, a d-tensor) if it has the property (6.5)
( ^ , ^1 , , ) ( , , , , , X ) , MUJ eX*(E),
MX €X(E),
is
(a = l,...,r;
b=l,...,s).
For instance, the components XH and Xv from (6.1) of a vector field X are d-tensor fields. Also the components LJH and uiv of an 1-form w, from (6.2) are d-l-form fields. Clearly, the set T[(E) of the d-tensor fields of type (r, s) is a module and the module T{E) = ® 77 is a tensor algebra. It is not difficult to see that any tensor field on E can be written as a sum of d-tensor fields. We express a d-tensor field T from (6.5) in the adapted basis I -r—•,— I and yox oy j adapted cobasis (dx\5yl). From (6.5) we get the components of T: (6.5)'
T}l;;Z(x,y) = T (dxh,...,6y^,
A - , • • • ,JL\ , (iu ...,{,, j u ...,Js =
So, T is expressed by (6.6)
T = T^X
{x,y)^®---®~
Taking into account the formulae (5.7) and (6.5)', we obtain: Proposition 1.6.2. With respect to (1.5) the components Tj\\"ja{x,y) of a d-tensor field T of type (r, s) are transformed by the rules:
The Geometry of Hamilton & Lagrange Spaces
20
But (6.7) is just the classical law of transformation of the local coefficients of a tensor field on the base manifold M. Of course, (6.7) characterizes the d-tensor fields of type (r, s) on the manifold E = TM (up to the choice of the basis from (6.6)). Using the local expression (6.6) of a d-tensor field it follows that < L—^i-pr-^ > , (i = 1, ..., n), generate the d- tensor { 6x' dyl) algebra T over the ring of functions F{E) Taking into account Lemma 1.5.2 it follows: Proposition 1.6.3. 1° IPjk and Vjk from (5.12), (5.13) are d-tensor fields of type (1,2). 2° The Liouville vector field C =
1.7
J/'-T-T
dy1
is a d-vector field.
N—linear connections
Let N be an a priori given nonlinear connection on the manifold E = TM. The adapted basis to N and V
Ii
s and adapted cobasis is its dual
Definition 1.7.1. A linear connection D (i.e. a Kozul connection or covariant derivative) on the manifold E = TM is called an N–linear connection if: 1° D preserves by parallelism the horizontal distribution N. 2° The tangent structure J is absolute parallel with respect to D, that is DJ = 0. Consequently, the following properties hold: (7.1)
{DXYH)V H
= 0, (DXYV)H H
(7.1)'
DX(JY )
= J(DXY );
(7.1)"
Dxh = 0, Dxv = 0.
= 0, DX(JYV)
=
We will denote (7.2)
DHXY = DXHY,
DVXY
Thus, we obtain the following expression of D: (7.3)
Dx = Dx' + Dx
= DxvY.
J{DXYV),
Ch.1.
The geometry of tangent bundle
21
The operators DH and Dv are special derivations in the algebra of d-tensor fields on E. Of course, DH, Dv are not covariant derivations, because D f / = XH f ^ Xf, D]cf = Xvf ¥" Xf- However the operators DH and Dv have similar properties to D. For instance, DH and Dv satisfy the Leibniz rule with respect of tensorial product of d-tensor fields. It is important to remark that DH and Dv applied to d-tensor fields give us the d-tensor fields, too. We can see these important properties on the local representtion of DH and Dv in the adapted basis I -^^>-^-^ I . DH and \6xl dy%J v D will be called the h-covariant derivation and v-covariant derivation, respectively. Remarking that D_£, = Dj^, D_s, = D ^ , we obtain: •Si*
6X>
By'
can be uniquely represented in the form:
(74)
gyt
aptedbas i s an N-linear connection D dxl dyl)
D
%h=L'*^h- °%h = hi>^h'
The system of functions DT(N) = {L'jk{x, y), C'jkix, y)) gives us the coefficients of the h-covariant derivative DH and of the v-covariant derivative Dv, respectively. Proposition 1.7.2. With respect to the changes of local coordinates on TM, the coefficients Lljk(x,y), Cljk(x,y) of an N-linear connection D are transformed as follows: /--
jh{X V)
ni
ar ' ~ dx1 dx~i d& ,- -, ^ dxS dxT r*
°»(x>y) = a* a& W
c
+
dx~r
WW'
"•
Indeed, the formulae (7.4) and (5.7) imply the rules transformation (7.5).
Remarks. 1° C*ffc(:c,2/) are the coordinates of a d-tensor field. 2° A reciprocal property of that expressed in the last proposition also holds. Let us now consider a d-tensor field T in local adapted basis, given for simplicity by
The Geometry of Hamilton & Lagrange Spaces
22
Its covariant derivative with respect to X = XH + Xv = X1 -j-^ + y J —- is given ox1 oy3
by (7.6)
DXT = (*'?*„ + y'rj | f ) A 0 ^_ g, ^
e
V,
where we have the /z-covariant derivative, D%T = X T ? * ! . — ® -^—
<5T-S
^ j't
r i rpls I
i
r 5 T^i^ r I rpis T^i^ r IU rpis lr jh ~ hjr (h ~
Lr 5 1
jl rpis hr jl-
1 j lL rpis
Therefore, "|" is the operator ofh-covariant derivative. Of course, T}Lr is a d-tensor field with one more index of a covariance. The v-covariant derivative of T is D\T = YlTis, —: ® 7 - - ® dx3' ® Syh, and ift|* ox1 oys the coefficients T", are as follows: 9T Here we denoted by "|" the operator of v-covariant derivative and remark that T", is a d-tensor field with one more index of a covariance. The operators "1" and "|" have the known properties of a general covariant derivatives, applied to any d-tensor field T, taking into account the facts: j-r = /|,-, -~ = /i. for any function / € F{E). An important application can be done for the Liouville vector field C = y1-^-dyl
The following d-tensor fields (7.9)
D l , = y% d*,- = y%
are called the h- and v-deflection tensor fields of the N-linear connection D. Proposition 1.7.3. The deflection tensor fields are given by (7.9)'
D'j = ysUsj - tfj, tfj = 5} + y*Clsj.
Indeed, applying the formulae (7.7), (7.8) we get the equalities (7.9)'. The d-tensor of deflections are important in the geometry of tangent bundle.
Ch.1. The geometry of tangent bundle
23
A N-linear connection D is called of Cartan type if its tensor of deflection have the property: Dl3 = 0, dtj = S'j.
From the last proposition, it follows Proposition 1.7.4 The N-linear connection DT{N) = (Vjk, C'jfc) is of Cartan type if and only if we have Nij = y'Litj,y'Ciaj
(7.10)
= 0.
We will see that the canonical metrical connection in a Finsler space is of Cartan type. We can prove [113]: Theorem 1.7.1. IfM is a paracompact manifold then there exist N-linear connections on TM.
1.8
Torsion and curvature
The torsion of a N–linear connection d is given by (8.1)
T{X, Y) = DXY - DYX - [X, Y], X, Y £ X{TM).
Using the projectors, h, and v associated to the horizontal distribution N and to the vertical distribution V, we find T(X, Y) = "![(XH, YH) + T(XH, Yv) + T(XV, YH) + TT(XV, Yv). Taking into account the property of skew-symmetry of T and the fact that [Xv, YV]H 0 we find Theorem 1.8.1. The torsion IT of an N-linear connection is completely determined by the following d-tensor fields: H
, YH) = D%YH - D$XH - [XH, YH]H,
(8.2)
hTT(XH, Yv) = -D^XH - [XH, Yv}», vTT(XH,Yv) = D\YV - [XH,YV]V, vT{Xv,Yv) = DVXYV - D\XV - [XV,YV]V,
X,Y € X{TM).
The Geometry of Hamilton & Lagrange Spaces
24
Corollary 1.8.1. The following properties hold: a) vT{XH,YH) = 0 <=*• HTM is an integrable distribution. b) h1T{XH, Yv) = 0<=^ vD»Yv = [XH, Yv]v, X, Y e X{TM), We shall say that hT(XH, YH) is /i(M)-torsion of D, thatvT(XH, YH) is torsion, etc. Since the Lie brackets of the vector field from the adapted basis are given by the formula (5.12), we obtain Theorem 1.8.2.
The local components, in the adapted basis I-—-l-^-}, F F [Sx1 dy'j torsion T of an N-linear connection are as follows:
ri
(8.3)
pi
fi
_ ft
=dN[_Li
ffi
gi
of the
J
(~
=(ji
_&
Proof. These local coefficients are provided by the five formulae (8.2) if we consider instead of X and Y the components of the adapted basis I —{>^-r 1. The curvature of a N–linear connection D is given by (8.4)
R{X,Y)Z = DXDYZ - DYDXZ - D[xy]Z, VX,Y,Z € X{TM).
It is not difficult to prove the following theorems: Theorem 1.8.3. The curvature tensor 1R of the N-linear connection D has the properties: ^8
vM(X, Y)ZH = 0, hJR(X, Y)ZV = 0, M(X, Y)Z = hlR{X, Y)ZH + vM(X, Y)ZV, MX, Y,Z e X{TM).
Theorem 1.8.4. The curvature of an N-linear connection D on TM is completely determined by the following three d-tensor fields: R(XH, Y»)ZH = £>£D?Z» - D«D»ZH - D?XH YH]ZH (8.6)
R(XV,YH)ZH
= DVXD§Z" - D«DXZ"
- Dfx^yn]ZH
R(XV,YV)ZH
= DXD^ZH - DVYD\ZH - D\xV
H
yV]Z
.
D^HYH]ZH, Djxv[yH]ZH,
Ch.1. The geometry of tangent bundle
25
Remark. Thecurvature H has six components. But the property J(M(X, Y)ZH) = M(X, Y)ZV shows that only three components, namely the one in (8.6) are essential. In the adapted basis, the local coefficients of the d-tensors of curvature are given
by
(8.7)
II IR, [ "
" | " = 5" * "
Now, using Proposition 1.7.1, we obtain: Theorem 1.8.5. In the adapted basis the d-tensors of curvature Rh.ljk,Phljk and Shljk of an N-linear connection DF(N) — (L'jk,C'jk) are as follows: n
Kh
i
hj
Q'-'hk _i_ rro j i
* ~ T^T ~ ~T~T +
. _ dClj
dUhk
L
hjLmk
m
rm
ji
, /~ii
~ LhkLmj + ChmX
rym
]k,
i
where | denotes, as usual, the h-covariant derivative with respect to the N-linear connection DY{N) = ( L ' ^ C V ) . The expressions (8.6) of the J-tensors of curvatureiR(XH, YH)ZH, R(XV, YH)ZH and M(XV, YV)ZH in the adapted basis lead to the Ricci identities satisfied by an N-linear connection D. Proposition 1.8.1. The Ricci identities of the N—linear connection {Lijh,Cijk) are:
(8.9) where X% is an arbitrary d-vector field. The Ricci identities for an arbitrary d-tensor field hold also.
The Geometry of Hamilton & Lagrange Spaces
26
For instance if 9ij{x,y) is a d-tensor field, then the following formulae of the commutation of second h- and v-covariant derivative hold: 9ij\k\h - 9ij\h\k — ~9sjRiSkh ' (8.10)
9ij\k\h ~ gij\h\k ~ ~9sjPiSkh
Applying the Ricci identities(8.9) to the Liouville vector field C = y'-^— we dyl deduce some fundamental identities in the theory of N-linear connections. Taking into account the h- and v-deflection tensors D'j — y%\j, d'j = y'i. we have from (8.9): Theorem 1.8.6. For any N-linear connection DF(N) = (L'jk,C'jk) identities hold: *-> k\h~ U h\k — y Its kh ~ D sl lfi k =
®*k\h ~~ d \
(8.11)
S
the following
kh ~ « sH- kh, s
y P<>'kh ~ D'sC kh.
— d'sPSkh,
Corollary 1.8.2. If DT(N) is an N-linear connection of Cartan type, then the following relations hold:
ysRs\h
(8.12)
= R\h, ysPs\h = P\h, y3Sa\h = Slkh.
The d-torsions and d-curvature tensors of an N-linear connection DF(N) — (L'jk,C'jic) are not independent. They satisfy the Bianchi identities[113], obtained by writting in the adapted basis the following Bianchi identities, verified by the linear connection D:
£[£xTT(y, Z) - B.{X, Y)Z + TT(T(X, Y),Z)} = 0, where £ means the cyclic sum over X, Y, Z.
1.9 Parallelism. Structure equations Consider an N-linear connection D with the coefficients DT(N) = (Lljk,Cljk) in the adapted basis
( 6
d\
—-i-^— .
\Sx' dy'J
Ch.1. The geometry of tangent bundle
27
If c is a parametrized curve in the manifold TM, c:tel-*c(t) - (xl(t),
y'(t))eTM, dc with the property Imc C -K~1(U) C TM, then its tangent vector field c = — can at be written in the form (6.3), i.e. i = 6P+
(9.1) K '
cv = ^.-L + dt 6x'
^.±.
dt dy*
Syi The curve c is horizontal if — = 0 and it is an autoparallel curve of the nonlinear Syi dxi connection N if — = 0, y' — —— dt dt We denote ny DX (9.2) — = DtX, DX = ^--dt,VXe X{TM). dt dt DX Here —— is the covariant differential along with the curve c of the N-linear condt nection D. Setting X = XH + Xv, XH = X1-^-, Xv = X{4~ we have 6xl dy% H V k DX^_DX DX _ ( t dx • Syk) 8 ~dT ~ dt + dt ( J f »+ Jfl-'-+
Let us consider Jj = Ujkdxk + C j f c V.
(9.4)
The objects wlj are called the "1-forms connection" of D. Then the equation (9.3) takes the form: DX
[dXi ,xm"lm\
{
+ x
5
+
UX*
+x
^
The vector X on TM is said to be parallel along with the curve c, with respect to DX N-linear connection D if —— = 0. A glance at (9.3) shows that the last equation dt DXH DXV is equivalent to —;— = —;— = 0. Using the formula (9.5) we find the following dt dt result: Proposition 1.9.1. The vector field X = X1— + X'-r— from X{TM) is parallel along the parametrized curve c in TM, with respect to the N-linear connection
28
The Geometry of Hamilton & Lagrange Spaces
Dr(N) = {Uj^C'jk) if and only if its coefficients X ! (x(t),y(f)) and X'(a;(t),y{t)) are solutions of the linear system of differential equations
A curve A curve
theorem of existence and uniqueness for the parallel vector fields along with a c on the manifold TM can be formulated. horizontal path of an N–linear connection D on TM is a horizontal parametrized c : I —> TM with the property Dtc = 0. dr'
•
fiv*
Using (9.1) and (9.5), with X1 = —-, X1 = -f- = 0 we obtain the following at at theorem: Theorem 1.9.1. The horizontal paths of an N-linear connection DT(N) = are characterized by the system of differential equations:
(L'jk,Cljk)
Now we can consider a curve cvXo in the fibre TXQM = 7r-1(a;o). It can be represented by the equations
xl = 4 , y' = y{(t), t e I.
The above cvXo is called a vertical curve of TM in the point x0 € M. A vertical curve c\a is called a vertical path with respect to the N-linear connection D if Divocvxo = 0 . Again the formulae (9.1), (9.5) lead to: Theorem 1.9.2. The vertical paths in the point XQ £ M, with respect to the N-linear connection DF(N) = (Lljk,C'jk) are characterized by the system of differential equations
Obviously, the local existence and uniqueness of horizontal paths are assured if the initial conditions for (9.6) are given. The same consideration can be made for vertical paths, (9.7). Considering the 1-form of connection w*,- from (9.4) and the exterior differential of the 1-forms from the adapted dual basis (dx\6yi) we can determine the structure equations of a N-linear connection D on the manifold TM.
Ch.1. The geometry of tangent bundle L e m m a 1.9.1. by
29
The exterior differentials of 1-forms 5yl — dy' + N2jdx^ are given dSy* = ]^Ri}mdxmAdx3
(9.8)
Bijm6ymAdxj
+
where
Indeed, a straightforward calculus on the exterior differential dSy1 leads to (9.8). Remark. B'jm from (9.8)' are the h-coefficients of an N-linear connection, called the Berwald connection. Lemma 1.9.2. With respect to a changing of local coordinate on TM, the following 2-forms d(dxl) -
dxmAujim
diStf) - SymA0Jim are transformed like a d-vector field and the 2-forms are transformed like a d-tensor field of type (1,1). Indeed, taking into account Lemma 1.9.1 and the expression of 1-forms of connection uilj the previous lemma can be proved. Now, we can formulate the result: Theorem 1.9.3. The structure equations of an N-linear connection DT(N) — (L'jk, C'jk) on the manifold TM are given by
(Q Q)
1
'
d{dxt)-dxmAujlm
=-
d(6yi)-6ymAuJim
=
dui'j - LjmjAu)'m (o).
=
(o) i
n
(1)
-ni -Q'j
(i).
where Q' and W are the 2-forms of torsion ni = j- TjkdxjAdxk 2
(9.10) (!)
+
CijkdxjASyk
1
1
J
2
n* = - RljkdxjAdxk + P^kdrfAdy" + -
Sijk8yjASyk
30
The Geometry of Hamilton & Lagrange Spaces
and the 2-forms of curvature fi*7 are expressed by (9.11)
Wj = l-Rjikhdxkhdxh + P/kltdxkA8yh +
^S/aS
Proof. By means of Lemma 1.9.2, the general structure equations of a linear connection on TM are particularized for an N-linear connection D in the form (9.9). Using 1-forms connection UJ1J from (9.4) and the formula (9.8), we can calculate, (o).
(i).
without difficulties, the 2-forms of torsion fll, and the 2-forms of curvature Hl7, obtaining the expressions (9.10) and (9.11). The geometrical theory on the manifold TM of tangent bundle will be used in next chapters for studying the geometries of Finsler and Lagrange spaces.
Chapter 2 Finsler spaces The notion of general metric space appeared for the first time in the disertation of B. Riemann in 1854. After sixty five years, P. Finsler in his Ph.D. thesis introduced the concept of general metric function, which can be studied by means of variational calculus. Later, L. Berwald, J.L. Synge and E. Cartan precisely gave the correct definition of a Finsler space. During eighty years, famous geometers studied the Finsler geometry in connection with variational problem, geometrical theory of tangent bundle and for its applications in Mechanics, Physics or Biology. In the last 40 years, some remarkable books on Finsler geometry were published by H. Rund, M. Matsumoto, R. Miron and M. Anastasiei, A. Bejancu, Abate-Patrizio, D. Bao, S.S. Chern and Z.Shen, P. Antonelli, R.Ingarden and M. Matsumoto. In the present chapter we made a brief introduction in the geometry of Finsler spaces in order to study the relationships between these spaces and the dual notion of Cartan spaces. In the following we will study: Finsler metrics, Cartan nonlinear connection, canonical metrical connections and their structure equations. Some special classes of Finsler manifolds as (a, /?)-metrics, Berwald spaces will be pointed out. We underline the important role which the Sasaki lift plays for almost Kählerian model of a Finsler manifold, as well as the new notion of homogeneous lift of the Finsler metrics in the framework of this theory.
2.1
Finsler metrics
At the begining we define the notion of Finsler metric and Finsler manifold. Definition 2.1.1. A Finsler manifold (or Finsler space) is a pair Fn = (M,F(x,y)) where M is a real n-dimensional differentiable manifold and F : TM —> IR a scalar function which satisfy the following axioms: 31
The Geometry of Hamilton & Lagrange Spaces
32
1° F is a differentiable function on the manifold TM = T M \ { 0 } and F is continuous on the null section of the projection -n : TM —» M. 2° F is a positive function. 3° F is positively 1-homogeneous on the fibres of tangent bundle TM. 4° The Hessian of F 2 with elements _ 1
d2F2
is positively defined on TM. Proposition 2.1.1. The set of functions 9ij(x,y) from (1.1) is transformed, with respect to (1.3) in Ch.l, by the rule
_ dxr
dxs
Indeed, in virtue off/3/(1.4)C/x*in Ch.l Iwe Or have: 9y' 95* (??/r 2 dy^dyi
OX
OX
I
dx? dxi 2
dyrdys
Consequently (1.2) holds. Because of (1.2) we say that gy is a distinguished tensor field (briefly d-tensor field). Of course, gij is a covariant symmetric of order 2 d-tensor field defined on the manifold TM. The function F(x, y) is called fundamental function and the d-tensor field gij is called fundamental (or metric) tensor of the Finsler space Fn = (M,F(x,y)).
Examples. 1° A Riemannian manifold (M, jij(x)) determines a Finsler manifold
Fn = (M,F(x,y)),
(1.3)
where F{x,y)
The fundamental tensor of this Finsler space is coincident to the metric tensor Jij(x) of the Riemann space (M,jij(x)). 2° Let us consider the function (1.4)
F(X, y) =
j
Ch.2. Finsler spaces
33
defined in a preferential local system of coordinate on TM. The pair Fn = (M, F(x,y)), with F defined in (1.4) satisfies the axioms 1-4 from Definition 2.1.1. So, F " is a Finsler space. The fundamental tensor field - atiX1 (ojj are positive constants), where L = {(y1)"1
+ (y2)m + ••• + (yn)m}i/m,
m > 3, m being even.
4° Randers metric. Let us consider the function of a Finsler space F " :
F(x,y)
=a(x,y)+0{x,y),
where a2 := aij(x)y'y:' is a Riemannian metric and P(x,y) := bi(x)y' is a differential linear function in yl. This metric is called a Randers metric and was introduced by the paper [135]. The fundamental tensor of the Randers space is given by [89]: rv-l-/?0
9ij = h
t
j
0
+didj, hij.= dij
00
00
liij, di := bi+ £i, £f=
rln
rr
and one can prove that the fundamental tensor field #y is positive definite under the condition b2 = a^bibj < 1 (see the book [24]). The first example motivates the following theorem: Theorem 2.1.1. If the base manifold M is paracompact, then there exist functions F : TM -»• R which are fundamental functions for Finsler manifolds. Regarding the axioms 1-4 formulated in Definition 2.1.1, we can prove without difficulties. Theorem 2.1.2. The system of axioms of a Finsler space is minimal. However, the axiom 4° of this system is sometimes too strong in applications of Finsler geometry in construction of geometrical models in other scientific disciplines, for instance in theoretical physics.
The Geometry of Hamilton & Lagrange Spaces
34
Let us consider the fibre TXoM in a point x0 e M of the tangent bundle It is known that TI0Af is a real n-dimensional vector space. The set of points (TM,TT,M).
/(*„) = {" € T I 0 M | F(u) < 1} C T I 0 M
(1.5)
is called the indicatrix of Finsler space F" in the point :r0 e Af. The restriction F(xo,y) of the fundamental function F to the fibre TX0M determines a Riemann metric gij(xo,y) in the submanifold T X0 M immersed in the manifold TM. Since gij{xo,y) is positively defined, the following property holds, [139]: Theorem 2.1.3. If a Finsler manifold Fn = (M,F(x,y)) has the property: in every point x0 € M, the indicatrix I(x0) is strictly convex, then the axiom 4° is satisfied. If one retains the axioms 1°, 2°, 3° from the Definition 2.1.1 of a Finsler manifold and add the following axiom 4´: 4 ' a. mnk\\gij(x, y)\\ = n on TM. b. signature of d-tensor field gij{x,y) is constant, it results the notion of Finsler manifold with semidefinite metric. If the axioms 1°, 2°, 3°, 4 ´ are satisfied on an open set •K~1{U) C TM we will say that we have a Finsler space with the semidefinite metric on the open set n~1(U). In general, Randers metric a + j3 (without the condition \\b\\ < ) give rise to Finsler spaces with semi-definite Finsler metrics. In the following sections of chapters of the present monograph we refer to the Finsler spaces with the definite or semidefinite metric without mention the difference between them.
2.2
Geometric objects of the space Fn
The property of 1-homogeneity of the fundamental function F(x, y) of the Finsler space Fn induces properties of homogeneity of the geometrical objects derived from it. Theorem 2.2.1. On a Finsler manifold Fn the following properties hold: 1° The components of the fundamental tensor field gij are 0-homogeneous, i.e.,
(2.1) 2° The functions
tf
| f = 0.
Ch.2. Finsler spaces
35
are 1-homogeneous. 3° The functions
'3
1
4 dy'dyWyk
are (-l)-homogeneous. In the same time we have some natural object fields:
Proposition 2.2.1. 1° The quantity Pi from (2.2) is a d-covectorfield. 2° The set of functions tensor field. 3° ||X|| 2 •-gij{x,y)XiXJ 4° (X,Y)
:^ gij{x,y)XiY^
from (2.3) is a covariant of order 3 symmetric dis a scalar field, if Xi is a d-vector field. is a scalar field, Xi:Yi
being d-vector fields.
The proof of previous properties is elementary. ||-\T||2 is called the square of norm of vector field X and (X,Y) is the scalar product (calculated in a point u e TM). ^Assuming ||X|| U / 0, ||y|| u jt 0 the angle
<X,Y>(u) \\X\\JYL • The number ip is uniquely determined, because in a Finsler space with the definite metric cos from (2.4) satisfy the condition — 1 < cosy? < 1. Other properties are given by Proposition 2.2.2. hold:
In a Finsler manifold Fn = (M, F) the following identities
1° ptf = F2 2° yi :- gtjy3 = Pi 3° Cojh = y'Cijh. = 0, CjQh = Cjho = 0
4°
F2{x,y)=gij{x,y)yiyK
The Geometry of Hamilton & Lagrange Spaces
36
Some natural object fields are introduced in the following: Theorem 2.2.2. In a Finsler manifold Fn = (M, F) we have the following natural object fields: 1° The Liouville vector field
2° The Hamilton 1-form uj=Pidx{.
(2.6) 3° The symplectic structure (2.7)
6 = du> = dpiAdx\
Proof. 1° The Liouville vector field C exists on the manifold TM independently of the metrical function F of the space Fn (cf. §1, Ch.l). 2° By means of Proposition 2.2.1 and on the fact that with respect to (1.1) it follows that w does not depend on the changing of local coordinates. 3° 6 is a closed 2-form on TM and rank||0|| = 2n = dim of TM.
q.e.d. Definition 2.2.2. A Finsler space Fn = (M, F(x, y)) is called reducible to a Riemannian space if its fundamental tensor field does not depend on the directional variables yl. The previous definition has a geometrical meaning,
since the equation
•^-j- = 2djk = 0 does not depend on the changing of local coordinates. Theorem 2.2.3. A Finsler space Fn is reducible to a Riemannian space iff the tensor Cijk is vanishing on the manifold TM. Proof. If 9ij does not depend on y1, thus from the identity
(2.8)
§ £ = 2Ctjk
Ch.2. Finsler spaces
37
dgu the condition -^-f = 0 implies Q,-* = 0. Conversely, Cijk = 0 on TM and (2.8) implies -^j — 0 on T M . Let us consider another geometrical notion: the arc length of a smooth curve in a Finsler manifold Fn = (M,F(x,y)). Let c be a parametrized curve in the manifold M: (2.9)
c : t e [0,1] — • c(t) eU CM
U being a domain of a local chart in M. The curve c has an analytical expression of form: (2.7)'
x{ = x\t), t € [0,1].
The extension c of c to TM is defined by the equations (2.7)"
x{ = xi(t) yi =
(t) t 6 [0 ll.
Thus the restriction of the fundamental function F(x, y) to c is
We define the "length" of curve c with extremities c(0), c(l) by the number
(2.10)
Hc)=JoF\x(t),-(t)jdt.
The number L(c) does not depend by the changing of coordinates on TM and, by means of 1-homogeneity of the fundamental function F, L(c) does not depend on the parametrization of the curve c. So L(c) depends on c, only. We can fix a canonical parameter on the curve c, given by the arclength of c. Indeed, the function s = s(t), t € [0,1], given by
is derivable, having the derivative:
The Geometry of Hamilton & Lagrange Spaces
38
So the function s = s(t), t € [0,1], is invertible. Let t — t(s) be its inverse. The change of parameter t -»• s, given by s = s(t), has the property (8)^ = 1.
F(X(8),^
(2.11) So, we have:
Theorem 2.2.4. In a Finsler space Fn = for any smooth curve c:t€[O, l]-> c(t)eUcM exists a canonical parameter s with the property (2.11).
2.3 Geodesies Let us consider a smooth parametrized curve c : t e [0,1] —> c(t) e U C M having the ending points c(0) and c(l). Its length is given by the formula (2.10). We will formulate the variational problem for the functional L(c). Consider a vector field V'(x{t)) along the curve c with the properties V*(c(0)) = V'^l)) = 0. Let c £ be a set of smooth curve given by the mappings c£:te[0,l]->ce(f)G£/, such that the analytical expression of cs(t) being with |e| small, e being real number. So, the curves ce{t) have the same end points c(0), c(l) and same tangent vectors in this points with curve c. The length of the curve ce is given by
The necessary condition for L(c) to be extremal value of L(ce) is as follows dL(ce) de
= 0. £=0
This equality is equivalent to
Integrating by parts the second term one obtains \dF
d fdF\]
„
„
,
dx{
Ch.2. Finsler spaces
39
Since V is an arbitrary d-vector field we get from the previous equation the following Euler-Lagrange equations dF
^
ft?
d
~S
Definition 2.3.1. The curves c = {(x{(t)), t € [0,1]} solutions of the Euler-Lagrange equations (3.1) are called geodesies of theFinsler space Fn. Since F(x,y) of equations 3
( '^
= Jgij{x,y)yiy^
the equations (3.1) are equivalent to the system
d (dF2\ dt [dyi)
dF2 _ dxi
2
dF_dF_ i_d^_ dt Qyi' y dt
Substituting F2 = gijy'yi we get the following form of the previous Euler-Lagrange equations
where Gi =
(3.3) the functions jljk This is (3 4)
±iijk{x,y)yiyk,
being the Christoffel symbols of the fundamental tensor field
yt(zu)--
xi
O
dxk
dxr (
dx\
C h a n g i n g n o w t o t h e canonical p a r a m e t e r s, w e have F \x,— \ = 1. T h e equations V ds • (3.2) become d2xl
,
/
dx\ dxi dxk
= 0.
Theorem 2.3.1. Geodesics in a Finsler space Fn in the canonical parametrization are given by the differential equations (3.5). A theorem of existence and uniqueness of the solutions of differential equations (3.5) can be formulated.
The Geometry of Hamilton & Lagrange Spaces
40
2.4
Canonical spray. Cartan nonlinear connection
For a Finsler space Fn = (M, F), we can define a canonical spray S and an important nonlinear connection. Noticing that the function F2 is a regular Lagrangian, we introduce its energy by:
Thus, integral action of the Lagrangian L(x, y) along with smooth parametrized curve c : [0,1] —¥ U C M is given by the functional
(4.1)
£(c)
=/>(*,§),«.
The Euler-Lagrange equations are given by: ,4.2)
0
*,,,.,:
Remark. Starting from the property that L(x,y) = F2(x,y) is energy function of the regular Lagrangian F2, we will prove in the next chapter the following two properties. Theorem A. Along with the integral curves of the Euler-Lagrange Q
equations
ffr'
Et (F2) = 0, y* = — we have: at dt Theorem B. (Noether) For any infinitesimal symmetry x'1 = xl + eVl(x,t), t' = = t + er(x,t) (e = const.) of the regular Lagrangian F2(x,y) and for any C°° -function
is conserved along with the integral curves of the Euler-Lagrange 0
equations
Ch.2. Finsler spaces
41
Proposition 2.4.1. The Euler-Lagrange equations (4.1) can be expressed in the form
where Gl is given by (4-4)
2
Proof. The same calculus as in the previous section, taking F2 = glj(x,y)ylyJ shows us that the equations (4.2) are equivalent to (4.3). Remarking that the equations (4.3), (4.4) give the integral curve of the spray
The vector field S is called canonical spray of the space. Proposition 2.4.2. The spray S of (4.5) is determined only by the fundamental function F(x,y). Its integral curve are given by the equation (4.3), (4.4). Consequently, we have: Proposition 2.4.3. In a Finsler space Fn = (M, F) the integral curves of the canonical spray are the geodesies in canonical parametrization. Indeed, the equations (3.5) of geodesics in the canonical parametrization are coincident with the equations (4.3), (4.4). Now, applying the theory from the section 4, ch.l, one can derive from the canonical spray S the notion of the nonlinear connection for the Finsler space Fn = (M, F). Definition 2.4.1. The nonlinear connection determined by the canonical spray 5 of the Finsler space Fn is called Cartan nonlinear connection of the Finsler space Theorem 2.4.1. The Cartan nonlinear connection N has the coefficients (4.5)'
Ni] =
1
-^
It is globally defined on the manifold TM and depends only on the fundamental function F(x,y).
The Geometry of Hamilton & Lagrange Spaces
42
s li s l 1 to the distributions 8x dy j N and V, determined only by the Cartan nonlinear connection. The coefficients of N*j from (4.5)' are 1-homogeneous functions with respect to yl and have the properties
^ y = g - = 2G{ = y'oo,
(4.6) where 7^0 := 7*j*yVWe have Theorem 2.4.2.
1) The horizontal curves w : i —>• t H respect to Cartan nonlinear connection are characterized by the following system of differential equations: (4.7)
x' = x\t),
t € / , 6-£ = ^
+ N^(x(t),y(t))~
= 0.
2) The autoparallel curves of the Cartan nonlinear connection are characterized by the system of differential equations:
<"»
2.5
Metrical Cartan connection
The famous metrical Cartan connection in a Finsler space JF" = (M, F) can be defined as an N-linear connection metrical with respect to the fundamental tensor field g,j and with h- and v-torsions vanish, N being Cartan nonlinear connection. Indeed, we have: Theorem 2.5.1. The following properties hold: 1) There exists a unique N-linear connection D on TM with coefficients DT(N) — = (Lljk,Cljk) satisfying the following axioms: Al D is h-metrical, i.e. g^ = 0. A2 D is v-metrical, i.e. g. .1. = 0. IJ I K
A3 D is h-torsion free, i.e. Tljk := L'jk - L\j = 0. A4 D is v-torsion free, i.e. S{jk := C{ik - C'kj = 0.
Ch.2. Finsler spaces
43
2) The coefficients DF(N) = (L'jk,C'jk) symbols: ri
Lj Ah
l
im (69mh , 6gjm
^9jh\
h m firi7~ + ox l)r1— firox IiI 1 \ ox
—Q
1L
are given by the generalized Christoffel
I
dyi
%im
dgjh\
J
3) D depends only on the fundamental function F of the Finsler space Fn. The proof of the theorem is made by using the known techniques. It was initiated by M. Matsumoto [88]. The connection CT(N) from the previous theorem will be called the canonical metrical Cartan connection. Taking into account this theorem one can demonstrate without difficulties the following properties of the Finsler spaces endowed with the canonical Cartan nonlinear connection and the canonical metrical Cartan connection. Proposition 2.5.1. The deflection tensor field of the Cartan metrical connection CT(N) satisfies the following equations: (5.2)
..
Remark. If we consider the following Matsumoto's system of axioms A1-A4 and the axiom A5
D{j = 0
we obtain the system of axioms which uniquely determined the Cartan metrical connection CT(N). Miron, Aikou, Hashiguchi proved the following result: The Matsumoto's axioms A1-A5 of the Cartan metrical connection CT{N) are independent. Proposition 2.5.2. connection:
The following properties hold with respect to Cartan metrical 1. * j f c = 0,
F\k = jyk,
2. F , fc = 0, F2\k = 2yk, 2
3.
Vi\k = 0,
y{\k = 9ik-
The Geometry of Hamilton & Lagrange Spaces
44
Proposition 2.5.3. The Ricci identities of the metrical Cartan connection CT(N) are: A ^ ^ - A |A|A. - A r
(5.3)
- A ^ 0 *ft- A |
r k h
where the torsion tensors are: ^ '
Cjk
P 3 k
'
-
and the curvature d-tensors are: I
hj
U* Ilk
.
'h
(5.5)
"a^
dChk
rps
T7>i
Ch\Pjkj
7T£Chjfc|j + 9Chlj
rpi
j-15
n
S
n«
r
»
r
'
a ^ r+ Ch JCSk ~ChkCs >•
Hereafter we denote the metrical Cartan connection DT(N) by CT(N) or by CV. Let us consider the covariant d-tensors of curvature Rijkh = 9jsRiSkh,
Proposition 2.5.4. dentities:
Pijkh = <7;s-Pi''kh, Sijkh = 9jsSiSkh-
The covariant d-tensors of curvature satisfy the following i-
Rijkh + Rjikh = 0,
Pijkh + Pjikh = 0,
Sijkh + Sjikh = 0,
-Rijfc/i + -Rijhfc — 0; Sjjfch + Sijhk = 0.
Indeed, the last two identities are evident. Applying the Ricci identities to the fundamental tensor
P
Po'hk = P\k-, P
:
S
ijk = Cyfc|0>
( vk = 9isP jk)
ajRijk)
{R>jk •=
(ijk)
= o,
9imRmjk)
£V hk = 0,
Ch.2. Finsler spaces
45
where (7 means the cyclic sum in the indices i,j,k.. (ijk)
Indeed, by applying the Ricci identities to the Liouville d-vector field yl and looking at the tensor of deflection Dlj = yly = 0 and d*,- = <5^-, we get the first identity (5.6). For the other identities, we will write the symplectic structure 9 — dptAdxl in the form 9 = 6ptAdxl and write that its exterior differential vanishes, d6 = 0.
2.6
Parallelism. Structure equations
Let CT be a metrical Cartan connection of the Finsler space Fn. The coefficients (L*jk,C'jk) of CT are given by the formula (5.1). As usually, the adapted basis 6 d\ 7-Ti—^ of Cartan nonlinear connection N and vertical connection V and the l
5x dy'J
dual adapted basis (dxl,5y') we can study the notion of parallelism of the vector fields in Finsler geometry. Let 7 : [0,1] —• TM, 11-> j(t) be a parametrized curve of the manifold TM and — be the tangent vector field along with the curve 7. Then, we can write nt ~db = ~dt Jx* + ~dt dy1'
^
As we know 7 is horizontal curve with respect to the nonlinear connection N if dy' Sy* —— = 0. Also, 7 is autoparallel curve of the nonlinear connection N if — = 0, dt dt yl = —— We denote the tangent vector field along with 7 by 7 = — and taking dt dt into account (6.1), we can set for the vector field X along with 7: , , DX DX 7 dt dt DX -^— is called the covariant differential along with the curve 7. dt Setting X = X" + Xv,
(6 3)
-
DX __ DX"
XH = X ' / r , Xv = X ' - ^ r , we get ox* ay1 DXV
* * * A - . „ *•
The Geometry of Hamilton & Lagrange Spaces
46 Let us consider
ul, = Ujkdxk + P^Sy".
(6.4)
where OJXJ are the 1-forms connection of D. Then the equation (6.3) can be written in the form: (6 5)
'
Definition 2.6.1. We say that the vector field X on TM is said to be parallel along with the curve 7 with respect to Cartan mectrical connection CT, if —— = 0. at DX DX By means of (6.3), the equation —— = 0 is equivalent with equations —•— = 0, * * DXV
1T = °-
From the formula (6.5), one obtains the following result:
Theorem 2.6.1.
The vector field X = Xl-r-~ + X1-^— is parallel along with the
parametrized curve 7 with respect to the metrical Cartan connection if and only if its coefficients X'(x,y), X'(x,y) are solutions of the linear system of the differential equations
A theorem of the existence and uniqueness for the parallel vector field along with a given curve on TM can be formulated. A horizontal path of the metrical Cartan connection D on TM is a horizontal parametrized curve 7 with the property Dji = 0. dx{ Using (6.5) for X1 = —— and taking into account the previous theorem, we get: at Theorem 2.6.2. The horizontal paths of Cartan metrical connection in Finsler space Fn are characterized by the system of differential equations , , (6.6)
d2xl ,,• . .dxJ dxk — + Uik(x, y)——
=
dyl Q,JL
+
.„• , .dxj N^(x, y ) - = 0.
If we describe the initial conditions of the previous system, we obtain the existence and uniqueness of the horizontal paths in the Finsler space Fn.
Ch.2. Finsler spaces
47
Now let us consider a curve 7^0 in the fibre ir~1(x0) of TM. It can be represented the equations: x* = xl0, y1 = yl{t), t € / . The curve 7^0 is called a vertical curve in the point x0 6 M. A vertical curve 7^0 is called a vertical path with respect to metrical Cartan connection CT if Dy> iZ0 = 0. Now, applying equation (6.5), we have Theorem 2.6.3. In the Finsler space Fn the vertical paths in the point x0 6 M with respect to metrical Cartan connection CT are characterized by the system of differential equations (17,
^
i
, ^
;
+ C.,(lD,s)f|^=0.
Now, taking into account the theory of structure equations of N—linear connection given in the section of chapter 1, we can apply it to the case of metrical Cartan connection CT. We get the following result: Theorem 2.6.4. The structure equations of the Cartan metrical connection CT(N) are given by (o). d{dxl) - dxmAu1m = - Q\ (6 8) dw'j - umjAujim
-
-fi'j,
(o). ( i ) .
where the 2-forms of torsion fl\ Ql and 2-form of curvature Q,lj are as follows: C*jkdx3t\5yh,
fl' = (6.9)
(i) i
l
O = -RljhdxJAdxh
&] = \RhmdxhAdxm
+ Pljhdx3A6yh, + Pj\mdxhA5ym
+
\sj\m5yhA5ym.
Now, the Bianchi identities of the metrical Cartan connection CT(N) can be obtained from the system of exterior equations (6.8) by calculating the exterior differential of (6.8), modulo of the same system (6.8) and using the exterior differential (o). ( i ) .
of 2-forms 0 ' , Ql and of 2-form of curvature Q.lj. We obtain:
The Geometry of Hamilton & Lagrange Spaces
48
Theorem 2.6.5. The Bianchi identities of the Cartan metrical connection CT(N) of Finsler spaces Fn are as follows:
(6.10)
(6 11)
M {ck% + c}\p\t - p/w> = o, (jk)
M {#jhCkht + PjhPKt + P{keij} = Bjjk - R\
(6.12)
M {RsljhCkhe + PS^u
+ P'skm = ~Ss\hRh
jk
(6-13)
M {PJKjCh - SjjhP\k where J\A means the interchange of the indices j , k and subtraction and O" means {jk)
UM)
the cyclic permutation of indices j,k,l and summation. Remark. The structure equations given in Theorem 2.6.4 are extremely useful in the theory of submanifolds of the Finsler manifold Fn.
2.7
Remarkable connections of Finsler spaces
Let us consider an N-linear connection D with the coefficients DF(N) = (L'jk, Cljk). To these coefficients we add the coefficients N'j of the nonlinear connection and we write D with the coefficients DV(Nlj,L'jk,Cljk). For metrical Cartan connection CV(N) we have the coefficients Cr(A^iJ, I}jk, C1^) given by the formulae (4.5)' and (5.1). To the metrical Cartan connection CT(Nlj,Lljk, C'jk) we associate the following N-linear connections:
Ch.2. Finsler spaces
49 dN' •
1° Berwald connection BY [ 7V%, - ^ . 0 I. 2° Chern-Rund connection RT (N^, Ujk, 0). BN* •
These remarkable connections satisfy a commutative diagram:
fir HI'
obtained by means of connection transformations [113]. The properties of metrizability of those connections can be expressed by the following table: v — metrical CT(N) h - metrical = BV(N) 9 W ~2Cyi|0 9
Rr{N) Hr(N)
«l* h - metrical 9 w = — 2Cijjfc|o a\k
ij\k
9 M - 2Cijk ij\k
v — metrical
Remark. It is shown that the Chern connection (introduced in [25], [42]) can be identified with the Rund connection (cf. M. Anastasiei [7]).
2.8
Special Finsler manifolds
Berwald space is a class of Finsler spaces with geometrical properties similar to those Riemann spaces. Based on the holonomy group of Berwald connection Z. Szabo made a first classification of Berwald spaces. Other important classes of special Finsler spaces are Landsberg spaces and locally Minkowski Finsler spaces. In this section we briefly describe some of the main properties of these spaces.
The Geometry of Hamilton & Lagrange Spaces
50
Definition 2.8.1. A Finsler space is called Berwald space if the connection coefficients G'ik = i.e.
, , dyk
of the Berwald connectionBV are function of position alone, dG'jk _ Q
We denote by Bn = (M, F(x, y)) a Berwald space. The space Bn can be characterized by the following tensor equation. Theorem 2.8.1. A Finsler space is Berwald space if and only if 0 ^
(8.1)
= 0.
Proof. It is not difficult to prove that the Cartan connection CV = (N'j, and Berwald connection BT - {G{j, are related by the formulae: {
' '
G\ = N^ = G'j0.
From here we obtain:
where
From the second expression of (5.5), we have
F*V = Phljfc + CJw -
(8.3)
It follows that the condition
C\rPrjk.
, J.h = 0 is equivalent to dyk
Eliminating the term Fhljk from (8.3) and (8.4), we find dP>
where we used geh
Qph .. ay
iJ
Qp . = -^~ oy
2CtkhPhij.
L'jk,
Ch.2. Finsler spaces
51
Permutating now the indices (ilj) in (8.5) and taking into account the identity 0". {Phkji} = 0, one obtain:
(hkj)
Contracting this by yl, we obtain (8.6)
3-^-f yl + CjiklQ = -3P ifcj + C ^ o = -2C jifc|0 = 0,
which means Pikj = 0. So the equation (8.5) reduces to (8.7)
Pm + Cak\i = 0.
On the other hand, in general, the hv-curvature tensor Phkji of CT can be written as, [88]: (8.8)
Phkji + Ckij\h ~ Chij\k + ChjTPTki ~ CkjrP*hi-
Taking into account of Pikj — 0, from (8.8) we obtain: (8.9)
Phkji = Ckij\h —
Hence, from (8.7), (8.9) we have Cijk\e = 0. Conversely, from Ciik\i = 0 we obtain P{jk = 0, Phkij = 0. Therefore, (8.3) gives F^jk = 0, and then Gh{ik = 0, i.e. &hi = G\j{x). q.e.d. Corollary 2.8.1. A Finsler space is Berwald space if and only if the hv-curvature ••= o r ) of BT vanishes identically. Another important class of Finsler space is given by the Landsberg spaces [88]: Definition 2.8.2. A Finsler space is called Landsberg space if its Berwald connection BV is h-metrical, i.e. (8-10)
g w = -2C ijfc | 0 = 0, ij | k
where the index 0 means contraction by y'. Theorem 2.8.2. A Finsler space is Landsberg space if and only if the hv-curvature tensor Phijk of Cartan connection CT(N) vanishes identically.
The Geometry of Hamilton & Lagrange Spaces
52
Proof. From Proposition 2.5.6, a Landsberg space is characterized by Pijk = 0. In general, from the equation P ^ = Ci3k\a we obtain P
ijk\h ~
P
hjk\i
=
\Cijk\r\h ~ '•} I
C
hjk\r\i) yT
v
'
Ci
M(
~
C
hjk\l
J I'/ | r
w h e r e w e u s e d t h e R i c c i i d e n t i t y ( 5 . 3 ) a n d C..,>, J
+
ijk\h
- C, , i . = 0 . h]k\i
Taking into account equation (8.8) and the above equation we obtain:
From (8.11), if P^k — 0 hold good, then we obtain Phijk = 0. Conversely, from the relation Po\k = P\k (cf. (5.6)), we can conclude the assertion. q.e.d. The following result is now immediate. Corollary 2.8.2. If a Finsler space is Berwald space, then it is a Landsberg space. The locally Minkowski Finsler space are introduced by the following definition. Definition 2.8.3. A Finsler space F" = (M, F(x, y)) is called locally Minkowskiif in every point x € M there is a coordinate system (x\ U) such that on 7r~1(C7) C TM its fundamental function F(x,y) depends only on directional variable y\ Such a coordinate system {x',U) is called adapted to a locally Minkowski space. Theorem 2.8.3. A Finsler space is locally Minkowski if and only if the covariant tensor of curvature Rjjhk of the Cartan connection CT vanishes and the tensorial equation Cxjk\h = 0 holds. Proof. In an adapted coordinate system, first two coefficients of the Cartan connection CT are given by Nlj = 0, U^ — 0. Hence from the definitions of a Rfi'jk, taking into account of (5.4), (5.5), respectively, we obtain Rhijk — 0. Next, we see easily that under the adapted coordinate system _ dChi] _ 1 8 (dghi ^~ dx" ~ 2 dy
Ch k
Conversely, if Rhijk = ® and Cyfc|* = 0, from (5.6) and (8.8) we obtain Pijk = Phijk = QHence from (8.8) we obtain Fhljk — :— }k J = 0. Hence, L\j = L\j(x). v y /
So,
Ch.2. Finsler spaces
53
the first equation of (5.5) reduces to
This implies Riemannian flatness. Hence there exists a coordinate system (xa) such thatlfe = 0, from which 7v£ = 0. Consequently, from the axiom of h-metrizability, we obtain dgab dgab jjd _ Td T T T •^37 = -g^J Nc + Labc + l~>bac = U, Labc •= 9bd^a^
from which we obtain gab — gab{y)q.e.d. Since we have an example given by Antonelli of a locally Minkowski space [11], that means: On the paracompact manifolds there exists locally Minkowski Finsler spaces. So, we obtain the following sequences of inclusions of special Finsler spaces: CD BD MD Mf)1Z, where C is the class of Landsberg spaces, B the class of the Berwald spaces, M the class of the locally Minkowski spaces, and 11 the class of Riemannian spaces. We remark that M n "R is the class of flat Riemannian spaces. In 1978, Y. Ichijyo has shown the geometrical meaning of the vanishing hvcurvature tensor using the holonomy mapping as follows: Theorem 2.8.4. (Ichijyo) Let us assume that (M, F) is a connected Finsler space with the Cartan connection CT. Let p and q be two arbitrary points of M, and let t be any piecewise differentiable curve joining p and q. In order that the holonomy mapping from TPM to TqM along with I, with respect to the nonlinear connection N, be always a C-affine mapping, it is necessary and sufficient that the hv -curvature tensor Ph'jk vanishes identically. It still remain an open problem: If there exists Landsberg space with vanishing hv-curvature tensor. In [68], Y. Ichijyo introduced the following interesting fundamental function (8.12)
F(i,dx) = F{aa), aa = af{x)dx\
where F is a fundamental Finsler function, the function is 1-positively homogeneous in aa, (a = 1,2, ...,n) and af(x)dx' are linearly independent differentiable 1-forms. Definition 2.8.4. The Finsler metric (8.12) is called 1-form Finsler metric and the space (M, F) is called 1-form Finsler space. There are some special 1-form metrics:
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1. Berwald-Móormetric: F = (y1y2-..yn)" which is atypical Minkowski metric in a local coordinat system. G.S.Asanov introduced a more general form: F = (a1a2...a")», where a a (a = 1,2, ...,n) are linearly independent 1-forms. 2. m-th root metric: F = {(al)m + (a2)"1 H h (a")m}m which was studied by Shimada [152] and Antonelli and Shimada [19] for the case n = 2. 3. A special Randers metric: F = {(a1)2 4- (a2)2 -f h (a") 2 }* + fea1, wherefc is a constant. This example was given by Y. Ichijyo. Using the Cartan connection, Matsumoto and Shimada proved the following result for a 2-dimensional 1-form Finsler space [91]: Theorem 2.8.5. If a 2-dimensional 1-form Finsler space is a Landsberg space, then it is a Berwald space. In order to introduce the notion of Douglas space, let us observe first that the geodesies of a Finsler space Fn can be written in the form: x'x* - xixi + 2Dij {x, x) = 0,
(8.13) where Dij(x,x) := G{xj -
Gji\
Definition 2.8.5. A Finsler space Fn is called Douglas space if the functions Dij(x,x) are homogeneous polynomials in se of degree three. This is equivalent to the fact that the Douglas tensors of Fn vanishes. M. Matsumoto and S. Bacso [23] proved: Theorem 2.8.6. If a Landsberg space is a Douglas space, then it is a Berwald space. Theorem 2.8.7. A Randers space Fn = (M, F - a + ft) is a Douglas space if and only if the differential 1-form is a closed form. We will describe in the sequel the Finsler spaces with constant curvature. Definition 2.8.6. The quantity K(x,y, X) given by K(x,y,X)
=
"""V
:*
,vk> VX G TXM, X ? 0, V(i, V ) € TM,
is called the scalar curvature at (x, y) with respect to X, where Hhljk is the h-curvature tensor of the Berwald connection BY [88].
Ch.2. Finsler spaces
55
One can remark that the h-curvature tensor of the Berwald connection BV is ([88][p.ll8]) Hhjk = Khjk + -M {Cj\\Q\k + Ch\\oCjT/t\o}U*]
From Definition 2.8.6 it can be said that the scalar curvature K(x,y,X) is defined as the sectional curvature of a 2-section spanned by y and X, with
Riok =
KF2hlk,
where Rijk is the torsion tensor of the Cartan connection CT. The left hand of (8.14) is also called flag curvature:
The flag curvature is one of the important numerical invariants because it lies in the second variation formula of arc length and takes the place of sectional curvature from the Riemannian case. In 1975 the following interesting result concerning scalar curvature was obtained: Theorem 2.8.9. (Numata [129]) Let Fn, n > 3, be a Berwald space of scalar curvature K. The n is a Riemannian space of constant curvature or a locally Minkowski space, according K ^ 0 or K = 0, respectively. Lastly in this chapter we remark that Finsler spaces with – metric were studied in the paper [89]. And using the invariants of a Finsler space, was made a classification of some Finsler spaces with (a, -metric, namely Randers class, Kropina class, Matsumoto class, etc. The classes are providing new concrete examples of (a, -metrics.
2.9
Almost Kählerian model of a Finsler manifold
A Finsler space Fn — (M, F) can be thought as an almost Kähler space on the manifold TM = T M \ { 0 } , called the geometrical model of the Finsler space Fn.
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In this section we present the Sasaki-Matsumoto lift of the metric tensor <7y of the space Fn and a new lift, given by R. Miron [109], which is homogeneous and allows us to study problems concerning the global properties of the Finsler spaces. In this way the theory of Finsler spaces gets more geometrical consistence. If we consider the Cartan nonlinear connection N'j of the Finsler space Fn = = (M, F), then we can define an almost complex structure IF on TM by:
It is easy to see that IF is well defined on TM, F 2 = —/ and it is determined only by the fundamental function F of the Finsler space Fn. Theorem 2.9.1. The almost complex structure IF is integrable if and only if the h-coefficients Rljk of the torsion of CT vanishes. Let (dx\Syt)
be the dual basis of the adapted basis |-^^>-^-^). Then, the
\Sxl dy'J
Sasaki-Matsumoto lift of the fundamental tensor ^ can be introduced as follows: (9.2) G = gijdxi^dxj + gijSyi<S>Syj. Consequently, G is a Riemannian metric on TM determined only by the fundamental function F of the Finsler space Fn and the horizontal and vertical distributions are orthogonal with respect to it. The following results can be proved without difficulties, [113]:
Theorem2.9.2.
(i) The pair (G, IF) is an almost Hermitian structure on (ii) The almost symplectic 2-form associated to the almost Hermitian structure (G, IF) is
(9.3)
0=
gij(z,y)Syi®dxj.
(iii) The space H2n — (TM;G,W) is an almost Kählerian space, constructed only by means of the fundamental function F of the Finsler space Fn. The space H2n = (TM; G, F) is called the almost Kählerian model of the Finsler space Fn. Theorem2.9.3. The N-linear connection D with the coefficients CT(N) = (F^.C^-jt) of the Cartan connection is an almost Kählerian connection, i. e.: (9.4)
DXG = 0, DXW = 0, VX 6 X(TM).
Ch.2. Finsler spaces
57
Hence, the geometry of the almost Kählerian model H2n can be studied by means of Cartan connection of the Finsler space F or instance, the Einstein equations n i are given by the Einstein equations in the previous model H2n. One can find it in [113]. Remarking that the Sasaki-Matsumoto lift (9.2) is not homogeneous with respect to y\ a homogeneous lift to TM of the fundamental tensor field gtj (x, y) was introduced by R. Miron in the paper [109]. We describe here this new lift for its theoretical and applicative interest. Definition 2.9.1. We call the following tensor field on TM: 0
.
G (x,y) = 9ij(x,y)dxl®dxi
(9.5)
a2
+ —^ \\y\\
l 9ij(x,y)5y ®6y>,
V(a:,y) € TM,
the homogeneous lift to TM of the fundamental tensor field &j of a Finsler space Fn, where a > 0 is a constant, imposed by applications (in order to preserve the physical dimensions of the components of norm of the Liouville vector field:
\\y\\2 = gij(x, y ) v V = yiy* - F*(x, y),
(9-6) with
) and where \\y\\2 is the square of the
ft
= g^
= -
j p
We obtain, without difficulties: Theorem 2.9.4. o 1° The pair (TM, G) is a Riemannian space. o 2° G is 0-homogeneous on the fibers ofTM. o 3° G depends only on the fundamental function F (x, y) of the Finsler space Fn. o 4° The distributions N and V are orthogonal with respect to G. o We shall write G in the form (9.7)
where (9-8)
h
n = TT2 \\y\\
Gv =
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Consequently, we can apply the theory of the (h, v)-Riemannian metric on TM investigated by R. Miron and M. Anastasiei in the book [113]. The equation F(x0, y) = a determines the so called indicatrix of the Finsler space Fn in the point XQ € M. Therefore, we have: o Theorem 2.9.5. The homogeneous lift G of the metric tensor g{j (x, y) coincides with the Sasaki-Matsumoto lift of 9i3{x,y) on the indicatrix = a, for every point XQ E M. A linear connection D on TM is called a metrical N-connection, with respect o o to G, if D G— 0 and D preserves by parallelism the horizontal distribution N. As we know,[113] there exist the metrical N-connection on TM. We represent a linear connection D, in the adapted basis, in the following form: ,
. '
S H. S ~ d m 6x1 Sxl dy' H 6 i 5 ~t d D-^-n ~ ^"j*T7l + Cl3k -3-7'1 a^ oxi 5x dy
I'^U^C1^
where (^1% Theorem 2.9.6. 0
d «• S v. d J J^ dyi Sxl oy* d « ; <5 v. d Da-—- 3 = Cljk-^—. + Cljic% -—-7 1 5^ oy ox% oy
C^^fr^C1^) are the coefficients of D.
There exist the metrical N-connections D on TM, with respect
to G, which depend only on the fundamental function F(x, y) of the Finsler space Fn. One of them has the following coefficients L'jk = L'jfc = C*jfc = C'jk = 0, H
(9.10)
i
_ pi
LTjk-Fjk-~g
_
l
nis
(dhsk
(69sk
^—
+
dh]S
&9js
—
dhj
W
where CT(N) = (F'jk, C'jk) is the Cartan connection of the Finsler space Fn, BV(N) — (B'jk,Q) is the Berwald connection and \\ means the h-covariant derivation with respect to BT(N). Of course, the structure equations of the previous connection can be written as in the books [112], [113]. In order to study the Riemannian space (TM, G) it is important to express the coefficients H.
~.
KS.
V.
H.
~ .
RJ
.
{Lljk, Lljk, Vjk, Lljk, C%jk, Cljk, C'jk,
V.
Cljk).
Ch.2. Finsler spaces
59
To this aim, expressing in the adapted basis the conditions: X G (Y, Z)- G (DXY, Z)- G (K, Z?XZ) = 0,
- [X, Y] = 0, V Jf, r , Z e X{TM) and using the torsions Rljk and Cljk of the Cartan connection CT(N), we get by a direct calculus: o Theorem 2.9.7. The Levi-Civita connection of the Riemannian metric G have in an adapted basis the following coefficients Ll rii
k
= - Qim (^0
\ Krrk \
£J
jk
H.
~
AT'
U Jb
UJb
— I uim f9hmj ~ 2 \ dyk «. .
V. •^ jk
+ ^ ^ - ^^1 , AT^/
U Jb
/
dhmk ^ dhjk dyi dym l .
1 — ~
^skj
~
1 X-t
jfci
The structure equations of the Levi-Civita connection (9.11) can be written in the usual way. Let us prove that the almost complex structure IF, defined by (9.1) does not preserve the property of homogeneity of the vector fields. Indeed, it applies the 1-homogeneous vector fields -—. (i =
onto the 0-homogeneous vector fields
d — ,(i =
l,...,n).
We can eliminate this incovenient by defining a new kind of almost complex structure F : X(TM)
(9.12)
-> X(TM),
* ( ' ) = - M \5x*)
a
setting:
» * ( « ) . • « ( < _ !
dyl
\dy>J
\\y\\ 5x*
„).
Taking into account that the norm of the Liouville vector field ||y|| and the Cartan nonlinear connection N are defined on TM, it is not difficult to prove:
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60
Theorem 2.9.8. The following properties hold: 1° F is a tensor field of type (1.1) on TM. o o 2° F o F = -I. o 3° F depends only on the fundamental function F of the Finsler space Fn. 4° The T{TM) -linear mapping W: X(TM) -^_X(TM) homogeneity of the vector fields form X{TM).
preserves the property of
o It is important to know when F is a complex structure. o __Theorem 2.9.9. F is a complex structure on TM if and only if the Finsler space Fn has the following property:
(9.13)
Jf y = 1
Proof. The Nijenhuis tensor Mo :
No (X, Y) =W2[X, Y\ + [W X,WY]-W
[F X, Y}~ F [X, F Y], X,Y€
X(TM),
vanishes if and only if the previous equations hold. Remark. If F " is a Riemann space, the equation (9.13) is a necessary and sufficient condition that it to be of constant sectional curvature. o o The pair (G, F ) has remarkable properties:
Theorem 2.9.10. We have oo
1° ( G , F ) is an almost Hermitian structure on and depend only on the fundamental function F of the Finsler space Fn. o 2° The associated almost symplectic structure 9 has the expression
(914)
l
'U6-
where 6 is the symplectic structure (9.3). 3° The following formula holds:
Ch.2. Finsler spaces
61
o o 4° Consequently, (G, F ) is a conformal almost Kählerian structure and we have o o d0=Q (modulo 9). Remarks. o o 1° The previous theorem shows that (G, F ) is a special almost Hermitian structure. 2° There exist the linear connection compatible with the conformal almost Käo o hlerian structure ( G , F ) .
oo
The conformal almost Kählerian space (TM, G, F ) is another geometrical model o of the Finsler space Fn. It is based on the homogeneous lift G, (9.5). The previous considerations are important for study the Finslerian gauge theory, [22], [177], and in general in the Geometry of Finsler space Fn. The importance of such kind of lift wasemphasized by G.S. Asanov [22]. Namely he proved that some (h,v) metrics on TM satisfy the principle of the Post Newtonian calculus. The o metric G belongs to this category, while Sasaki-Matsumoto lift has not this proper-
ty.
The theory of subspaces of Finsler spaces can be found in the books [112], [113].
Chapter 3 Lagrange spaces The notion of Lagrange space was introduced and studied by J. Kern [76] and R. Miron [96]. It was widely developed by the first author of the present monograph [106], Since this notion includes that of Finsler space it is expected that the geometry of these spaces to be more rich and applications in Mechanics or Physics to be more important. We will develop the geometry of Lagrange spaces, using the fundamental notions from Analytical Mechanics as: integral of action, Euler-Lagrange equations, the law of conservation of energy, Noether symmetries, etc. Remarking that the Euler-Lagrange equations determine the canonical spray of the space, we can construct all geometry of Lagrange space by means of its canonical spray, following the methods given in the Chapter 1. So the geometry of Lagrange space is a direct and natural extension of the geometry of Finsler space. At the end of this chapter, we emphasize the notion of generalized Lagrange spaces, useful in the geometrical models for the Relativistic Optics.
3.1
The notion of Lagrange space
At the beginingwe define the notions of differentiable Lagrangian using the manifolds TM and TM, where M is a differentiable real manifold of dimension n. Definition 3.1.1. A differentiable Lagrangian is a mapping L : (x,y) € TM -> L(x, y) € R, of class C00 on manifold TM an d continuous on the null section 0 : M ->• TM of the projection n : TM -> M. The Hessian of a differentiable Lagrangian L, with respect to y\ has the elements:
63
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64
Exactly as in the Finsler case we prove that gij{x, y) is a d-tensor field, covariant of order 2, symmetric. Definition 3.1.2. A differentiable Lagrangian L(x, y) is called regular if the following condition holds: rankHftjOr,?/)!! = n, on TM.
(1.2)
Now we can give: Definition 3.1.3. A Lagrange space is a pair Ln = (M, L(x, y)) formed by a smooth real n-dimensional manifold M and a regular Lagrangian L(x, y) for which d-tensor g^ has a constant signature over the manifold TM. For the Lagrange space Ln = (M, L(x, y)) we say that L(x,y) is fundamental function and gij(x,y) is fundamental (or metric) tensor. We will denote, as usually, by gij the contravariant of the tensor g y .
Examples. 1° The following Lagrangian from electrodynamics [112], [113] (1.3)
L(x, y) = mnij(x)yy
+ ^ A^y*
+ U(x)
where 7y (x) is a pseudo-Riemannian metric, Ai (x) a covector field and U(x) a smooth function, m, c, e being the known constants from Physics, determine a Lagrange space Ln. More general: 2° The Lagrangian (1.4)
L(x, y) = F2(x, y) + A W
+ U(x)
where F(x,y) is the fundamental function of a Finsler space Fn = (M,F(x,y)), Ai(x) is a covector field and U(x) a smooth function gives rise to a remarkable Lagrange space, called the Almost Finsler-Lagrange space (shortly AFL-space). In particular, (-Aj(x) = 0, U(x) = 0) the pair L" = (M, F2(x, y)) is a Lagrange space. In other words, Proposition 3.1.1. Any Finsler space Fn — {M,F(x,y)) is a Lagrange space Ln = 2 n = (M,F ). Conversely, any Lagrange space L — (M, L(x,y)) for which fundamen1 d2L(x y) tal function L(x, y) is positive and 2-homogeneous with respect to y% and ' 2 is positive definite determine a Finsler space Fn — (M, JL(x,y)).
Ch.3. Lagrange spaces
65
The previous example proves that: Theorem 3.1.1. If the base manifold M is paracompat then there exist regular Lagrangians L(x, y) such that the pair Ln = (M, L(x, y)) is a Lagrange space.
3.2
Variational problem Euler-Lagrange tions
equa-
The variational problem can be formulated for differentiable Lagrangians and can be solved in the case when we consider the parametrized curves, even if the integral of action depends on the parametrization of the considered curve. Let L : TM -> R be a differentiable Lagrangian and c : t e [0,1] -> (z*(t)) 6 U C M a curve (with a fixed parametrization) having the image in the domain of a chart U on the manifold M. The curve c can be extended to ir~l(U) C TM as [0,1] —> (*'(*), ^
(*)) € TT 1
Since the vector field — (t), t e [0,1], vanishes nowhere, the image of the mapping c* belongs to TM. The integral of action of the Lagrangian L on the curve c is given by the functional (2.1)
/(«)
Consider the curves (2.2)
ce:te
[0,1] —> (i''(t) + eV ! (t)) € M
which have the same end points ^(0), xx{x) as the curve c, Vl(^) = Vl(x{t)) being a regular vector field on the curve c, with the property Vl(0) = V l (l) = 0 and e a real number, sufficiently small in absolute value, so that Im ce c U. The extension of curves ce to TM is given by
c; : t e [0,1] —> ^{t)+eVi(t),~+e^pj
e TT" 1 ^).
The integral of action of the Lagrangian L on the curve ce is given by (2.2)-
/(*)
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66
A necessary condition for I(c) to be an extremal value of I(ce) is (2.3)
de
= 0. £=0
Under our condition of differentiability, the operator — is permuting with operator fe of integration. From (2.2)' we obtain
(2 4)
-¥
-
A straightforward calculus leads to:
dv e=0
\ dxl
dt dyi J
dL dVi ~dy{ ~~dt
_ 8L_ "~ dx*
dt \ dyl
/'
y
dt
Substituting in (2.4) and taking into account the fact that V'(x(t)) is arbitrary, we obtain the following. Theorem 3.2.1. In order that the functional I(c) be an extremal value o is necessary that c be the solution of the Euler-Lagrange equations:
f it
E
>{L):=d*-Itdy->=0'y=lif
(2 5)
"
Some important properties of the Euler-Lagrange equations can be done. Introducing the notion of energy of the Lagrangian L(x, y), by: E
^-y%-L
(2.6)
we can prove the so called theorem of conservation of energy: Theorem 3.2.2.
The energy Ei of the Lagrangian L is conserved along to every dx' integral curve c of the Euler-Lagrange equation EAL) = 0, yl — —— at
dx{ Indeed, along to the integral curve of the equations EAL) — 0, —— at have: dt
L
~ dt dyl+V dt Vdy'J
L__±d *
V
dt d
dx'
dt dy* ~
we
Ch.3. Lagrange spaces
67
Consequently, we have —— = 0. at
q.e.d.
Remark. A theorem of Noether type for the Lagrangian L(x, y) can be found in the book [106].
3.3
Canonical semispray. Nonlinear connection
A Lagrange space Ln = (M, L(x,y)) determines an important nonlinear connection which depends only on the fundamental function L(x,y). Remarking that E{(L), from (2.5) is a d-covector and that the fundamental tensor of the space, #,_,, is nondegenerate we can establish: Theorem 3.3.1. If Ln = (M, L) is a Lagrange space then the system of differential equations 9ljE3(L) = 0, y{ =
(3-1)
^
can be written in the form: dx\ where 2Gi(x,y)=1-9v Indeed, the formula dL
d dL
dL
f d2L
h
dyh)
dxi
holds. Hence (3.1) is equivalent to (3.1)', 6" being expressed in (3.2). Theorem 3.3.2. semi-spray
(3-3)
The differential equation (3.1)' gives the integral curves of the
S=
where Gl(x,y) are expressed in (3.2).
tf*A-2C(x>tf)A,
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68
Indeed, the differential equations (3.1)' do have a geometrical meaning and therefore it follows that Gl are the coefficients of a semispray S from (3.3). The integral curves of S are given by
hence the differential equations (3.1)' are satisfied.
q.e.d.
The previous semispray is determined only by the fundamental function L(x, y) of the space Ln. It will be called canonical semispray of Lagrange space Ln. Corollary 3.3.1. The integral curves of the Euler-Lagrange equations E{{L) = 0, dxi —— = y' are the integral curves of the canonical semispray S from (3.3). Indeed, we can apply Theorems 3.3.1 and 3.3.2 to get the announced property. As we know, (Ch.1), a semispray S determines a nonlinear connection. Applying Theorem 1.4.3, we obtain: Theorem 3.3.3. In a Lagrange space Ln = (M, L) there exists the nonlinear connections which depend on the fundamental function L. One of them has the coefficients
JV N J
[
~ dy>• " 4 W \9
a
(
\dy*dxh
v
V
dx*
Proposition 3.3.1. The nonlinear connection N with coefficients Nlj (3.4) is invariant with respect to the Carathéodory transformations (3-5)
L'(x>y) = L(x,y)
^
where y(x) is an arbitrary smooth function. Indeed, we have Ei(L'(x,y)) = Ei (L(X,V) + ^ \ =Ei(L(x,y)).
So, Ei(L'(x,y)) = Q
determine the same canonical spray with Ei(L(x,y)) = 0. Thus, the previous theorem shows that the Carathéodory transformation (3.5) does not change the nonlinear connection N. Because the coefficients of N are expressed by means of the fundamental function L, we say that N is a canonical nonlinear connection of the Lagrange space Ln.
Ch.3. Lagrange spaces
69
Example. The Lagrange space of electrodynamics, Ln = (M,L(x,y)), L(x,y) being given by (1.3), with U(x) — 0, has the coefficients Gl(x,y) of the canonical semispray S, of the form: G\x, y) = ^'i* W t f * - 9liFik{x)yk,
(3-6)
where Yjk(x) are the Christoffel symbols of the metric tensor gij(x) = meyij(x) of the space Ln and Fjk is the electromagnetic tensor
Therefore, the integral curves of the Euler-Lagrange equation are given by solution curves of the Lorentz equations: dx
/o o \ (3 8)
'
^
+
i
. .dxj {x)
dxk
1* -a7 IT
^
The canonical nonlinear connection of L" has the coefficients (3.4) of the form (3.9)
rfjix,
y) = Yjk(xhk ~ 9ik(x)Fkj{x).
It is remarkable that the coefficients Nlj of the canonical nonlinear connection N of the Lagrange spaces of electrodynamics are linear with respect to y\ This fact has some consequences: 1° The Berwald connection of the space
, has the coefficients Yjki^)-
2° The solution curves of the Euler-Lagrange equation and the autoparallel curves of the canonical nonlinear connection N are given by the Lorentz equation (3.8). In the end part of this section, we underline the following theorem: Theorem 3.3.4. The autoparallel curves of the canonical nonlinear connection N are given by the following system of differential equations:
where N'j is expressed in (3.4). This results from Section 3, Ch.1.
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70
3.4
Hamilton—Jacobi equations
Let us consider a Lagrange space Ln — (M,L(x,y)) and N(Nlj) its canonical nonlinear connection. The adapted basis
f6
d\
-r—,—— to the horizontal distribution N
and the vertical distribution V has the vector fields -r— : 6x* — =— -
(4 1)
Wt—-
Its dual is (dx',5y'), with
8yl=dyi+Nijdxj.
(4.1)'
The momenta pt of the space Ln can be defined by
Thus Pi is a d-covector field. We can consider the following forms (4.3) u = pidxi, e
(4.4)
Proposition 3.4.1. The forms w and 6 are globally defined on TM, and we have 0 = du.
(4.5)
Proof. 1° Indeed, p,dx^, and 8 from (4.4) do not depend on the transformation of local coordinates on TM. 2° duj = dPlAdx' = -d^Adx 2 dyl
(
0
= ( | ^ 2 \6xm dyl
+
^ dymdyl
( ) Because of a direct calculus shows that 5 dL 5 dL _ Q <52;m 9?/* dxi dym Theorem 3.4.1. The 2-form 9, from (4.4), determines on TM a symplectic structure, which depends only on the fundamental function L(x,y) of the space Ln.
Ch.3. Lagrange spaces
71
Proof. Using theprevious proposition results that 0 is integrable, i.e. dO = 0, and
rank||0|| = dimTM.
q.e.d.
Corollary 3.4.1. The triple (TM, 8, L) is a Lagrangian system. The energy EL of the space Ln is given by (2.6), Ch.3. Denoting % -
-EL,
£ = - £, we get from (2.6), Ch.3: H=Piyl-C(x,y).
(4.6)
But, along the integral curve of the Euler-Lagrange equations we have d7i dC dx{ ~~ dx{ ~
d dC dpi dt dy{ ~ ~dt
And from (4.6), we have also dU _ i __ dxi So, we obtain: Theorem 3.4.2. Along to the integral curves of Euler-Lagrange equations we have the Hamilton-Jacobi equations
dtf__ an djH_ _mt_ dt
dpi dt
ox*
Corollary 3.4.2. The energy EL is conserved along to every integral curve of the Hamilton-Jacobi equations.
3.5
The structures P and F of the Lagrange space Ln
The canonical nonlinear connection N determines some global structures on the manifold TM. One of them is the almost product structure P . It is given by the difference of the projectors h and v (5.1)
W = h-v.
It follows (5.2)
P 2- /.
The Geometry of Hamilton & Lagrange Spaces
72
Theorem 3.5.1. The canonical nonlinear connection N of the Lagrange spaceLn = — (M,L(x,y)) determines an almost product structures P , which depends only on the Lagrangian L(x, y). The eigensubspace of P corresponding to the eigenvalue -1 is the vertical distribution V and eigensubspace of P corresponding to the eigenvalue +1 is the horizontal distribution N. Conform to a general result, we get from Ch.1. Theorem 3.5.2. The almost product structure P determined by the canonical nonlinear connection N is integrable if and only if the horizontal distribution N is integrable. This condition is expressed by the equations Rik{x,y) = Q<mfM,
(5.3) where
Rijk=^nk Sx b~xi Proof. It is not difficult to see that the Nijenhuis tensor of the structure P vanishes if and only if [XH,YH}H = 0. Taking XH,YH in the adapted basis we have that (5.4)
—,_
= RhijTr-ji = 0 hold iff i?Ay = 0. But Rhij is a J-tensor field, so the
condition Rhtj = 0 is verified on the manifold TM.
q.e.d.
On the manifold TM there exists another important structure defined by the canonical nonlinear connection N : It is the almost complex structure, given by the T(TM) -linear mapping F : X{TM) —» X{TM)
or by tensor field (5.5)'
W
= -^®dxi ay'
+ ^ox%
It follows: Theorem 3.5.3. We have: 1° IF is an almost complex structure globally defined on the manifold T M. 2° F is determined only by the fundamental function L(x, y) of the Lagrange space Ln.
Ch.3. Lagrange spaces
73
Indeed, 1° The form (5.5)' of F shows that the F is a tensor field defined on TM. From (5.5) we prove that F is an almost complex structure. This is that we have (5.6)
FoF=-/.
This can be proved by means of (5.5), using the adapted basis
(6 d\ -r~,—-^ . \ox' ay1)
2° Clearly, F is determined only by the canonical nonlinear connection, which depends only on L(x, y). q.e.d. Theorem 3.5.4. The almost complex structure F is a complex structure (i.e. F is integrable) if and only if the canonical nonlinear connection N is integrable. Indeed, the Nijenhuis tensor A/F vanishes if and only if 1° the distribution N is integrable, i.e., Rljk = 0;
2° the J-tensor tjk
3Nl 8N\ = -—± k - —4 = 0. dy
dyi
But N{j = — • So, it follows fjk = 0.
3.6
q.e.d.
The almost Kählerian model of the space Ln
Following the construction of the almost Kählerian model from geometry of Finsler space, we extend for Lagrange spaces the almost Hermitian structure (G,F) determined by the lift of Sasaki type G of the fundamental tensor field „• and by the almost complex structure F . The metric tensor gij(x,y) of the space Ln = (M,L(x,y)) and its canonical nonlinear connection N(Nj) allows to introduce a pseudo-Riemannian structure G on the manifold TM, given by the following lift of Sasaki type: (6.1)
G(i, y) = gtj{x, y)dx%dxi + gij{x, y)6y'®6yj
We have: Theorem 3.6.1. 1° G is a pseudo-Riemannian structure on the manifold TM determined only by the fundamental function L(x, y). 2° The distributions N and V are orthogonal with respect to .
The Geometry of Hamilton & Lagrange Spaces
74
Indeed, 1° The tensorial character of #„, dx1 and Syl shows that G does not depend on the transformations of local coordinates on TM.
q.e.d. For the pair (G, F) we get: Theorem 3.6.2. 1° The pair (G, F) is an almost Hermitian structure on TM, determined only by the fundamental function L(x, y). 2° The almost symplectic structure associated to the pair (G, F) is given by (4.4), i.e. e
(6.2)
3° The space (TM, G, F) is almost Kählerian. Proof. 1° is evident. G from (6.1) and from(5.5) depend only on L(x,y) and we have, G(WX, FY) = G(X, Y), VX, Y e X(TM) hold. 2° Calculating in the adapted basis 0(X,Y) = G(WX,Y) we obtain (6.2). 3° Taking into account Theorem 3.4.1, it follows that 9 is a symplectic structure. q.e.d. The space K2n = (TM,G,W) is called almost Kahlerian model of the Lagrange space Ln = (M,L(x,y)). We can use it to study the geometry of Lagrange space Ln. For instance, the Einstein equations of the Riemannian space (TM, G) can be considered as "the Einstein equations" of the space Ln. G.S. Asanov showed [22] that the metric G given by the lift (6.1) does not satisfy the principle of the Post-Newtonian calculus. This fact is because the two terms of G has not the same phyisical dimensions. This is the reason to introduce a new lift [109] which can be used in a gauge theory. Let us consider the scalar field: M 2 = <Mz,!/)l/V-
(6-2)'
It is determined only by L(x,y). We assume ||t/||2 > 0. As in the case of Finsler geometry (cf. Ch.2), the following lift of the fundamental tensor field gtj(x,y) : o
(6.3)
a2
•
G (x, y) = gi3(x,y)dxt®dx3 + r--^9ij(x, y)5y'®6y3 \\y\\
Ch.3. Lagrange spaces
75
where a > 0 is a constant, imposed by applications in theoretical Physics. This is o to preserve the physical dimensions of the both members of G . Let us consider also the tensor field on TM :
(6.4)
F=-M_^
and 2-form
where 8 is given by (6.2). As in the Finslerian case we can prove: Theorem 3.6.3. We have oo __ 1° The pair (G, i s an almost Hermitian structure on TM, depending only on the fundamental function L(x, y). o 2° The almost symplectic structure 6 associated to the structure (G,F) is given by the formula (6.5). o oo 3° 8 being conformal to symplectic structure 6, the pair (G, F ) is conformal almost Kählerian structure. 0
0
0
We can remark now that the conformal almost Kählerian space K = (TM, G, F ) can be used for applications in gauge theories which implies the notion of the regular Lagrangian.
3.7
Metrical N-linear connections
Now applying the methods exposed in the first chapter, we will determine some metrical connections compatible with the Riemannian metric G determined by the formula (6.1). Such kind of metrical connection will give the metrical N-linear connections for the Lagrange space L". These connections depend only on the fundamental function L(x, y) and this is the reason for the N-metrical connection to be called canonical. Applying the theory of N—linear connection from Chapter 1, one proves without difficulties the following theorem: Theorem 3.7.1. On the manifold TM there exists the linear connection D which satisfy the axioms:
The Geometry of Hamilton & Lagrange Spaces
76
1° D is metrical connection with respect to G i.e. DG = Q.
(7.1)
2° D preserves by parallelism the horizontal distribution of the canonical nonlinear connection N. 3° The almost tangent structure J is absolute parallel with respect to D, i.e., DJ = 0. From this theorem it follows that we have D — DH + Dv and the h-component D and v-component Dv have the properties H
D%G = Q, DXG = Q,
(7.2) Moreover,
(7.3)
\ {DXYH)V
= 0, {DXYV)H
D%J = 0,
= 0, V X e
X{TM).
DxJ = Q.
Consequently, D is an N-linear connection and has two coefficients DY(N) = (L'jk, C'jk) which verify the following tensorial equations: gljlk
(7.4)
= 0 , g..\k = 0
where | (|) is the h (v)-covariant derivative with respect to DT(N) respectively. And conversely, if an N-linear connection with the coefficients DG = {Lljk,Cljk) verifies the properties (7.3), at (7.4) then it is metrical with respect to G, i.e. the equations (7.2) are verified. But if (7.2) are verified then the equation DG = 0 is verified also. So we can refer the (7.4). We shall determine the general solution (Lljk, Cljk) of the tensorial equations (7.4). First of all we prove Theorem 3.7.2. 1° There exists only one N-linear connection DT(N) — (Lljk, Cljk) which verifies the following axioms: A\ N is canonical nonlinear connection of the space Ln. A2 9ij\k — 0 (it is h-metrical). A3 ..i. = 0 (it is v-metrical). Ai Tljk = 0 (it is h-torsion free), A5 Sljk = 0 (it is v-torsion free).
Ch.3. Lagrange spaces
77
2° The coefficients L'jk and C'jk are expressed by the following generalized Christoffel symbols: Ti i
L
(7.5)
k
X > 9
'2
C jk = ~Q
l°9Tk +_L °9rJ °9ik (lxT ^~-Sx^ I -7T- +
3° This connection depends only on the fundamental function L(x, y) if the Lagrange space Ln.
This theorem can be provided in the usual way (see [113]). And this metrical N-linear connection will be called canonical and will be denoted by CT(N). Similar as in the case of Finsler spaces one can determine all N-linear connections which satisfy only the axioms Ai, A%, A3 from the previous theorem. Now we can study the geometry of Lagrange space Ln by means of canonical metrical connection or by the general metrical connections which satisfy the axioms A\, A2, A3. In this respect we can determine as in the Finsler geometry, the structure equations of the metrical N-linear connection DT(N). Moreover, Ricci identities and Bianchi identities can be written in the usual manner. Therefore, by means of §.9, Ch. 1, the connection 1-forms uilj of the canonical metrical N-connection CT(N) are Ji = Djkdxk + Cljk5yk,
(7.6)
where L%jk,C%jk are given in (7.5).
Theorem 3.7.3. The connection 1-forms of the canonical metrical N-connection CT(N) satisfy the following structure equations d{dxl) - dxkAcu*k = -
(0).!
ft
and du/j - w^-Aw'k = -S
(7.8)
(o). ( i ) .
where the 2-forms of torsion fil, fi* are as follows
(
7
"
9
)
<
fi« = - Rljkdx]Adxk
<
»
•
!
•
•
+ Pljkdx3A5yk *
•
•
*
The Geometry of Hamilton & Lagrange Spaces
78 and the 2-forms of curvature fi'j are I?,- = l- Rjikhdxk*dxh
(7.10)
+ Pj'khdxkASyh
+
^S/kh6yk
the d--tensors of curvature Rj'kh, Pjlkh, Sj'kh and d-tensor of torsion Rljki P%jk having the known expression (see Ch.1, 5). We notice that starting from the canonical metrical connection CT(N) = (£*,•*, C"j*)' other remarkable connections like the connection of Berwald Chern-Rund RT(N) and Hashiguchi HT(N) have the coefficients
^ / , 0 J , RT(N) = (L'jk,0), HF(N) = respectively. The following commutative diagram holds. KT(N) CT{N)
BJ(N)
The corresponding transformations of connections from this diagram may be easily deduced from the Finslerian case. Some properties of the canonical metrical connection CT are given by
Proposition 3.7.1. We have: 1° ^2 Rijk = 0, Pijk = 9isPsjk is totally symmetric. &L
1 '•>"•
4
Q
3° The covariant curvature d-tensors Rijkh, Ptjkh, SVjfch (with Rijkh = 9jsRiskk, etc.) are skew-symmetric in the first two indices. These properties can be proved using the property dd = 0, where 8 is the symplectic structure (6.2), the Ricci identities applied to the fundamental tensor gtj of the space and the equations &,-)* = 0, g..\ = 0. By the same methods we can study the metrical N-linear connections DT{N) = = (L jkjC^jk) which satisfy the axioms A\,A^,Az and have a priori given d-tensors of torsion T \ t and 5\*-.
79
Ch.3. Lagrange spaces Theorem 3.7.4. 1° There exists only one N-linear DT(N) = (Vjk,C1jk) fies the following axioms:
connection which satis-
A\ N is canonical nonlinear connection of the space Ln, A-2 9i3\k — 0 ( is h-metrical). ^3 9 It = 0 ( is v-metrical). A\ A'5
l]\K
The h-tensor of torsion T1^ is apriori given. The v-tensor of torsion S jk is apriori given.
2° The coefficients H jk and (Tjk of the previous connection are as follows
(7-H)
\ vJk = ujk + \ 9th (9jrrkh+9krr]h .
{ '
+
\
- ghTr
\
(Lljic,Cljk) being the coefficients of canonical metrical N-connection CT(N). The proof is similar with that of Theorem 3.7.2. From now on T*jfc, 5*^ will be denoted simply by T*'^, Sljk.
Proposition 3.7.2. The Ricci identities of the metrical N-connection DT(N) are given by: = XmRmijk (7.12)
X i \j\k
vi ~
k\j \k\j
V"
-
P
~
jk-
\i\k
Indeed, we can apply Proposition 1.8.1. Of course, the previous identities can be extended to a d-tensor field of type (r, s) on the ordinary way. Denoting (7.13)
D'j = y\,
d'j = y
}
we have the h-deflection tensor field £>'. and v-deflection tensor field dlj. The tensors Dlj and dlj have the known expressions: (7.13)'
D^ = ysL\j
- A ^ ; d1,- = S1, +
y'Cs'j.
Applying the Ricci identities (7.12) to the Liouville vector field yl we obtain:
The Geometry of Hamilton & Lagrange Spaces Theorem 3.7.5. identities hold (7.14)
If DT(N) is a metrical N-linear connection then the following
D
\\k ~ d
m
d .^ - d\\. = y Sm<jk -
i
m
d mS jk.
We will apply the previous theory in next section, taking into account the canonical metrical ,/V-connection CT(N) and taking Txjk = 0, Sljk = 0. Of course, a theory of parallelism of vector field with respect to the connection DT{N) can be done taking into account the considerations from §9, Ch.1.
3.8
Gravitational and electromagnetic fields
Let us consider a Lagrange space Ln = (M, L(x, y)) endowed with the canonical metrical ,/V-connection CF(N) = (Lljk, Cljk). The covariant deflection tensors D{j and d^ can be introduced by Dij=gitD'j,
(8.1)
dij = gisdsj.
Obviously we have and analogous for the v-covariant derivation. Then, Theorem 3.7.5 implies: Theorem 3.8.1. The covariant deflection tensor fields Dij and dij of the canonical metrical N-connection DT(N) satisfy the identities: Dij\k - Dik]j = ysRSiik D
disRsjk,
i,\k ~ dW = »'P»>* ~ D»C'i"
~
di pS k
° i>
!
Some considerations from the Lagrange theory of electrodynamics, lead us to introduce: Definition 3.8.1. The tensor fields (8.3)
Ftj = I(Ai - Dji), hi = \{da - da)
are called h- and v-electromagnetic tensor field of the Lagrange spaces L", respectively.
Ch.3. Lagrange spaces
81
Proposition 3.8.1. With respect to the canonical metrical N-connection the v-electromagnetic tensor jtj vanishes.
CT(N),
The Bianchi identities of CT(N) and the identities (8.2) lead the following important result. Theorem 3.8.2. The following generalized Maxwell equations hold: (8.4)
to*)
where Cioj = y'Citj. Remarks. 1° If Lagrange space L" is a Finsler space Fn then Ci0$ = 0 and equations (8.4) simplifies. 2° If the canonical nonlinear connection N is flat, i.e. the distribution N is integrable, Rhij = 0 and the previous equations have a simple form. If we put F*' = gisgjrFsr
(8.5) and
hJi = F^\j,vJi = Fi^.,
(8.6) then we can prove:
Theorem 3.8.3. The following laws of conservation hold (g
where Rij is the Ricci tensor of the curvature tensor Rihjk. Remark. The electromagnetic tensor field Fy,/^- and the Maxwell equations were introduced by R. Miron and M. Radivoiovici, [113]. Important contributions have M. Anastasiei, K. Buchner, R. Ro§ca (see Ref. from the book [113]). The curvature J-tensors of the connection CT(iV), Rihjk,Pihjk,Sihjk following Ricci and scalar curvatures
{
p/i
c
Oh
J i y — -n-i jh, Oij — Oi jh,
Ip
r^
nh
— tTi jh,
up
p/i
rij — r~i h.j)
have the
The Geometry of Hamilton & Lagrange
82
Spaces
n v 1 2 ( 6 d \ Let us denote by T,3, T,j, Tijt Tij the components in the adapted basis 7~i''a~i I of the energy momentum tensor. Using the almost Kählerian model K2n = (TM, G, F) of the Lagrange space and taking into account the canonical metrical connection CT(N) (see Theorem 3.7.2) weobtain[113]: Theorem 3.8.4. 1° The Einstein equations of the almost Kählerian space K2n, (n > 2), endowed with the canonical metrical connection CT(N) are the following
_ 1
_
V
H
_
2
where k is a real constant. H
V
2° The energy momentum tensors T^ and T^ satisfy the following laws of conservation (8.9)'
« f ,-,,• = - \ [PlhjsR\i
+ 2Rsi]P\)
,KT.,.=0
The physical background of the previous theory was discussed by S. Ikeda in the last chapter of the book [112]. All this theory is very simple if the Lagrange spaces Ln have the property Pj'kh = 0. Corollary 3.8.1. If the canonical metrical connection Pihjk = 0 then we have
has the property
1° For n > 2, the Einstein equations of the Lagrange space Ln have the form:
(8.10) H.
V.
2° The following laws of conservation hold: T1^ = 0, T* .L = 0. In the next section, we apply this theory for the Lagrangian of Electrodynamics.
Ch.3. Lagrange spaces
3.9
83
The Lagrange space of electrodynamics
The Lagrange spaces L% = (M, L0(x, y)), with the Lagrangian from Electrodynamics (9.1)
•
L0{x,y) = mcYijixjy'y3
2e
+ — Ai{x)yl TTv
which is obtained from the Lagrangian L(x, y), (1.3) for U = 0 is a very good example in our theory. It was studied in the book [113]. We emphasized here only the main results. The space L™ with the fundamental function (9.1) is called the Lagrange space of electrodynamics. The fundamental tensor of the space LQ is gij(x,y)=mcyij(x)
(9.2) and its contravariant is glJ(x,y)
— —jli{x). me The canonical spray of the space LQ is given by the differential equations
*•§) §= where (9.3)'
(?(x, y) = \iirs{x)yrys + —fj
Fik (x)yk
and
Fjk(x) =
(9.3)"
±{djAk-dkAj).
Let us denote Fij{x) = gu(x)F,j{x).
(9.3)'"
The canonical nonlinear connection N has the coefficients Nij=-rijk(x)yk-Fij.
(9.4) As we remarked already we have:
Theorem 3.9.1. The autoparallel curve of the canonical nonlinear connection (9.4) are the solutions of the Lorentz equations d2xi
i
dxi dxk
o
dx>
The Geometry of Hamilton & Lagrange Spaces
84
The canonical metrical connection CT(iV) has the coefficients Lijk{x,y) = fjk{x),
C > = 0.
It follows that the curvature d-tensors are Rj\h{x,y) = r/khix), Pj\h = 0, S/kh = 0, where rj\h{x) is the curvature tensor of the Levi-Civita connection Y The covariant deflection tensors are o Dij{x,y) =Fij (x), di:j{x,y) = Proposition 3.9.1. The h- and v-electromagnetic tensor field of the Lagrange space of electrodynamics are Fij =Fijt
hi = 0.
Consequently, the v-covariant derivative of the tensor Fy- vanishes. We obtain Theorem 3.9.2. The generalized Maxwell equations (8.4) of the Lagrange space of electrodynamics reduce to the classical ones. Since P/kh = 0 implies 'P y - = " Ptj = 0 and C*^ = 0 implies S{j = 0, S = 0, we get from Corollary 3.8.1. Theorem 3.9.3. The Einstein equations (8.10) in LQ reduce to the classical Einstein equations associated to the Levi-Civita connection.
3.10
Generalized Lagrange spaces
The generalized Lagrange spaces were introduced by the first author, [95], and then was studied by many collaborators (see [113]). The applications in the general Relativity or in the relativistic optics was treated too [115]. A detailed study of these spaces one finds in the books [113]. In this section we give a short introduction in the geometrical theory of generalized Lagrange spaces, since they will be considered in the geometry of Hamilton spaces. Definition 3.10.1. A generalized Lagrange space is a pair GL" = (M,gij(x,y)), where gij{x,y) is a d-tensor field on the manifold TM, covariant, symmetric, of rank n and of constant signature on TM.
Ch.3. Lagrange spaces gij is called fundamental tensor or metric tensor of the space GLn. Evidently, any Lagrange space Ln = (M, L(x, y)) is a generalized Lagrange space, 2 1 dL whose fundamental tensor field is gu = - . 1.„ 3 But not any generalized Lagrange 2 ay ay
space GLn = (M,
d2L
2 dydyi
n ()
y
'
has solutions with respect to L(x, y). In order to a space GLn be reducible to a Lagrange space is necessary that the d-tensor field -—^ be totally symmetric. So we have dyk Proposition 3.10.1. A generalized Lagrange space GL" =
for which
the tensor field -=-^is not totally symmetric is not reducible to a Lagrange space. dyK Example 1. The space GLn = (M,gij(x,y)) with 9ij{x,y)=e2«(x
(10.2)
'
with nonvanishing d-covector field ^-r and jij(x) a Riemannian metric is not reducible to a Lagrange space. This example is strongly related to the axioms of Ehlers-Pirani-Schield and theory Tavakol-Miron, from General Relativity [121]. This example was also studied by Watanabe S., Ikeda, S. and Ikeda, F. [171]. Example 2. The space GLn = (M,gij(x,y)) (10-3)
gi,(x,y) =
with the fundamental tensor field
ltj(x)
where Jij(x) is a Riemann or Lorentz metric tensor, yi = riij{x)y:> and n(x,y) > 1 is a refractive index, is not reducible to a Lagrange space. This metric was introduced in Relativistic Optics by J.L. Synge [156]. It was intensively studied by R. Miron and his collaborators [113], [115].
The Geometry of Hamilton & Lagrange Spaces
86
These two examples show that the nonlinear connection with the coefficients
7'jfc(i) being Christoffel symbols of the metric tensor 7y(i), can be associated to the fundamental tensor 5^ of the space GLn. But, generally, we cannot derive a nonlinear connection N from the fundamental tensor gtj. So, we adopt the following postulate: The generalized Lagrange space GLn = (M, g,j(x, y)) is endowed with an a priori given nonlinear connection N. In this situation, we can develop the geometry of the pair (GL n , N) by using the same methods as in the case of Lagrange space Ln. /
X
£i
\
For instance, considering the adapted basis I rr-i-^— I to N and V, respectively, \ox% oy%) we can prove: Theorem 3.10.1. 1° For a generalized Lagrange space endowed with a nonlinear connection N, there exists a unique N-linear connection DT(N) = {L'jk,C*jk) satisfying the following axioms: 2° DT(N) has the coefficients given by the generalized Christoffel symbols: ji (1
°'4)
1 is (^9sk
"
&gjs
3gjk\
'*U4_dg21_dg1A
For more details we send the reader to the books [113]. Let us end this chapter with the following important remark. The class of Riemannian spaces Rn is a subclass of the Finsler spaces Fn and this is a subclass of the class of the Lagrange spaces Ln. Moreover, the class of Lagrange spaces Ln is a subclass of the generalized Lagrange spaces GLn. Hence, we have the following inclusions:
{Rn} C {Fn} c {Ln} c {GLn}.
This sequence of inclusions is important in applications to the geometric models in Mechanics, Physics, Biology, etc., [18].
Chapter 4 The geometry of cotangent bundle The geometrical theory of cotangent bundle (T*M,M) of a real, finite dimensional manifold M is important in the differential geometry. Correlated with that of tangent bundle (TM, n, M) we get a framework for construction of geometrical models for Lagrangian and Hamiltonian Mechanics, as well as, for the duality between them - via Legendre transformation. The total space T*M can be studied by the same methods as the total space of tangent bundle TM. But there exist some specific geometric objects on T*M. For instance the Liouville-Hamilton vector C*, Liouville 1-form w, the canonical symplectic structure and the canonical Poisson structure. These properties are fundamental for introducing the notions of Hamilton space or Cartan space. In this chapter we study some fundamental object fields on T*M: nonlinear connections, N-linear connections, structure equations and their properties. We preserve the convention that all geometrical objects on T*M or mappings defined on T*M are of C°°-class.
4.1 The bundle(T*M, TT*, M) Let M be a real n-dimensional differentiable manifold and let (T*M, M ) be its cotangent bundle [175]. If (ar') is a local coordinate system on a domain Uofa chart on M, the induced system of coordinates on ir*~l(U) are (xl, p^, (i,j,k,... = 1, ...,n). The coordinates pit ...,pn are called " momentum variables". 87
The Geometry of Hamilton & Lagrange Spaces A change of coordinate on T*M is given by
x1' = x*(x\...,xn),ra.nk I f - I = n, dxi
(1.1)
j
dx
d
Therefore the natural frame
(1.2)
d
TT^'"^— l
I a r e transformed by (1.1) in the form
\dx dpij [ d__djc?_ _d_^dpj_ _d_ dx* ~~ dx1 d _ dx' d dpi dx* dpj
Looking at the second formula (1.2) the following notation can be adopted
Indeed (1.2) gives us:
d' =
(1.3)'
.
The natural coframe (dx\dpi) is changed by (1.1) by the rule:
dx*
dx^
d^x^
The Jacobian matrix of change of coordinate (1.1) is
J(u) =
It follows
dx%
dpj dxj \ dx* dx{ I
det J(u) = 1
We get: Theorem 4.1.1. The manifold T*M is orientable. Like in the case of tangent bundle, we can prove:
Ch.4. The geometry of cotangent bundle
89
Theorem 4.1.2. If the base manifold M is paracompact, then the manifold T*M is paracompact, too. The kernel of the differential dir* : TT*M —> TM of the natural projection n* : T*M —> M is the vertical subbundle VT*M of the tangent bundle TT*M. Associating to each point u e T*M the fibre Vu of VT*M we obtain the vertical distribution: V : u £ TM —> Fu C TUTM This distribution is locally generated by the tangent vector field (d1,...,^71). So it is an integrable distribution, of local dimension n. Noticing the formulae (1.2) and (1.2)' we can introduce the following geometrical object fields: (1.4)
C = Piti
(1.5)
u=pidx*
(1.6)
0 = du = dpiAdx'
Theorem 4.1.3. The following properties hold: 1° C* is a vertical vector field globally defined on T*M. 2° The forms u and 6 are globally defined on T*M. 3° 6 is a symplectic structure on T*M. Proof. 1°. By means of (1.1) and (1.3)'it follows that C* belongs to the distribution V and it has the property C* = Pid' = pid\ 2°. u) and 6 do not depend on the changes of coordinates on T*M. 3°. 0 is a closed 2-form on T*M and rank||0|| = 2n = dim T*M. C* will be called the Liouville-Hamilton vector field on T*M, u is called the Liouville 1-form and 0 is the canonical symplectic structure on T*M. The pair (T*M, ) is a symplectic manifold.
4.2
The Poisson brackets. The Hamiltonian systems
Let us consider the Poisson bracket { , } on T*M, defined by
The Geometry of Hamilton & Lagrange Spaces
90 Proposition 4.2.1. We have
Proof. By means of (1.1), (1.2), f(x,p)
= f{x,p) and
df_ _ dx>_ df dpj_ d]_ df _ dx> df { { ~dx ~~ ~dx ~dx~i + ~dx{ dfj' dPj ~ dx1 dp{' Therefore we deduce df 8g dpi dxi df dg dpj dxi
df dx* (dxm dg_ dpj dxi \ dxi dxm df dg dpm Qpj gpm Qxi
dpm dg dx xl
df dg dpj dxi
because —Jr = 0. Consequently, {/, g} = {/,§}• dx1
q.e.d.
Theorem 4.2.1. The Poisson bracket { , } has the properties 1° {/,} = -{,/} 2° {f, g } is R-linear in every argument
4° {
By a straightforward calculus, using (2.1), l°-4° can be proved. Therefore { , } is called the canonical Poisson structure on T*M. The pair ( { , }) is a Lie algebra, called Poisson-Lie algebra. The relation between the structures 6 and { , } can be given by means of the notion of Hamiltonian system. Definition 4.2.1. A differentiable Hamiltonian is a function H:T*M-tR which is of class C°° on T*M = T*M\{0} and continuous on the zero section of the projection TT* :T*M -t M. Definition 4.2.2. A Hamiltonian system is a triple (T*M, H) formed by the manifold T*M, canonical symplectic structure 6 and a differentiable Hamiltonian H.
Ch.4. The geometry of cotangent bundle
91
Let us consider the ^(T*M)-modules: X(T*M) vectors and covectors fields on T*M, respectively. Thus, the following J r (T*M)-linear mapping:
and X*(T*M)
- of tangent
S$ : X{T*M) —> X*(T*M) can be defined by: (2.2)
St(X) = ix0,
VXeX(T*M).
Proposition 4.2.2. S is an
isomorphism.
Indeed, Se is a F(T*M)-linear
mapping and bijective, because rank||0|| = 2n. -^— e 1
ox
o f is sent by Sg in the
dpij
local base (dx\dpi) of X*{T*M) by the rule (2.3) We get also (2.3)'
59(C*) = w.
We can apply this property to prove: Theorem 4.2.2. The following properties of the Hamiltonian system (T*M, 6, H) hold: 1° There exists a unique vector field XH 6 X(T*M) having the property (2.4)
iXlie = -dH
2° The integral curves of the vector field XH are given by the Hamilton-Jacobi equations:
^
<M_ = dH_dv1 dt dpi' dt
=
_dH_ dx{
The Geometry of Hamilton & Lagrange Spaces
92
Proof. 1°. The existence and uniqueness of the vector field XH is assured by Proposition 4.2.2. It is given by XH =
(2.6)
S;l(-dH)
Using (2.3), XH is expressed in the natural basis by dH 2°. The integral curves of from
d
dH d
(2.6)' are given by the equations (2.5).q.e.d.
XH is called the Hamilton vector field. Corollary 4.2.1. The function H(x,p) is constant along the integral curves of the Hamilton vector field XH • dH Indeed,
at
= {H, H\ = 0.
The structures 6 and { , } have a fundamental property given by the theorem: Theorem 4.2.3. The following formula holds: (2.7) {/,g} = v(Xf, Xg), V (f,g) G !F{1 M), VX G Xyl Proof. From (2.6)' we deduce
M).
{/,} = xf9 = -x9f = -df(x3) = (iXfe)(xg) = e(xf,xg). q.e.d. Corollary 4.2.2. The Hamilton-Jacobi equations can be written in the form ^-^{H,^}, ^ = {H,Pi}. at at One knows, [167], the Jacobi method of integration of Hamilton-Jacobi equations (2.5). Namely, we look for a solution curve ^(t) in T*M, of the form
(2-8)
(2 Q\
<£* = xHt)
V —
(x(t))
where S G T(M). Substituting in (2.5), we have dz* _ 8H_
(t])dS_(
dpi__
d2S
dH_ _ _dH_
Ch.4. The geometry of cotangent bundle It follows
/ dS\
93
(dH dH
dH dH\
Consequently, H (a;>-rdx I = const., which is called the Hamilton-Jacobi equation
V J
(of Mechanics). If integrated, it defines S and j(t) is obtained by the integration of the first equation (*). Remarks. 1° In the next chapter we will define the notion of Hamilton space Hn = (M, H(x, p)), H being a differentiable regular Hamiltonian and we shall see that the Legendre mapping between the Hamilton space Hn and a Lagrange spce Ln (M, L(x,y)) sent the Euler-Lagrange equations into the Hamilton-Jacobi equations. This idea is basic for considering the notion of £-duality [66], [67], [105]. 2° The Poisson bracket { , } is also basic for quantization. The quantization of a Mechanical system is a process which associates operators on some Hilbert space with the real functions on the manifold T*M (phase space), such as the commutator of two such operators is associated with Poisson bracket of functions (Abraham and Marsden [3], [167]).
4.3
Homogeneity
The notion of homogeneity, with respect to the momentum variables pi, of a function / G !F(T*M) can be studied by the same way as the homogeneity of functions defined on the manifold TM (see Ch.l, §2). Let Hp be the group of homotheties on the fibres of T*M: Hp = {ha : (x,p) G T*M —> (x,ap) G TM | a € R+} The orbit of a point u0 = (xQ,p0) by Hp is given by xi = 4 , Pi = ap°i, VaE R+ The tangent vector at the point UQ = hi(u) is the Liouville-Hamilton vector field CT(«o).
__
A function / € IF{T*(M), differentiable on T'M and continuous on the zero section is called homogeneous of degree r (r 6 , with respect to the variables pi (or on the fibres of T*M), if (3.1)
foha
=
arf,VaeR+
The Geometry of Hamilton & Lagrange Spaces
94
This is f(x,ap) = arf(x,p), Va€R+
(3.1)'
Exactly as in the case of the homogeneity of the function f : TM —> R, one can prove: Theorem 4.3.1. A function f : differentiableon T*M and continuous on the zero section of •K* : T* M —> M is r-homogeneous with respect to Pi if and only if we have Ccf = rf,
(3.2)
where £ c - is the Lie derivation with respect to the Liouville-Hamilton vector field C\ But (3.2) can be written in the form (3.2)'
Pt |=r/.
A vector field X 6 X{T*M) is r-homogeneous with respect to
if
Xoha = ar-lh*aoX, Va£R+ It follows : Theorem 4.3.2. A vector field X 6 X{T*M) is r-homogeneous if and only if we have: £c.X =
(3.3)
(r-l)X.
Evidently, CcX = [C*,X]. Consequently: 1°
-^-T>-T— 1
ox dpi
are 1-and 0-homogeneous with respect to
2° If i s F{T*M) s -homogeneous and X € X(T*M) is r-homogeneous then fX is s + r homogeneous. 3° A vector field X given in the natural frame by
(3.4)
X=X*£1% + Xi£dx
dp
Ch.4.
The geometry of cotangent bundle
95 (0)
is r-homogeneous with respect to pi, if and only if X * are r — 1 homogeneous (i)
and X{ are r-homogeneous with respect to p%4° C* = pi —— is 1-homogeneous. 5° If X is r-homogeneous vector field on T*M and f is a function, s-homogeneous, then Xf is r + s — -homogeneous function. 6° -T— are s — 1 homogeneous an dpi mogeneous.
-—d— are s — 2 homogeneous, if f is s-hodpidpj
A q-form ui e A'(T*M) is called s-homogeneous with respect to pi (or on the fibres of T*M) if: Loah*a = aroj,
(3.5)
VaeR+.
All properties given in Ch. 1, §2, for q-forms defined on the tangent manifold TM, concerning their homogeneity are valid in the case of q-forms from A9(T*M). We get: Theorem 4.3.3. A q-form OJ on T*M is s-homogeneous on the fibres of T*M if and only if £c-w = sw.
(3.6)
Consequences. 1° w e Aq(T*M), w' 6 Aq'(T*M), s + -homogeneous.
s- respectively s -homogeneous, imply
2° u s A*(T*M), -homogeneous and X,--,X (i)
(?)
r-homogeneous vector fields de-
termine the function ui(X, •••,X) r + s - 1 homogeneous. (1)
3° dx\dpi
(9)
are 0-respectively 1-homogeneous.
4° The Liouville 1-form u> is 1-homogeneous. 5° The canonical symplectic structure 6 is 1-homogeneous. The previous consideratiosn will be applied especially, in the study of Cartan spaces.
The Geometry of Hamilton & Lagrange Spaces
96
4.4
Nonlinear connections
On the manifold T*M there exists a remarkable distribution V : u 6 T*M —)• Vu : TUT*M. As we know V is integrable, having the adapted basis (<9\ ..., d") and it is of local dimension n. Definition 4.4.1. A nonlinear connection on T*M is a differentiable distribution N : e T*M — : C TUT*M which is supplementary to the vertical distribution V, i.e.: TUT*M = NU® Vu, Vu € T'M
(4.1)
Consequently, the local dimension of the distribution N is N — i s M. N will be called also the horizontal distribution. If N is given, there are uniquely determined a system of functions i n every domain ofa local chart IT*'1 (U), such that the adapted basis to the distribution N has the form
The functions Nij(x,p) are called the coefficients of the nonlinear connection N. Theorem 4.4.1. A change of coordinate (1.2) on T*M transforms the coefficients Nij(x,y) ofa nonlinear connection N by the rule: ,A ,
(«)
~ ._ ,
N
^
dxs
dx\r N
+
Conversely, a system of functions Nij(x,y) defined on each domain of local chart from T*M, which verifies (4.3) with respect to (1.1), determines a nonlinear connection. Proof. By means of (1.1), (1.2), it follows from (4.2) the formula (4.3). Therefore, < 7—;-, • • •.-— > generate a distribution N, which is supplementary to the vertical {ox1 oxn} distribution V, V being generated by — • q.e.d. opi orfield s give us an adapted basis to the distributions
ox' dp,/
N and V. The changes of coordinates (1.1) has the effect: . S dxj 6 d dx{ d lt K ' ' Sxi dxi 6xi dpi dx> dpj
Ch.4. The geometry of cotangent bundle
97
-^-i-r— I is given by (dx\ 8pi), (i = 1,..., n) with
dx' opij
(4.5)
8pi - dpi - Nji dxj
and the transformations of coordinates (1.1) transform the "adapted" dual basis, (dx',5pi) in the form
(4.4)'
d~x> = d£dx\ Sp, = g ^ .
Theorem 4.4.2. If the base manifold M is paracompact, then on the manifold T*M there exist the nonlinear connections. Proof. M being paracompact, Theorem 4.1.2 affirms that T*M is paracompact, too. Let G be a Riemannian structure on T*M and N the orthogonal distribution, to the vetical distribution V with respect to G. Thus, the equality (4.1) holds.
q.e.d.
Consider a nonlinear connection N with the coefficients Nij(x,p) and define the set of functions (4-6)
rij =
^(Nij-NJi)
Proposition 4.4.1. With respect to (1.1), T,J is transformed by the rule ~
T
dxT
dx
°
« = d¥ W
T
-
Indeed, the formulae (4.3) and (4.6) lead to (4.7). Consequently, r^ is a distinguished tensor field, covariant of order two, skewsymmetric. Tjj is called the tensor of torsion of the nonlinear connection N. The equation Ty = 0 has a geometrical meaning. In this case the nonlinear connection N is called symmetric. Theorem 4.4.3. With respect to a symmetric nonlinear connection N, the canonical symplectic structure 6 and the canonical Poisson structure { , }, can be written in the following invariant form (4.8) 6 = SpiAdxi
The Geometry of Hamilton & Lagrange Spaces
98
Proof. Using (4.2), (4.5) and (4.6) we have 6 — SpiAdx' + m dx'Adxj i I 9 )
dPi &xi
dPi 6x>
l}
dPi dxi
But these formulae, for r y = 0, give us the announced formulae (4.8) and (4.9). Now, by means of (4.4) and (4.4)' it follows the invariant form of 6 and { , }.q.e.d. Of course, we can define the curvature tensor field of a nonlinear connection N. Indeed, d9 = 0 imply d(5pi)Adxl — 0. But d(5pi) is given by Proposition 4.4.2. The exterior differential of Spi are given by (4.11)
d(SPi) = -
(^Rlj
where
Indeed, d(6pi) = -dNn/\dxJ
6NdN and dNH - —J1mdxm + -r-^Spm which determine 6x dpm
together (4.11) and (4.12). It follows, without difficulties:
Proposition 4.4.3. By means of (1.1) we obtain: ~ A
__ dxr dxs dxp
- — — —ffN — dV sh + ~dx7dx^ ~d¥ di7 dx So, we can say that Rijh is a d-tensor field called the d-tensor of curvature of and d'Njh is a d- connection determined by N called the Berwald connection. Analogously, we can study the Lie brackets of the vector fields from adapted Jh
basis hr7'-5— \6x* d Proposition 4.4.4. The Lie brackets of the vector fields ox*
are as follows (4.14)
dpi
Ch.4. The geometry of cotangent bundle
99
Now we can give a necessary and sufficient condition for the integrability of the distribution N called the integrability of the nonlinear connection N. Theorem 4.4.4. The horizontal distribution N is integrable if and only if the dtensor of curvature Rijh vanishes. Indeed the Lie brackets of the vector fields from adapted basis <5; of N belongs to N if and only if Ri]h = 0 (cf. (4.14)).
4.5
Distinguished vector and covector fields
Let N be a nonlinear connection on T*M. It gives rise to the direct decomposition (4.1). Let h and v be the projectors defined by supplementary distributions N and V. They have the following properties (5.1) If X £ X(T*M) (5.2)
h + v = I, h2 = h, v2 = v, h o v = v o h = 0. we denote XH = hX, Xv = vX.
Therefore we have the unique decomposition (5.3)
X = X" + Xv
Every component XH and Xv is called a distinguished vector field. Shortly a dvector field. In the adapted basis (<5j,d') we get (5.3)'
XH = Xi{x,p)6u
Xv = Xi{x,
With respect to (1.1), we have, using (4.4),
But, these are the classical rules of transformations of the local coordinates of vector and covector fields on the base manifold M. Therefore Xl(x,p) is called a d-vector field, too and Xi(x,p) is called a d-covector field on T*M. For instance C* = p,9' is a d-vector field and pt is a d-covector field on T*M. W e h a v e C * " = 0, C*V = C.
The Geometry of Hamilton & Lagrange Spaces
100
A similar theory can be done for distinguished 1-forms. With respect to the direct decomposition (4.1) an 1-form w € X*(T*M) can be uniquely written in the form u=u
(5.4) where
uiH = w o h, UJV = w o v.
(5.4)'
In the adapted cobasis (dx^Spt), we get UJH = u)i(x,p)dx\ uv = u>l(x,p)Spi
(5.5)
u>H and are called, every one, d-1 form. The coefficients uji(x,p) and w'(:c,p) are d-covector and d-vector fields, respectively. Now, let us consider a function / 6 T he 1-form df can be written in the form (5.4) (5.6) df = [df)H + {df)v, where (df)" = ^-dx\
(5.6)'
(df)v =
^8Pi
Acurve 7 : / C R —> T*M, having Im 7 c 7r*(f/) has the analytical representation: xl = xi(t),
(5.7)
&=&(£), t € l .
The tangent vector —-> in a point of curve 7 can be set in the form dt
dt
=(\
\dt)
+(
\dt)
=
dt 5xl
+
dt dpi
where
d"f Id-y\ The curve 7 is called horizontal if — = ~r , V * e /. at \ at y Theorem 4.5.1. An horizontal curve 7 w characterized by the system of differential equations (5.9)
*' = *'(*), ^
= 0.
Clearly, if the functions a;'(t) are given the previous system of differential equations has local solutions, when a poin x'o = x%(t0), p°, t e is given.
Ch.4. The geometry of cotangent bundle
4.6
101
The almost product structure P. The metrical structure G. The almost complex structure F
A nonlinear connection N on T*M can be characterized by some almost product structure P . Let us consider the -linear mapping P : X{T*M) —• X{T*M) defined by
(6.1)
F{XH) = XH, P(XV) = -Xv,
\fX 6 X(T*M).
Thus P has the properties: PoP = ; (6.2)
P = / - 2 u = 2/i-J rank P = In.
Theorem 4.6.1. A nonlinear connection N on T*M is characterized by the existence of an almost product structure P on the manifold T*M whose eigenspaces corresponding to the eigenvalue -1 coincide with the linear spaces of the vertical distribution V on T*M. Proof. If N is a nonlinear connection, then we have the direct sum (4.1). Denotingh and v the projectors determined by (4.1) we get P = = I—2v. Then P has the property (6.1). P is an almost product structure, for which JP(XV) = -Xv. Conversely, if P 2 = / and P ( ^ v ) = ~XV, then v = \{I - P) and /i = I ( / + P) satisfy (5.1). Therefore, N = Ker v and it follows N © V = TT*M. q.e.d. Proposition 4.6.1. The almost product structure P is integrable if and only if the nonlinear connection N is integrable. Indeed, the Nijenhuis tensor of the structure P , given by (6.3)
MP(X, Y) = P 2 [X, Y] + [PX, JPY] - P[PX, Y] - P[X, PK]
The Geometry of Hamilton & Lagrange Spaces
102 vanishes if and only if[XH,YH]v
= 0, VX, Y e X{T*M),
because
Remark. Of course we can consider a general almost product structure on T*M. It will determine a general concept of connection on T*M. This idea was developed by P. Antonelli and D. Hrimiuc in the paper [14] and will be studied in a next chapter of this book. Let us consider a Riemannian (or metrical) structure G on the manifold T*M. The following property is evident: G uniquely determines an orthogonal distribution N to the vertical distribution V on T*M. Therefore N is a nonlinear connection. Let Nij(x,p) be the coefficients of N and h, v the supplementary projectors defined by N and V. Then G can be written in the form (6.4)
G = GH + Gv, GH(X,Y)
= G{XH,YM),
GV{X,Y)
=
G{XV.XV).
Or, in the adapted basis, (6.5)
G=
i
9ij{x,p)dx
® dxj + tij(
where gy is a covariant nonsingular, symmetric tensor field and ti' is a contravariant nonsingular, symmetric tensor field. Of course, the matrix ||<7y(a:,p)||, ||/i y '(a;,p)|| are positively defined. The Riemannian manifold (T*M, ) can be studied by means of the methods given by the geometry of the manifold T* M. If a tensor gij(x,p) covariant, symmetric and positively defined on T*M and a nonlinear connection N with coefficients Nij(x,p) are given, then we can consider the following Riemannian structure on T*M: (6.6)
G(x,p) = gij(x,p)dx% ® dxj + gtj(x,p)Spi ® Spp
The tensor G is called the N-lift to T*M of the d-tensor metric gij(x,p). Assuming that the d-tensor metric gij(x,p) and the symmetric nonlinear connection N(Nij) are given let us consider the following J"(T*M)-linear mapping F : X(T*M) —> X{T*M) (6.7) where (6.7)'
^
It is not difficult to prove, by a straightforward calculus:
Ch.4. The geometry of cotangent bundle
103
Theorem 4.6.2. 1° F is globally defined on T*M. 2° F is the tensor field
F = -gijd1 ® dxj + gij6i ® Spj
(6.8)
3° F is an almost complex structure: (6.9)
FoF = -/.
We have, too Theorem 4.6.3. In order to the almost complex structure F be integrable is necessary that the nonlinear connection N be integrable. Proof. The Nijenhuis tensor ftfjp (Si,5j) = 0, imply, firstly R,jk = 0. It implies also some conditions for g^ and N. q.e.d. The relations between the structures G and F are as follows Theorem 4.6.4. 1° The pair (G, F), G given by (6.6) and F given by (6.7), is an almost hermitian structure. 2° The associated almost symplectic structure to (G, F ) is the canonical symplectic structure 6 = Proof. 1°. We get from (6.6), (6.7), in the adapted basis ( £ , # ) , G(FX,FY) = G(X,Y). 2°. e(X,Y) = G(WX,Y). q.e.d.
4.7
d-tensor algebra. N-linear connections
As we know, a nonlinear connection N determines a direct decomposition (4.1) with respect to which any vector field X on T*M can be written in the form X = XH + Xv. And any 1-form u on T*M is uniquely decomposed: w = uH + UJV. The components XH,XV are distinguished vector fields and A re distinguished 1-form field. Briefly, d-vector field, d-l-form field.
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104
Definition 4.7.1. A distinguished field on T*M (briefly d-tensor field) is a tensor field T of type (r, s) on T*M with the property: T(^...^X,...,X)
(7.1)
=
is
T(^,...,uv,XH,...,Xv) I s
for any i,...,w € X*{T"M) and for any X,-,X H
v
H
€ X(T*M).
v
X = X , X = X , or u) = u) , u - u are examples of d-vector fields or d-l-form fields. In the adapted basis (<$i,dl) and adapted cobasis (dx',6pi) T can be written, uniquely, in the form (7.2)
T = Til;;* {X, p)6tl ® • • • ® # • ® dx» ® • • • ® fer.
It follows that the addition of d-tensors and tensor product of them lead to the d-tensor fields. So, the set (1, ) generates the algebra of d-tensors over the ring of functions F(T*M). Other examples Sif, dlf are d-covector and d-vector fields, respectively. Clearly, with respect to (1.1) the coefficients T£;;.£(a:,p) of a d-tensor fields are transformed by the classical rule: (7 1) K
'
rpil-lr
K-''
-.
Br'1
Br'r drkl
UX
UX
UX
dxhi
dxhr dxh
.
ftr>
OX
rphl...hr
d&'kl"kt'
the notion of N-linear connection can be defined in the known manner (see Ch.l). In the following we assume N is a symmetric nonlinear connection. Definition 4.7.2. A linear connection D on T*M is called an N-linear if:
connection
1° D preserves by parallelism every distribution N and V. 2° The canonical symplectic structure 8 = SpiAdx' has the associate tensor £ absolute parallel with respect to D: (7.4)
D0 = O.
The following properties of an N-linear connection D are immediate: (7.5)
Dxh = Dxv = 0; £> X P = 0
Indeed, (Dxh){Y) = DX(YH) - {DXY)H. For Y = YH we get H (Dxh)(Y ) = 0 and for Y = Yv we get (Dxh)Yv = 0. Similarly, Dxv = 0 andDxJP = DX(I -2v) = 0.
Ch.4. The geometry of cotangent bundle
105
For an almost Hermitian structure ( G , F ) given in the previous section, any N-linear connection D, with the property DG — 0, has the property DW = 0, too. , then DXY
(7.6)
=
We find new operators of derivation in the algebra of d-tensors defined by: DX~
(7.7)
These operators are not the covariant derivatives in the d-tensor algebra, since
DHxf = XHf + Xf, Dvxf = Xvf ± Xf. Theorem 4.7.1. 1° D%f - XHf,
The operators Dx, D^ have the following properties: Dvxf =
Xvf.
Dx(fY)=Xvf-Y [ D%( 3
°
BX(
4° D»+Y = D$ + D»,
Dx+Y
6° D f B = 0, DX6 = 0 7° The operators DX,D% have the property of localization. The The will be Dx,
(
•
proof of the previous theorem can be done by the classical methods [113]. operator Dx will be called the operator of h-covariant derivation and Dvx called the operator of v-covariant derivation. Dx act on the 1-form u on T* M by the rules
'
Consequently, the actions of Dx and Dx on a tensor field T of type (r, s) on T*M are well determined.
The Geometry of Hamilton & Lagrange Spaces
106
4.8
Torsion and curvature
The torsion T of an N-linear connection D is expressed as usually by (8.1)
T{X,Y) =
DXY-DYX-[X,Y].
It can be characterized by the following vector fields T{XH,YH),TT{XH,YV),
r
W{Xv,Yv)
Taking the h- and v- components of these vectors we obtain the d-tensors of torsion: (8.2)
TT(XH, YH) = hTT(X", YH) + vTT(XH, YH), T(XH, Yv) = hTT(XH, Yv) + vT(XH, Yv), TT(XV, Yv) = hTF(Xv, Yv) + vTT(Xv, Yv).
Since V is an integrable distribution (8.1) implies hT{Xv, Yv) = 0
(8.3)
Taking into account the definition (4.8.1) of the torsion IT we obtain: Proposition 4.8.1. The d-tensors of torsion of an N-linear connection D are:
(8.4)
hTT(X", YH) = DlY" - D»XH - [XH\ YH]H hT(XH, Yv) = -(D%XH + [XH, YV}H) vT(Xa, YH) = ~[XH, YH]V vT{XH,Yv) = D%YV - [XH,YV]V vT{Xv.Yv) = DVXYV - D^XV - [XV,YV]V.
The curvature IR of an N-linear connection D is given by (8.5)
U(X, Y)Z = (DXDY - DYDX)Z - D[X}Y]Z
Remarking that the vector field , Y)ZV is vertical one, we have (8.6)
1R(X, Y)ZH is horizontal
one and
v(R(X, Y)ZH) = 0, h(]R(X, Y)ZV) = 0
We will see that the d-tensors of curvature of the v-linear connection D are: (8.7)
TR(X",YH)ZH,
Therefore, by means of (8.5)
JR{XH,YV)Z",
1R(XV,YV)ZH
Ch.4. The geometry of cotangent bundle
107
Proposition 4.8.2. 1° The Ricci identities of an N-linear connection Dare [D
(8.8)
2° The Bianchi identities are given by £ (XYZ)
(8.9)
{{DXT)(Y,
Z) - M(X, Y)Z + T(TT(X, Y), Z)} = 0
J2 {(DxB){U,Y,Z)-B.{T(X,Y),Z)U} = 0
{XYZ)
where
J^
means the cyclic sum.
{XYZ)
4.9
The coefficients of an N-linear connection
Let D be an N-linear connection and <$• = -r^> 9* = ^r— the adapted basis to N 5xl dpi n and V. Then Dx = JD^ + Dx • I order to determine the coefficients of D in the adapted basis, we take into account the properties: DSj = D%, Dbi = Dl-
(9-1)
Theorem 4.9.1. 1° An N-linear connection D on T*M can be uniquely represented in the adapted basis {Si,dl) in the following form Di}6t (9.2) 2° With respect to a change of coordinate (1.1) on T*M, the coefficients transform by the rule:
while Cjh{x,p)
Hljh(x,p)
is a d-tensor field of type (2, 1).
3° Conversely, if N is an apriori given nonlinear connection and a set of function (Hlik,Cjh), verifying 2° is given, then there exist a unique N-linear connection D with the properties (9.2).
The Geometry of Hamilton & Lagrange Spaces
108 Proof. 1°. Setting
and taking into account that DO = 0 where tensor 9 of the 2-form 0 = SpiAdx1 have the components
BiSiJi) = 0, 0(6it&) = -61, 6$,&i) = S{, 0(#,#) = 0 it follows that DX9 = 0, imply H^ = H^. Analogously, setting D&Si = C?j6h, Dfrfr = -Clhjdh, from Dx0 = 0, it follows C^ = C; j . 2°. By a straightforward calculus, taking into account the formulae (4.4), it ~ rtrT
follows (9.3) and C*§L
Br{
=
g_
r)ri
^
.
3°. One demonstrates by the usually methods. The pair DY{N)={H)h,C?)
is called the system of coefficients of D.
Proposition 4.9.1. The following formula holds: Ds,dx' = Dd* Let us consider a d-tensor T, of type (r, s) expressed in the adapted basis in the form (7.2), and a horizontal vector field X = XH — X'Si. Applying Theorems 4.7.1 and 4.9.1 we obtain the h-covariant derivation Dx of T in the form:
where (9.5)'
T
^
m
=
^ " ^
^
^
^
*'"
+
T H
^ ^~
The operator "|" is called h-covariant derivative with DT(N) = (H)h,Cih). Now, taking X = Xv = Xid', DVXT has the following form (9.6)
DVXT = XmT>z
respect
to
Ch.4. The geometry of cotangent bundle
109
where (9.0) _rpil...ir I
fh.m
hJ2-j,L/j,
rph--ir/~thm i
•"
ji...h°i, •
The operator "|" will be called the v-covariant derivative with respect to DF(N) =
t
Proposition 4.9.2. The following properties hold: 1° T j H ; | m is a d-tensor of type (r, s + ). 1° 7j 1 1 "£| m is a d-tensor of type ( + s). Proposition 4.9.3. The operators "|" and "|" satisfy the properties: 1° f, - —^-, f\m - ' 1
J\m — r _ m ' J I Y'i| m ^ am
{y. i /
\ I Wj| m = d m Wj — Wft-H , m ,
LOi\ —O U>i — UhCi
2° "|" and "|" are distributive with respect to the addition of the d-tensors of the same type. 3° "|" and "|" commute with the operation of contraction. 4°
"|" and "|" verify the Leibniz rule with respect to the tensor product of dtensors.
As an application let us consider the deflection tensors (9-8)
Ay=pjy,^=ftp
Using (9.7) we get
(9.9)
A y = Ni3 - PmHmt}, % = %-
PmCr
In the particular case, Ay = 0, 5- = 5{, DV(N) is called an N-linear connection of Cartan type. It is characterized by (9.10) We have:
JVy = PmHmij)
j
Pmcr
= 0.
The Geometry of Hamilton &Lagrange Spaces
110
Proposition 4.9.4. Ifthe pair {Hijk(x,p),Cfk(x,p)) is given and H'jk(x,p) satisfy (9.3), then Nij = pmHij are the coefficients of a nonlinear connection N, and (Hljk(x,p),Cfk(x,p)) are the coefficients of an N-linear connection D. The proof is not difficult if we apply the point 3° from Theorem 4.9.1. Based on the previous property we can prove: Theorem 4.9.2. If the base manifold M is paracompact then on T*M there exist the N-linear connections. Proof. Let g(x) be a Riemannian metric on M and Jjh(x) its coefficients of LeviCivita connection. Then Njh = Pmijh are the coefficients of a nonlinear connection N on T*M. So, the pair {~i)h{x), 0) gives us the coefficients of an N-linear connection DT(N) on T*M.
4.10
The local expressions of d-tensors of torsion and curvature
In the adapted basis (Si, dl) the Ricci identities (8.8), using the operators "|" and "|", lead to the local expressions of the d-tensors of torsion and curvature. Theorem 4.10.1. For any N-linear connection DT(N) = (H'jk,C{h) Ricci identities hold:
(10.1)
~ -A I \j
Ijl
X
i{j ^ Yi\h\j I — -A | |
=
^
"m j
T*
and Rkjh,Pkljh,Sklih
(10.3)
are
the
j
—
I
m
J
vmo iih V i | C J jh. — A Om J ' A | Om .
T~T*
f/*
I"i m
where the coefficients Tljh,Rmj)l,Cjmh,Phmj fin o\
~~
the following
Q^
and Sm>h are the d-tensors of torsion: r** J^
c1 ^
p*
d-tensors of curvature:
pk%h
Sktjh = frCk-i - &Ckih + CkmjCmlh
-
PT*
Ch.4. The geometry of cotangent bundle
111
Proof. By a direct calculus, using (9.7) we get X'^h - X'Wj = = [Sk, S^X* + X^H^
- SjH^ + H'mjH\h - H^HiJ -
X^T™^.
Taking into account the formula [6h,5j]X% — RmfljdmX\ the previous equality leads to the first Ricci formula, with the coefficients Tjh,Rkljh from (10.2) and (10.3). Etc. Proposition 4.10.1. The following formule hold: (10 4) and
The proof can be given by a direct calculus, using the formula (8.5) and the equations (8.6). As usually, we extend the Ricci identities for any d-tensor field, given by (7.2). For instance, if g*j(x,p) is a d-tensor field, the Ricci identities for g'i, with respect to N-linear connection DV(N) are 9l]\k\h ~ 9lJ\h\ -9ij\mTmkh-gij\mRmkh,
(10 5)
^Vl"-^lV=
. 9
5 11 ~9 I I
ij
/-» mh jm^/c
— ff "m
ij
m
5
p/i ^A:
+ 5 "m
~ 5 I "m •
In particular, if the N-linear connection DT(N) satisfy the supplementary conditions: gijlk = 0, gij\h = 0
(10.6)
then (10.5) lead to the equations -,771? D
9
(10.5)'
J
i
_| «IT7l D
Hm kh + 9
J
Krnkh ~
0
u
The Geometry of Hamilton & Lagrange Spaces
112
Such kind of equations will be used for the N-linear connections compatible with a metric structure G of the form (6.6). The Ricci identities applied to the Liouville-Hamilton vector field C* = Pid1 give us some important identities. To this aim, we take into account the deflection tensors Ay = pqj, 5j3 = pi\3.
Theorem 4.10.2. Any N-linear connection D satisfies the following identities
(10.7)
Ay |fc - Si\ sj k
= -PmPimjk
sk\j _
„
qmjk
- &imCjmk ~ _ xrnc
SrP'mj
jk
In particular, if the N-linear connection D is of Cartan type, i.e. Ay = 0, $}• = 8*, then we have Proposition 4.10.2. Any N-linear connection of Cartan type satisfies the following identities (io.8)
PmRTjk + Rijk = o, pmPimjk + P\J = o, PmSi^k
+ sjk = o.
Finally, we remark that we can explicitly write the Bianchi identities, of an TVlinear connection DF{N) — [H%jk,C{k) if we express in the adapted basis (<5j,94) the Bianchi identities (8.9).
4.11
Parallelism. Horizontal and vertical paths
Let D be an TV-linear connection, having the coefficients DT(N) = = {Hijk,Clk) in adapted basis (<&<,#). Consider a smooth parametrized curve 7 : / -> T*M, having the image in a domain of a chart of T*M. Thus 7 has the analytical expression of the form: (11.1)
x i = xi{t),pi
=
pi{t),tel.
The tangent vector field 7 = — can be written in the frame (Si, d1) (see §5, (5.8)), as follows
where
113
Ch.4. The geometry of cotangent bundle We denote ) DiX = -r-,DX at
(11.4)
=
DX ^-dt, at
DX DX is called the covariant differential of the vector field X and —— is the covariant at differential along the curve g. If the vector field X is written in the form X = X" + Xv = X{5t + Xi&, and 7 = j
H
+j
v
,
then we can write
dx n - nH -i-nv ' n 4. Sp* n A straightforward calculus leads to m (11.5) DX = (dX' + X w'm)6i + (dXi - Xmui?)&
where (11.6)
w'j = H{jkdxk + Cf8pk.
UJ'J are called 1-forms of connection of D. Setting Mi RV (11 6)
-
^
-dJ
w
dxk
= Hjk
^F
4-r**^*
+ Cj
DX the covariant differential —r- can be written: at
It
DX The vector field X is called parallel along the curve 7 : / —> T*M, if —— = 0, Vt€ I. We obtain Theorem 4.11.1. The vector field X = X'6i+Xid' is parallel along the parametrized curve 7, with respect to the N-linear connection D, if and only if its coordinates X*,Xi are solutions of the differential equations
The Geometry of Hamilton & Lagrange Spaces
114
The proof is immediate means of (11.7). A theorem of existence and uniqueness for the parallel vector fields along a given parametrized curve 7 in the manifold T*M can be formulate in the classical manner. Let us consider the case of vector field X and N-connection D, for which DX = 0, for any curve 7. Remark that DX = 0 is equivalent to dXl + X " V m = 0, dXt - XmLjmi = 0.
(*)
But dX' = SjX'dxi+dJX'Spj, equations, equivalent to (*)
together with (11.6) lead to the system of differential
Xl{jdxj + Xl\jSPj = 0, Xi[3dxj + XtfSpj = 0 Since dx',6pi are arbitrary, it follows
f X\, = 0, Xl\j = 0
I
(11.9)
"
'.
Using the Ricci identities (10.1), and taking into account (11.9) we obtain the necessary conditions for a vector field X = XlSi + Xid' be absolute parallel:
The h-connection D is called with "absolute parallelism of vectors if, (11.10) is verified for any vector X. It follows: Theorem 4.11.2. The N-linear connection D is with the absolute parallelism of vectors, if and only if the curvature of D vanishes, i.e., we have (11.11)
^ 4 = 0, P f c ? = 0, 5fc«* = 0.
Definition 4.11.1. The curve 7 : / -> T*M is called autoparallel curve with respect to the N-linear connection D if Djj = 0. Using (11.2) it follows m 12)
( dt ~\dt*
+
dt
dtj'ydt
6+(
The previous formula leads to the property:
dt
dt dt
Ch.4. The geometry of cotangent bundle
115
Theorem 4.11.3. The curve (11.1) is autoparallel with respect to the N-connection D if and only if the functions x'(t),pi(t), t £ I, are the solutions of the following system of differential equations: Ml 13) dt2
+
dt dt
= 0
' dt dt
dt
dt
= 0
Starting from (11.13) we can ennounce a theorem of existence and uniqueness for the autoparallel curve can be formulated as in the classical manner. dj
In Section 5, we introduced the notion of horizontal curve, the condition -7- = dit "7 \ -7• Theorem 4.5.1 gives us a characterization of the horizontal curves by means dt J of the system of differential equation (5.9), i.e. x1 = x'(t), —^ = 0. dt Definition 4.11.2. An horizontal path of an N-linear connection D is a horizontal autoparallel curve with respect to D. Theorem 4.11.4. The horizontal paths of an N-linear connection D are characterized by the system of differential equations: , . d?x' ,„• , . dxj dxk dpi (11.14) ^() | Indeed, the equations (11.13) and -7 1 = lead to (11.14). dt A parametrized curve 7 : / —> T*M is called vertical in the point x0 £ M, if its tangent vector field 7 belongs to the vertical distribution V. That means 7 belongs to the fibre of T* M in the point Xo € M. Evidently, 7 is a vertical curve in the point x0 e M, has the equations (11.1) of the form (11.15)
xi =
xi,Pi=Pi(t),tel.
Definition 4.11.2. A vertical path in the point XQ € M is a vertical curve 7 in the point XQ which is autoparallel with respect to the N-linear connection D. Theorem 4.11.3 implies: Theorem 4.11.5. The vertical paths in the point x0 £ M with respect to the Nlinear connection D are characterized by the system of differential equations:
Of course, it is not difficult to formulate a theorem of existence and uniqueness for the vertical paths in T*M at the point x0 G M.
The Geometry of Hamilton & Lagrange Spaces
116
4.12
Structure equations of an N-linear connection. Bianchi identities
Let us consider an N-linear connection D with the coefficients DT{N) = (Hljk,Cijk). Its 1-form of connection LJ'J are given by (11.6).
=
Proposition 4.12.1. With respect to a change of coordinate (1.1) on the manifold T*M we have ax
(12.1)
i
_< _ dx ~~ dx7
WJ
°X dx r dxs d2x{ U + ~dx~i " ~dx~i dxsdxr s
r
X
Indeed, the expression of o/;- from (11.6) and the rules of transformations of and Wjk, djk leads to (12.1). Now, it is not difficult to prove:
dx\Spi
Lemma 4.12.1. The following geometrical object fields d(dx') - dxmAuim,
dSpi + <5pmAwmi and dujlj - wmJ-Awim
are a d-vector field, a d-covector field and a d-tensor field of type (1, 1), respectively. Using this lemma, we can prove by a direct calculus a fundamental result. Theorem 4.12.1. For any N-linear connection D with the coefficients DT(N) = (Hljk,Cijk) the following structure equations hold: d{dxl) - dxmAijim
= -fi'
(12.2)
where fil, f2j are the 2-forms of torsion:
(12.3)
Q" = -Tjkdx3Adxk \ 0, = -Riikdx3Adxk
+
CjikdxjASpk
+ Pki3dx1A8pk
+
and where Cl'j is the 2-form of curvature: (12.3)'
SVj =
l
-R3lkmdxkAdxm
+ PiikmdxkA8pm
+ ^S
Ch.4. The geometry of cotangent bundle
117
Proof. £2* = dxmAu)lm and (11.6) imply the first equation (12.2); n{ = = -(d(Spi) + 8pmAujmi) and (4.11) lead to the second equation (12.2). Finally, the formulae (11.6), (4.11) dojlj = dH\mAdxm + dCjimA5pm + Cjmd(Spm), 3
3
dH'jm = 8sHjmdx + d H)Jps,
m
and
m
dC} = 8sC} dx3 + dsC*m8Ps q.e.d.
give us the last equation (12.2), with Olj from (12.3)'.
Remark. The previous theorem is extremely important in the geometry of the manifold T*M and, especially in a theory of submanifolds embedded in T*M. Now we remark that the exterior differential of the system (12.2), modulo the same system determines the Bianchi identities of an N-linear connection DT(N) = Theorem 4.12.2. The Bianchi identities of an N-linear connection DT(N) are as follows: S {T\T]S + T\mTmrs - &„ + RmrhCr) S {Skk't-SkhmSm"
(12 4)
=0
+ Skh"} = 0
S {RjTh\s + TmhrRjms + PmjSRmrh)
=0
{h,r,s)
S {R/hrU + Rj\mTmTS + Pj\mRmsr} = 0
(h,r,s)
(12.5)
5 {Sjihr\3 + A {Ph\T + ChmT\m + C, ir | s - ChimPrms} - T\sf
+ TmhsCmir = 0
(h,s)
(12.6)
^ % ^ W
+^
m
^
=0
Rk\s + Rksrf + SkmhRmsT - P\nTmrs
+ A {RksmCTmh(•>••')
— Ph, i — Ph Pm, \ — ft * kr\s * mr* ksj — u
The Geometry of Hamilton & Lagrange Spaces
118
A { A \ r - P\s\ ~ PhkmCsmr + SkhmPrm3}+ i pm c rh _ crh _ n 'r ks^m ^k \s — u>
i-i Q y\
pi |fc _ p i kfrn i c »fcm p i •n-j ra| -"j m -* r i T " j "msr T
+ A {Rj\nfirmk + P/Ar + P^P'rus} = 0, (r,s)
fl2 8)
S lSk
i \r + Pi\mSmkS + A {PjVf + Pfm'Cr"1" + SjismPkmr)
= 0-
In the applications we will consider the cases: a. Tjk = 0, Sjk = 0 (r) .,
b. T*iit = 0, d>k = 0. etc.
Of course, S is the symbol of cyclic sum and A is the symbol of alternate {h,r,s) (r,s) sum.
Chapter 5 Hamilton spaces Based of the conception of classical mechanics, the notion of Hamilton space was defined and investigated by R. Miron in the papers [97], [101], [105]. It was studied by D. Hrimiuc and H.Shimada [63], [66] et al. The geometry of Hamilton spaces can be studied using the geometrical method of that of cotangent bundle. On the other hand, it can be derived from the geometry ofLagrange spaces via Legendre transformation, using the notion of -duality. In this chapter, we study the geometry of Hamilton spaces, combining these two methods and systematically using the geometrical theory of cotangent bundle. We start with the notion of generalized Hamilton space. And then we detect from its geometry the theory of Hamilton spaces.
5.1
The spaces GH n
Definition 5.1.1. A generalized Hamilton space is a pair GHn = (M,gli(x,p)), where M is a real n-dimensional manifold and g^{x,p) is a d-tensor field of type (2.0) symmetric, nondegenerate and of constant signature on T*M. The tensor gij is called as usual the fundamental (or metric) tensor of the space GHn. In the case when the manifold M is paracompact, then there exist metric tensors g^(x,p) positively defined such that (M, gtj) is a generalized Hamilton space. Definition 5.1.2. A generalized Hamilton space GHn = (M,gtj(x,p)) is called reducible to a Hamilton one if there exists a Hamilton function H(x, p) on T*M such that (1.1)
5«
= \
{
119
The Geometry of Hamilton & Lagrange Spaces
120 Let us consider the d-tensor field
Cljk =
(1.2)
--dkgtJ.
In a similar manner in the case of generalized Lagrange spaces (cf. Ch.3) we can prove: Proposition 5.1.1. A necessary condition that a generalized Hamilton space be reducible to a Hamilton one is that the d-tensor Cljk be totally symmetric. Theorem 5.1.1. Let gij{x,p) be the fundamental tensor of a space 0 -homogeneous. Thus a necessary and sufficient condition that GHn be reducible to a Hamilton space is that the d-tensor field C y * be totally symmetric. Indeed, in this case the Hamilton function H(x,p) = glj(x,p)piPj satisfies the conditions imposed by Definition 5.1.2. Remarks. 1. The definition of Hamilton spaces is given in a next section of this chapter. 2. Let jij(x) be a Riemann metric tensor. It is not difficult to prove that the space GHn with the fundamental tensor gi3(x,p) = e-2aMyij(x),
a e T{T*M)
is not reducible to a Hamilton space. The covariant tensor g^ determined by gtj is obtained from the equations (1.3)
gikgkj =
Let us consider the following tensor field: (1.4)
CJ* = -Igisi&g** +
Proposition 5.1.2. The d-tensor Cjk gives the v-coefficients of a v-covariant derivation with the property: (1.5)
gijf = dkg'j + Cikgsj + C{kgis = 0
The proof can be obtained easily. We use the coefficients (1.4) in the theory of metrical connections with respect to the fundamental tensor g'J.
Ch.5. Hamilton spaces
5.2
121
N-metrical connections in GHn
In general, we cannot determine a nonlinear connection from the fundamental tensor
{8k = dk +
Nkj&,di}.
And an N-linear connection D has the coefficients DV(N) =
(Hljk,C^k).
Definition 5.2.1. An N-linear connection DT(N) is called metrical with respect to the fundamental tensor glj of the space GHn if: (2.2)
0
In the case when gl* is positively defined we obtain the geometrical meaning of the conditions (2.2). In this respect we can consider "the length" of a d-covector uii(x,p), given by \\w\\ as follows IMI2 = gii(x,p)wiwv
V(x,p) € f^M.
We can prove without difficulties the followings: Theorem 5.2.1. An N-linear connection DF(N) is metrical with respect to the fundamental tensor g%i of the space GHn if, and only if, along any smooth curve 7 : / —> T*M and for any parallel d-covector field uii: i.e., —— = o) we have ,., I, dt d\\\\ Using the condition (1.3), the tensorial equations (2.2) are equivalent to: (2-2)' Now, by the same methods as in Ch.3 we can prove a very important theorem. We have: Theorem 5.2.2. 1) There exist an unique N-linear connection DT(N) = (H'jk, Cijk) having the properties: 1° The nonlinear connection N is a priori given. 2° DV(N) is metrical with respect to the fundamental tensor g^ of the space GHn.
The Geometry of Hamilton & Lagrange Spaces
122
3° The torisons T%jk and 5, jfc vanish. 2) The previous connection has the v-coefficients Cjk from (1.4) and the h-coefficients Hljk expressed by Wjk = l-gis(5i9ak + 5k9js -
(2.3)
6sgjk).
For a generalized Hamilton space GHn ~ (M,
(2-4)
n# = i ( ^ +
ghkg»).
We can prove: Theorem 5.2.3. The set of all metrical N-linear connections DV{N) = with respect to the fundamental tensor gtj is given by
wAere Dr(iV) = {W^Ci*) fields.
is from (1.4), (2.3) andX{jk, Zjk
(Htjk,Ciik)
are arbitrary d-tensor
It is important to remark that the mappings DT{N) —>• DY(N) determined by (2.5) and the compsotion of these mappings is an Abelian group. From the previous theorem, we deduce: Theorem 5.2.4. There exist an unique metrical N-linear connection DV(N) — (H j k , Ci*k) with respect to the fundamental tensor g%:> having a priori given torsion d-tensor fields T1^ (= - 2 % - ) , S^k (= -Sikj). The coefficients of DT(N) have the following expressions: (2.6)
\
2
5kgjs - 6s9jk) + \gia{g3hThik 22
- gjhThsk +
gkhThjs)
+ 5V* - da9
Moreover, if we denote Rhljk = ghsRsljk, etc. and apply to fundamental tensor g' the Ricci identities taking into account the equations (2.2) we obtain: j
Proposition 5.2.1. For any metrical N-linear connections DY(N) with respect to the fundamental tensor glj of the space GHn the following identities hold:
Ri]Hk + Rj\k = 0, P V + P ' V = 0, S«** + S*** = 0 Rj\k + Rj'kh = 0, Siihk + SSkli = 0.
Ch.5. Hamilton spaces
123
The expressions of the curvature tensors are given in the formulae (10.3), Ch.4. The notions of parallelism, horizontal or vertical paths, as well as the structure equations for an metrical N-linear connection with respect to the fundamental tensor glj can be studied, using the results obtained in the last section of Chapter 4.
5.3
The N-lift of GHn
Let GHn — (M,gli) be a generalized Hamilton space and N (iVy) an apriori given non-linear connection on the manifold T*M. Thus (<5t,<9') from (2.1) is an adapted basis to the distributions N and V. Its dual (dx',6pi) basis is expressed by the 1-forms dx% and by Spi = dp, -
(3.1)
Njtdxj
Definition 5.3.1. The N-lift of the fundamental tensor (3.2)
G = g{j{x,p)dxi
is:
® dxj + gij{x, p)Spi
Theorem 5.3.1. We have: 1° The N-lift G is a tensor field of type (0,2) on
symmetric, nonsingular
depending only on g'i and on the nonlinear connection N. 2° The pair (T*M, G) is a pseudo-Riemannian space. 3° The distributions N and V are orthogonal with respect to G. Indeed, every term from (3.2) is defined on T*M and is a tensor field. In the adapted basis (2.1) the tensor field has the components (3.3)
G ( M ; ) = Sy, Gft.d') = 0, G ( # , # ) = gij.
It follows that G is symmetric, nondegenerate of type (0,2) tensor field. Obviously wehave G(X H ,Y v ) = 0, VXH,Yv. q.e.d. Now, assuming that the nonlinear connection TV" i s symmetric, i.e. Ntj — = NjU we consider the F{f*~M) -linear mapping F : X(f*M) —> X(f*M)defined in (6.7), Ch.4 (3.4)
The Geometry of Hamilton & Lagrange Spaces
124 Theorem 5.3.2.
1° F is globally defined on T*M. 2° F is the tensor field F = -gtjd1 ® dxj + gijS, ® Sp,. 3° F is an almost complex structure determined by the fundamental tensor g%i and by the nonlinear connection N. 4° The pair (G, F ) is an almost Hermitian structure determined only by g'^ and N. 5° The associated almost symplectic structure to (G, F ) is the canonical symplectic structure 9 = 6piAdx% = dpiAdx1. It follows that the space (f*M, (G, F ) ) is almost Kählerian. It is called the almost Kählerian model of the generalized Hamilton space GHn.
5.4
Hamilton spaces
A Hamilton space Hn = (M,H(x,p)) is a particular case of a generalized Hamilton space GHn — (M,g'j) in the sense that the fundamental tensor derived from a regular Hamilton function H :T*M —> R. Since the triple (T*M, 6, H) forms a Hamiltonian system, we can apply the theory from Chapter 4. Definition 5.4.1. A Hamilton space is a pair Hn = (M, H(x, p)) where M is a real n-dimensional manifold and H is a function on T*M having the properties: 1° H : (x,p) e T*M —> H{x,p) e R is differentiable on the manifold f*M and it is continuous on the null section of IT* : T*M —> M. 2° The Hessian of H (with respect to the momenta ) , given by the matrix ||<7lJ(:r, y)\\ is nondegenerate: (4.1)
g^(x, y) = -d^H,
rank||^(x,p)| = n on f^M.
3° The d-tensor field gli{x,p) has constant signature on T*M.
Ch.5. Hamilton spaces
125
Of course, gli from (4.1) is a d-tensor field, cf. §5, Ch.4. It is called the fundamental tensor, or metric tensor of the space Hn = (M, H) and the Hamilton function H is called the fundamental function for Hn. From the previous definition we obtain: Theorem 5.4.1. space.
Every Hamilton space Hn = (M, H) is a generalized Hamilton
Indeed GHn = (M, , where g%j is given by (4.1), is a generalized Hamilton space. The converse is not true (see Proposition 5.1.1). Theorem 5.4.2. If the base manifold is paracompact, then there exists a Hamilton function H on T*M, such that Hn = (M,H) is a Hamilton space. Indeed, M being a paracompact manifold, let gl*{x) be a Riemannian metric tensor on M. Then we can consider the Hamilton function on T*M: H { )
i
i
{ )
where m ^ 0, and c is speed light. The properties l°-3° from the last definition can be proved directly. q.e.d. Let us consider the canonical symplectic structure 6 on T*M: (4.2)
6 = dpiAdxi
By means of Definition 4.2.2, we obtain that the triple (T*M, 6, H), where H is the fundamental function of a Hamilton space Hn, form a Hamilton system. Thus the mapping Sg : X{T*M) —• X*{T*M) defined by (2.2), Ch.4, SB(X) = ix6is an isomorphism. Applying the Theorem 4.2.2 we obtain: Theorem 5.4.3. For any Hamilton space H" = (M,H(x,p)) ties hold:
the following proper-
1° There exists a unique vector field XH € X{T*M) with the property: (4.3)
iXH0 = -dH.
2° The integral curves of the vector field XH are given by the Hamilton-Jacobi equations:
The Geometry of Hamilton & Lagrange Spaces
126 (4.4)
dxl dt
dH dpi dpi dt
dH dxi
Indeed, from (2.6)', the Hamilton vector field XH can be written as: (4 5)
'
XH
dH
d dH d ~ ~dn~ M " a * dpi'
Also, we have: Corollary 5.4.1. The fundamental function H(x, p) of the space Hn is constant along the integral curve of the Hamilton vector field Xu. Now it is easy to observe that the following formula holds: (4.6)
{f,9}=0{Xf,Xg),
VX e X{T*M).
Therefore, by means of Poisson brackets, the Hamilton-Jacobi equations can be written in the form:
We remark that the Jacobi method of integration of Hamilton-Jacobi equations, mentioned in §2, Ch.4, works in the present case as well. One of the important d-tensor field derived from the fundamental function H of the Hamilton space Hn is: (4.7)
* = -!-&& &H =
-\frji.
Proposition 5.4.1. We have: 1° The d-tensor field Cw'fc is totally symmetric. 2° C'jk vanishes, if and only if the fundamental tensor field gli(x,p) does not depend on the momenta Pi. Let us consider the coefficients CV* from (1.4) of the v-covariant derivation. From (4.7) it follows (4.8)
CS = i
Of course, these coefficients have the properties (4.9)
51JT = 0, Sijh = 0.
In the next section we will use the functions Cjk as the v-coefficients of the canonical metrical connection of the space Hn.
Ch.5. Hamilton spaces
5.5
127
Canonical nonlinear connection of the space
An important problem is to determine a nonlinear connection TV (/%) for a Hamilton space Hn = (M,H(x,p)) depending on the fundamental function H, only. A method for finding a nonlinear connection with the mentioned property was given by R. Miron in the paper [97]. This consists in the transformation of the canonical nonlinear connection N of a Lagrange space Ln = (M, L(x,y)), via Legendre transformation: Leg: L" —> Hn, into the canonical nonlinear connection of the Hamilton space Hn. This method will be developed in the Chapter 7, using the notion of -duality between the spaces Ln and Hn. Here, we give the following result of R. Miron, without demonstration, which can be found in §2 of Chapter 7. We have:
Theorem 5.5.1. 1° The following set of functions (5.1)
Nrj = ~{9i}, H) - determines the coefficients of a nonlinear connection N of the Hamilton space Hn.
2° The nonlinear connection with the coefficients N,j, depends only on the fundamental function H of the Hamilton space Hn. The brackets { } from (5.1) are the Poisson brackets (2.1), Ch.4. Indeed, by a straightforward computation, it follows that, under a coordinate change on the total space of cotangent bundle T*M, JVy from (5.1) obeys the rule of transformation (4.3), Ch.4. Hence, the point 1° of Theorem 5.5.1 is verified. Taking into account the expressions of the coefficients iVy, given in (5.1), the property 2° is also evident. The previous nonlinear connection will be called canonical. Remark. If the fundamental function H(x,p) of the space Hn is globally defined on T*M, then the canonical nonlinear connection has the same property. Proposition 5.5.1. The canonical nonlinear connection N of the Hamilton space Hn = (M,H(x,p)) has the following properties (5.2)
Ty = i ( W y - J V , - 0 = 0
(5.3)
Rljk + Rjki + Rkl] = 0
The Geometry of Hamilton & Lagrange Spaces
128 where Rijfl is given by (4.12), Ch.4.
Proof. 1° The proof follows directly by (5.1); 2° is a consequence of the formula (4.12), Ch.4. q.e.d. Taking into account the fact that the canonical nonlinear connection N is a regular distribution on T*M with the property: Tu'Md = ATU © F u , Vu 6
(5.4)
f*M,
it follows that {Si,d') is an adapted basis to the direct decomposition (5.4), where 5( = % + Nyd*.
(5.5)
The dual basis of (6i,dl) is (dx',6pi), where
5pi =
(5.5)'
dpi-Njidxj.
Therefore, we can apply the theory of N-metrical connection expounded in §2 of the present chapter for study the canonical case.
5.6
The canonical metrical connection of Hamilton space Hn
Let us consider the N-linear connections DT(N) = (Hljk,Cijk) with the property that N is the canonical nonlinear connection with the coefficients (5.1). It can be studied by means of theory presented in §5.2. Consequently, we have: Theorem 5.6.1. 1) In a Hamilton space Hn = (M, H(x,p)) there exists a unique N-linear connection DT(N) = (Hijk,Ci>k) verifying the axioms: 1° N is the canonical nonlinear connection. 2° The fundamental tensor <7tJ is h-covariant constant ji]k
(6.1) tj
3° The tensor g (6.2) 4° DT(N)
= 0.
is v-covariant constant, i.e. ff«f
= 0.
is h-torsion free, i.e. T'jk
= Hljk — H'kj
= 0.
Ch.5. Hamilton spaces 5° DT(N)
129
is v-torsion free, i.e. V* = Cijk - dkj
= 0.
2) The connection DT(N) has its coefficients(Nij, Hljk,Cijk) by the following generalized Christoffel symbols: Ri
ik = «9is{6j9sk + h9js -
(6.3)
2
3
gisi&g i& "
hj
+ dg d
°
given by (5.1) and
Ssgjk), k
&) &•?")
3) This connection depends only on the fundamental function H of the Hamilton space Hn. The proof of this theorem follows the ordinary way. A such kind of connection, determined only by the fundamental function H is called canonical and is denoted by CY{N) = (Jf^, Cjk) or by CT = {!*&, CV*). Now we can repeat Theorems 5.2.3 and 5.2.4, using the canonical metrical connection CT. Proposition 5.6.1. The Ricci identities, with respect to the canonical connection CT are given by (6.4) where the d-tensors of torsion are T'jk = 0, S^1* = 0, Cjk and (6.5)
Rijh = 6hNj{ - SjNhi, Phij = Hhij -
d^N^
respectively, and the d-tensors of curvature of CT are given by (10.3), Ch.4. The Ricci identities can be extended as usual to any d-tensor field. For instance, in the case of a d-tensor field t^(x,p) we have
(6.4)'
*«ifcf - tt^\k ^
=
tmjPm\h
PmSmjkh.
Applying the identities (6.4)' to the fundamental tensor gtJ of the Hamilton space H we obtain n
The Geometry of Hamilton & Lagrange Spaces
130
Proposition 5.6.2. The canonical connection CT has the properties (2.7), i.e. (6.6)
Rlihk + R'\k = 0, P V + Pjihk = 0, Sijhk + Silhk = 0
The Ricci identities applied to the Liouville-Hamilton vector field C* = p,dl lead to some important identities. Proposition 5.6.3. The canonical metrical connection CT of the Hamilton space Hn = (M, H) satisfies the following identities ik\j — i.k
it
_
Pm*H pm
„
^jlfc _ ^.k|7 : _ _ „
jk k
°i *Wnjk A
Q mk
impk
Q.mjk
where Ay- and 5k are the deflections tensors of CT(N): (6.7)'
Aij=PiU;
^=Pjr
are given by (9.9), Ch.4. The canonical metrical connection CT is of Cartan type if Ay- = 0, 3j• = Jj. From (6.7) we obtain the following Proposition 5.6.4. If the canonical metrical connection of the space Hn is of Cartan type, then the identities (10.8), Ch. 4, hold good. Applying again the Ricci identities to the fundamental function H of the Hamilton space H", we obtain: Proposition 5.6.5. The following identities hold: (6.8)
Hi\^-Hfi\j = = 0.
5.7
Structure equations of CT(N). Bianchi identities
For the canonical connection CT = (//'•,*, Cjk) the 1-form of connection u'j are given by (11.6), Ch.4: (7.1)
w'j = H
Ch.5. Hamilton spaces
131
The structure equations of CT are given by Theorem 4.12.1. In this case, we have: Theorem 5.7.1. The canonial metrical connection CT of the Hamilton space Hn = = (M, H) has the following structure equations d{dx{) - dxmA(Jim = -fi*, d(6pi) + 8pmAuimi = —ft,,
(7.2)
du>lj — wmjAu}lrn
= — Qlj,
where fi\fij are the 2-forms of torsion: (7.3)
ft!
= CjikdxjA6pk,
Qi = -RijkdxjAdxk
+ Pk{
and where Qlj are the 2-forms of curvature: (7.4)
ft*,-
= -R/kmdxkAdxm
+ PjikmdxkA6pm + -S/k
The previous theorem is very useful in the geometry of Hamilton spaces and especially in the theory of subspaces of Hamilton spaces. We can now derive the Bianchi identities of the canonical metrical connection CT, taking the exterior differential of the system of equations (7.2), modulo the same system. We obtain a particular case of Theorem 4.12.2: Theorem 5.7.2. The canonical metrical connection CT of the Hamilton space Hn satisfies the Bianchi identities(12.4)-(12.8), Ch.4, with Tjk = 0, SSk = 0.
5.8
Parallelism. Horizontal and vertical paths
The notion of parallelism of vector fields in the Hamilton spaces Hn = (M, H(x,p)), endowed with the canonical metrical connection CT = (iVy, H%jk, Cjk) can be studied as an application of the theory presented in §11, Chapter 4. Let 7 : I —> T'M be a parametrized curve with the analytical expression (8.1)
x* = xl{t),
Pi
= pi(t), t e /, rank||ft(*)|| = 1.
Then it results
cty
dx\
5pi-{
The Geometry of Hamilton & Lagrange Spaces
132
where
£ i
dt~
1
dt~
N
Njt
^
dt •
Here ATy are the coefficients (5.1) of the canonical nonlinear connection. We say OtJ'
that the curve 7 is horizontal if = 0,Vt€/. For a vector field X on TM, given in adapted basis by
X = X% + Xidi, along of the curve 7, we have the covariant differential D of CT of the form (11.7), Ch.4: DX
where LJ'J are 1-forms connection of CF. It follows: Theorem 5.8.1. The curve 7, with respect coordinates X' and Xi ujlj{x{t),p{t)) are the
vectorfield = X'Si + Xid' is parallel along the parametrized to the canonical metrical connection i f ;/ and only if its are solutions of the differential equations (11.8), Ch.4, where 1-forms connection of CT'.
In particular, (see Theorem 4.11.2), we have: Theorem 5.8.2. The Hamilton space Hn = (M, H), endowed with the canonical metrical connection CY, is with absolute parallelism of vectors if and only if the d-curvature tensors of CT Rtjkh^ Pj'kh and Sijkh respectively vanish. Dj We say that the curve 7 is autoparallel with respect to CT if —— = 0, W € / . dt Taking into account Theorem 4.11.3, we have Theorem 5.8.3. A curve 7 : / —> T*M is autoparallel with respect to the canonical metrical connection CT if and only if the functions (x'(t),pi(t)), t £ I are the solutions of the system of differential equations (11.13), Ch. 4, in which OJ1J are the 1 -forms connection of CT. By means of Definition 4.11.2, a horizontal path of the canonical metrical connection CT of Hamilton space Hn, is a horizontal autoparallel curve. Theorem 4.11.4 gives us:
Ch.5. Hamilton spaces
133
Theorem 5.8.4. The horizontal paths of the Hamilton space Hn, endowed with the canonial metrical connection CT are characterized by the system of differential equations:
(8.4)
% + HWt),Plt))^
^ = 0, f - N , ^ ) ^ - 0
W e recall t h a 7 : t € / —> i ( t ) £ T*M is a vertical c u r v e i n t h e p o i n t i j E M dz' if —— = 0. Hence, its analytical representation in local coordinates is of the form at x' = xl, Pi=Pi(t),
tel.
Thus, a vertical path in the point x0 e M is a vertical curve 7 in the point x0 which is autoparallel with respect to CT. Theorem 4.11.5 leads to: Theorem 5.8.5. The vertical paths in the point XQ € M with respect to the canonical metrical connection CV of the Hamilton space Hn = (M, H) are characterized by the system of differential equations x
~x°> dT
In next section we apply these results in some important particular cases.
5.9
The Hamilton spaces of electrodynamics
Let us consider some important examples of Hamilton spaces. 1) Gravitational field The Lagrangian of gravitational field L = mc^ij(x)ylyj (see Ch.3) is transformed, via Legendre transformation ,in the regular Hamiltonian H=—^(x)PiPj.
(9.1)
TTIC
n
Therefore the pair H = (M, H) is a Hamilton space. Its fundamental tensor field is given by (9.2)
fl«
= —7y(x). me
The Geometry of Hamilton & Lagrange Spaces
134 It follows that (T*M.0,H) obtain
is a Hamiltonian system. Using Theorem 4.2.2, we
Proposition 5.9.1. The Hamilton- Jacobi equations of the space Hn, with the fundamental function (9.1), are:
Let us denote Yjk(x) obtain:
the Christoffel symbols of the metric tensor ^{x)
we
Proposition 5.9.2. The canonical nonlinear connection N of the space Hn = (M, H), (9.1) has the following coefficients
Nii(x,y)=Tkij(x)pk.
(9.4)
Indeed, the coefficients iVy are given by the formula (5.1). Therefore (9.4) holds. Proposition 5.9.3. The canonical metrial connection has the coefficients
of the Hamilton space Hn
Wjk = -fit, Qjk = 0.
(9.5)
Now we can apply to this case the whole theory from the previous sections of this chapter.
2) The Hamilton space of electrodynamics The Lagrangian of electrodynamics (9.1), Ch.3, is transformed via Legendre transformation in the following regular Hamiltonian (9.6)
H(x,p) = ±-Jij(x)PiPj TrtC
~ ~2Al{x)Pi TTt/C
+
—3Ai(x)Ai(x), /TIC
where m, c, Yj(x) have the meanings from (9.1), e is the charge of test body, Ai(x) is the vector-potential of an electromagnetic field, and A1 (x) = jx:i(x)Aj(x). The first thing to remark is: Proposition 5.9.4. The pair Hn = (M, H), (9.6) is an Hamilton space.
Ch.5. Hamilton spaces
135
Indeed, a direct calculus leads to the d-tensor gV = - U y -
(9.7)
TTbC
This is the fundamental tensor the space Hn. Its covariant is gij(x) = rrurfij(x). The Hamilton-Jacobi equations, (2.2), Ch.4, can be easily written. The velocity field of this space can be written: (9.8)
,
*tf
=
^
) P i
And the canonical nonlinear connection is given by Proposition 5.9.5. The canonical nonlinear connection of the space Hn with the Hamiltonian (9.6) has the following coefficients: Nij = jkij{x)Pk
(9.9) where Ai]k = - ^
-
+ ^(Ai[k +
Ak]i),
Asyik.
Indeed, to prove the formula (9.9) we apply the formula (5.1) to the fundamental function (9.6). Now, r e m a r k i n g that -z-% = mc-—^,K we get: oxK ox P r o p o s i t i o n 5 . 9 . 6 . The canonical metrical connection CT of the Hamilton space Hn with the fundamental function H(x,p) with the Hamiltonian (9.6), has the following coefficients: H'jk = fjk, Ct* = 0.
These geometrical object fields: H, gij, N^, Hljk, CVfc allow to develop the geometry of the Hamilton space of electrodynamic. 3) In the case when we take into account the Lagrangian 2e t :i
L(x,y) = mcjlj{x)y y
H
TTti
Ai(x)y' + U(x),
where U(x) is a force function, applying Legendre mapping we get the following Hamiltonian 1 2e j l H{x,p) = f ( ) A (jp-pi) +
2 C 3 f( m + +->\
V
c2,
The Geometry of Hamilton & Lagrange Spaces
136 It follows
Proposition 5.9.7. The pair Hn = (M, H) with the Hamiltonian (9.10) is a Hamilton space, having the fundamental tensor field: g*> = c22j
(m+
V
We can prove also: Proposition 5.9.8. The canonical nonlinear connection of the space Hn, (9.10) has the coefficients
Proposition 5.9.9. The canonical metrical connection of the space Hn, (9.10), has the coefficients CT = (#Vjfc, C,Jk): H% -u — V
11
-L
jk — ; jki
C-3* — 0 ^i
—
u
-
Now the previous theory of this chapter can be applied to the Hamilton space Hn, (9.10).
5.10
The almost Kählerian model of an Hamilton space
Let H" = (M, H(x,p)) be a Hamilton space and glJ'{x,p) its fundamental tensor field. The canonical nonlinear connection N has the coefficients (5.1). The adapted basis to the distributions ./V and Vis ( - r - = £ = <% + Nad3, d' I and its dual basis
\5x%
sis (dx\ dp, = dpi — Nji Thus the following tensor on T*M: (10.1)
J
J
G = gtj {x, p)dxi ® dxj + gij (x,
gives a pseudo-Riemannian structure on T*M, which depends only on the fundamental function H(x, p) of the Hamilton space Hn. These properties are the
Ch.5. Hamilton spaces
137
consequences of Theorem 5.3.1 and of the fact that Ny is the canonical nonlinear connection. The tensor G is called the N-lift of the fundamental tensor glK The distributions N and V are orthogonal with respect to G, because the formulae (3.3) hold. Taking into account the mapping F : X(T*M —> X{T*M)defined in (3.4) with respect to the canonical nonlinear connection, we obtain the following properties: 1° F is globally defined on
f*M.
2° F is an almost complex structure: F o F = -I
n T*M.
3° F is determined only by the fundamental function H(x, p) of the Hamilton space Hn. Finally, we obtain a particular form of Theorem 5.3.2:
Theorem 5.10.1. 1° The pair ( G , F ) is an almost Hermitian structure on the manifold T*M. 2° The structure ( G , F ) is determined only by the fundamental function of the Hamilton space Hn.
H(x,p)
3° The associated almost symplectic structure to (G, F ) is the canonical symplectic structure 6 — dpiAdx1 = SpiAdx'. 4° The space (f*M, G, F ) is almost Kählerian. The proof is similar with that from Lagrange spaces (cf. Ch.3). The equality SpiAdxi = (dp, - Njidx:')Adxi = dpiAdxi = 6 follows from the fact that the torsion r^ = -(A/y- - Nji) of the canonical nonlinear connection vanishes (cf (5.1)). _ The space (T'M, G, F ) is called the almost Kählerian model of the Hamilton space Hn. This model is useful in applications.
Chapter 6 Cartan spaces The modern formulation of the notion of Cartan spaces is due of the first author [97], [98], [99]. Based on the studies of E. Cartan, A. Kawaguchi [75], H. Rund [139], R. Miron [98], [99], D. Hrimiuc and H.Shimada [66], [67], P.L. Antonelli [21], etc., the geometry of Cartan spaces is today an important chapter of differential geometry. In the previous chapter we have presented the geometrical theory of Hamilton spaces Hn = (M,H(x,p)). In particular, if the fundamental function H(x,p) is 2-homogeneous on the fibres of the cotangent bundle (T*M, TT*, M) the notion of Cartan space is obtained. It is remarkable that these spaces appear as dual of the Finsler spaces, via Legendre transformation. Using this duality several important results in the Cartan spaces can be obtained: the canonical nonlinear connection, the canonical metrical connection etc. Therefore, the theory of Cartan spaces has the same symmetry and beauty like Finsler geometry. Moreover, it gives a geometrical framework for the Hamiltonian theory of Mechanics or Physical fields.
6.1
The notion of Cartan space
As usually, we consider a real, n-dimensional smooth manifold M, the cotangent bundle (T*M, M) and the manifold T*M = T*M\{0}. Definition 6.1.1. A Cartan space is a pair Cn = (M,K(x,p)) lowing axioms hold:
such that the fol-
1° K is a real function on T*M, differentiable on T*M and continuous on the null section of the projection n*. 2° K is positive on T*M. 3° K is positively 1-homogeneous with respect to the momenta Pi4° The Hessian of K2, with elements 139
The Geometry of Hamilton & Lagrange Spaces
140
g»(X,p) = lM#
(1.1) is positive-defined.
It follows that gli{x,p) is a symmetric and nonsingular d-tensor field, contravariant of order 2. Hence we have rank|<^(z,/>)||=n, on f^M.
(1.2)
The functions g'j(x,p) are 0-homogeneous with respect to the momenta pi. For a Cartan space Cn = (M,K(x,p)) the function K is called the fundamental function and gij the fundamental or metric tensor. At the beginning we remark: Theorem 6.1.1 If the base manifold M is paracompact, then on the manifold T*M there exist functions K such that the pair (M, K) is a Cartan space. Indeed, if M is paracompact, then T*M is paracompact too. Let g*j(x,p) be a Riemann structure on M. Considering the function K(x,p) = {gij(x)PiPjy/2
(1.3)
on T*M
we obtain a fundamental function for a Cartan space C". Examples. 1. Let (M,
) be a Riemannian manifold and
Assuming /3 > 0 on an open set U C T*M it follows that (1.4)
K(x,p)=a*+p*
(1-4)'
K(x,p) =
^
are the fundamental functions of Cartan spaces. The first one, (1.4), is called Randers metric and the second one, (1.4)´, is called the Kropina metric. They will be studied in Chapter 7. Remark. More generally, we can consider the Cartan spaces with (a*,/?*)-metric. They are given by the definition 6.1.1, with K(x,p) = K{a*{x,p), /3*(x,p)), if being a function 1-homogeneous with respect to a nd /3*.
141
Ch.6. Cartan spaces 2. The pair Cn = (M, K(p)), where (1.5)
K(p) = {(Pl)m
+ ( P 2 ) m + • • • + ( p m ) m } 1 / m , (m = 2r,r>
1).
in the preferential charts of an atlas on T*M, is a Cartan space. The function (1.5)'
K(x,p) = e"^K(p)
and K(p) from (1.5) is the so-called Antonelli ecological metric [11]. A general remark is imporant: Theorem 6.1.2. Every Cartan space Cn — (M,K(x,p)) Hamilton space: Hm = {M,K2(x,y)).
uniquely determines a
Indeed, by means of the axioms I ' M 0 from Definition 6.1.1 it follows that the pair (M , ) is a Hamilton space. We can apply the theory from previous chapter. So, considering the canonical symplectic structure 6 on T*M: (1.6)
8 = dpiAdx*
we deduce that the triple (T*M, ) is a Hamiltonian system. We can apply Theorems 4.4.3 and 5.4.3. Therefore we can formulate: Theorem 6.1.3. For any Cartan space Cn = (M,K(x,p)), hold: 1° There exists a unique vector field XK* G X(T*M) (1.7)
(
the following properties with the property
iXK2e = -dK2.
2° The integral curves of the Hamilton vector field XK2 are given by the Hamilton-Jacobi equations dx^^OK^ dPi = dK2 ' ' dt dpt' dt dx* ' Indeed, XKi is of the form
^ _ dK2 d dpi dx* and (1.8) gives us its integral curves.
dK2 d dx' dpi
Corollary 6.1.1. The function K2 is constant along the integral curves of Hamilton vector field XKi. The Hamilton-Jacobi equations (1.8) of the Cartan space Cn are fundamental for the geometry of C". They are the dual of the Euler-Lagrange equations of the Finsler space. Therefore (1.8) are called the equations of geodesies of the Cartan space C". This theory will be developed in the next chapter.
The Geometry of Hamilton & Lagrange Spaces
142
6.2
Properties of the fundamental function K of Cartan space Cn
Proposition 6.2.1. The following properties hold: 1° p' — -d'K2 2°
is 1-homogeneous d-vectorfield on T*M.
i s &>p' 0 - ' K
3° Cijk = --didi^K'i
2
is
homogeneous d-tensor field.
is (-1)-homogeneous, symmetric d-tensorfield.
Indeed, iff(x,p) is r-homogeneous with respect of p i ; then -— =
is r- 1-
OPi
homogeneous. Therefore l°-3° follows. Let gij be the covariant tensor of gij, i.e.: Proposition 6.2.2. We have the following formulae: (2.1) jf = 9'3Pj, Pi - gap' (2.2) K2 = gVpM = Pip> (2.3)
C^PK = CikiPk = C*hk = 0.
These formulae are consequences of l°-3° from the previous proposition. The fundamental tensor gx'(x,p) depends only on the point x = ir*(x,p) 6 M if and only if dhgij = 0. In this case the pair (M, ) is a Riemannian space. So we have Proposition 6.2.3. The Cartan space Cn = (M,K(x,p)) only if, the d-tensor field C':'k(x,p) vanished.
is Riemannian if, and
Indeed, frg^ = -2Cijh = 0 holds, if and only if gij(x,p) = gij(x). Let us consider the coefficients Cjk from (1.4), Ch.5, of the v-covariant derivation. They are given by (4.8) Ch.5 for a Cartan space, i.e.: (2.4)
C^ =
Naturally, these coefficients have the following properties: (2.5) g^ = 0, 5,J* = 0 (2.6) plf = 6l«. Other properties of the fundamental function K of the Cartan space C" will be expressed by means of the canonical nonlinear connection and canonical metrical connection.
Ch.6. Cartan spaces
6.3
143
Canonical nonlinear connection of a Cartan space
Since the Cartan space Cn = (M, K) is Hamilton space Hn — (M, K2), the canonical nonlinear connection of the space has the coefficients A^, from (5.1), Ch.5. If we consider the Christoffel symbols of gli(x,p) given by (3.1)
Yjk = \gih{dkghj
+ dj9hk -
dh9jk),
then the following contractions by Pi or pl: (3-2)
l% = YJkPi,l% = lljkPiPh
lead us to Theorem 6.3.1. (Miron [98], [99]) The canonical nonlinear connection of the Cartan space C" = (M, K) is given by the following coefficients Nv = 7° ~
(3-3)
\ll^9ty
Proof. By means of formula (5.1), with H = K2, and K2 = gijPiPj we obtain Ntj = 7?- + \d*gij But 7°0 = —-djgkhpkph- The two last equations imply the formula (3.3).
q.e.d.
Remark. The coefficients (3.3) can be obtained from the coefficients of the Cartan nonlinear connection of a Finsler spaces by means of the so called -duality (see Ch.7). Let us consider the adapted basis {Si,dl) to the distributions N and V where N is determined by the canonical nonlinear connection of the Cartan space Cn. Its dual basis is (dx',Spi). Then the d-tensor of integrability of N is (4.12), Ch.4: (3.4) By a direct calculus we have:
Rijh = 6 ^ - SjNhl.
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Proposition 6.3.1. In a Cartan space Cn the following properties hold: = l-{Nij - N3i) = 0
(3.5)
Tij
(3.6)
Rljk + Rjkl + Rklj = 0.
Proposition 6.3.2. The distribution N determined by the canonical nonlinear connection of a Cartan space Cn is integrable if and only if the d-tensor field Rijh vanishes. Other consequence of the previous theorem is given by Proposition 6.3.3. The canonical nonlinear connection of the Cartan space Cn depends only on the fundamental function K.
6.4
The canonical metrical connection
Let us consider the N-linear connection DV(N) = (Hljk,Cljk) of the Cartan space Cn = (M, K(x,p)) in which N is the canonical nonlinear connection, with the coefficients Nij from (3.3). The h- and v-covariant derivatives of the fundamental tensor 9lj = -drd^K2 of the space Cn are expressed by
(41)
\g»f
= frgv + gsjCslk + giaCjk
In particular, in the case of Cartan spaces, Theorem 6.6.1 implies: Theorem 6.4.1. 1) In a Cartan space Cn = (M,K(x,p)) there exists a unique N-linear connection CT(N) = {Hljk, C,jfc) verifying the axioms: 1° N is the canonical nonlinear connection ofth espace Cn, 2° The equation gij)k = 0 holds with respect to
CT(N),
3° The equation gi]f = 0 holds with respect to
CT(N),
4° CT(iV) is h-torsion free: T{jk = 0, 5° CT(iV) is v-torsion free: Stlk = 0.
Ch.6. Cartan spaces
145
2) The connection CF(N) has the coefficients given by the generalized Christoffel symbols: Hi
jk
(4.2)
= ^9 2
Si.(£y + &9js - 3V*) = S«
3) CF(N) depends only on the fundamental function K of the Cartan space Cn. The connection CT(N) from the previous theorem will be called the canonical metrical connection of the Cartan space Cn. The connection CF(N) is called metrical since the conditions g%i\k = 0, g^f = 0 hold good. But these two conditions have a geometrical meaning. ^__ Let us consider "the square of the norm" of a d-covector field on T*M: (4.3) and a parametrized curve c : t € I —> T*M, given in a local chart of T*M by x i = x i ( t ) , pi = pl(t),
t £ l .
We have the folowing Theorem 6.4.2.
An hi-linear connection
DT(N) = {Hijk,Cijk)
on T*M has
the property — — II AM = 0, along any curve c, and for any parallel covector field 2
\i(x(t),p(t))
on c, if and only if the following equations g%:!\k = 0, g'J\ = 0 hold.
Proof.
But Or, the curve c being arbitrary we get Dgij = 0 «=*> {gij\k — 0, gijf
= 0}.
Conversely, if g'h =
2
0, 5ijf=
= 0, then ^
- 0, ^ i = Oand (*) imply ^||A|| = 0. at at q.e.d. We will denote CT(iV) = (iFj*.CV fc )by CT = {Ni:j, Hijk,Ci>k),pointed out all coefficients of the canonical metrical connection. The first property of CT is as follows: dr
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Proposition 6.4.1. The coefficients {Nij,Hljk,Cjk) of the canonical metrical connection CT are homogeneous with respect to momenta pi, of degree 1,0,-1, respectively. Indeed, glj are 0-homogeneous imply 7 ^ are 0-homogeneous, 7^ are 1-homogeneous and, using (3.3), it results Ntj are 1-homogeneous etc. Proposition 6.4.2. The canonical metrical connection CT of the Cartan space Cn is of Cartan type. Namely its deflection tensors have the property
(4.4)
^ = ^ = 0, t>=Pi\i = 8i.
Proof. First of all we remark that the h-coefficients H*^ of CT give us (4-5)
PtH'jk
= 7°* -
l
-NOrdrgjk.
But NOj can be calculated from (3.3). We obtain NOj = 7 ^ . Therefore ™ = 7° - ^to^ga Similarly, % = 5) ~ pmCimj
- (-Tii - \N0hdh9ii)
= 0 q.e.d.
= S{.
The first equality (4.4) is an important one. It can be substituted with the axiom 1° from Theorem 6.4.1. We obtain a system of axioms of Matsumoto type (cf. 2.5, Ch. 2) for the canonical metrical connection ofthe Cartan space C". Theorem 6.4.3. 1) For a Cartan space Cn = (M, K(x,p)) there exists a unique linear connection DT = (Nij,Hljk,C^k) which satisfies the following axioms Ai A^ = 0 (h-deflection tensor field of DT vanishes) M 9l\k — 0 {DT is h-metrical) A3 gvf = 0 {DT is v-metricat) A4 TS,. = JJ^fc - #%• = 0 Ah Sijk = Cjk - dk> = 0 2) The previous metrical connection is exactly the canonical metrical connection CT. Proof. Assume that the nonlinear connection N, with the coefficients Nij satisfies the first axiom At then the connection DT{N) — (H'jk, C,3*) satisfies the axioms A2~A5 given by Theorem 6.4.1. Now, using Hljk from (4.2) we obtain psHsjk given by (4.5). So, for Ay = 0, we get N%] - p ^ = 0 and
()
w
£jva
r
Ch.6. Cartan spaces
147
Contracting by p\ we deduce NOj = 7^. Substituting in (*) we have
But these are the coefficients of the canonical nonlinear connection. The uniqueness of the connection CT = (Nij,Hljk,Cjk), which satisfies the axioms A\-A5 can be obtained by usual way [88]. q.e.d. One can prove that the axioms Ax-Ab are independent [88]. Finally of this section we obtain without difficulties Proposition 6.4.3. The canonical metrical connection = (M,K(x,p)), has the properties:
CT of the Cartan space Cn =
1° KU = 0, K\> = I 2° K2\j = 0, K2\i = 2pi 3° Pi\j = 0,
ptf=H
4° f\j = 0, p l | J ' = glj. Proposition 6.4.4. The d-torsions of the canonical metrical connection given by the following: (4.6)
/ ^ = 6kNij - 8jNik, Qjk, Tjk
CT are
= 0, # * = 0, Pljk = Wjk - V
Of course, we have (4.7)
Rijk = -Raj,
P{jk = P* w> djk
=
dki.
Proposition 6.4.5. The d-tensors of curvatures of the canonical metrical tion are given by the formula (10.3), Ch.4, where (4.8)
Sk^h = CkmhCji
Indeed, by means of Cklj - gkmCmij
and
-
h
CkmjCmih.
Cm^ -
Ckm}Cmih),
we obtain the expression (4.8) of the d-tensor of curvature Skljh.
connec-
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Applying the Ricci identities (10.5), Ch.4 to the fundamental tensor field g'j, and denoting as usual R^kh = 9wRs3kh, etc. we obtain: Theorem 6.4.4. The d-tensors of curvature of the canonical metrical connection CY have the properties: (4.9)
Rijkh + R?\h = 0, P V + Pi{kh = 0, Sijkh + Sjikh = 0.
The Ricci identities applied to the Liouville covector field p, and taking into account the equations: p ^ = 0, p ^ = 6\ we get some important identities. Theorem 6.4.5. The canonical metrical connection CY of the Cartan space C" satisfies the identities: (4.10)
R°jk + 1%* = 0, P ^ / + P * y = 0, Sp" = 0.
We derive from (4.10): Corollary 6.4.1. The canonical nonlinear connection CY has the property: (4.11)
ROjk = 0, PkOj = 0.
Of course, the index "0" means the contraction by pt or p \
6.5
Structure equations. Bianchi identities
Taking into account the general theory of structure equations and Bianchi identities i n of a general N-linearconnection DY(N) = the case of Cartan spaces n C = (M,K(x,p)), we obtain, for the canonical metrical connection CY(N) with the coefficients (4.2), the following results. The 1-form connections of CY are: (5.1)
uij = Hijhdxh + Cjih6ph.
Taking into account the fact that the torsion tensors Tljk and Sjk vanish, we obtain: Theorem 6.5.1. The structure equations of the canonical metrical connection CY of the Cartan space Cn = (M,K(x,p)) are: d{dx{) - dxm A u/ m = -fi* d(SPl)
Ch.6. Cartan spaces
149 du'j - wf Aw'™ = - f i ' j
(5.3)
2-forms of torsion: (5.4)
k
Q
+ PPkkijdxiA8pk
and fl lj is the 2-form of curvature:
-R/
(5.5)
Applying Theorem 4.12.2, we get: Theorem 6.5.2. The Bianchi identities of the canonical metrical connection CY of Cartan space Cn = (M,K(x,p)) are the following: „ ~ RmrhCsim}
= 0,
S
(5.6)
(h,T,S)
+
- 0,
5 {Rj\r\s + Pj\
= 0,
(5.7)
= 0,
^ (*,»)
(5.8)
h frJ - A {dsChir ft
A
+ ChmrCmis}
+ A {RksmC^
= 0, -
= 0,
P\
( (5.9)
A tft) = 0,
and (5.10) S is of cyclic sum and A
is for alternate
sum.
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6.6
Special N-linear connections of Cartan space Cn
Let Cn = (M, K) be a Cartan space, N its canonical nonlinear connection and CT(JV) the canonical metrical connection of Cn. The set of all N–linear connections DT(N) — (1Tjk,Ciki) which are metrical with respect to the fundamental tensor field g'i of the Cartan spce Cn is given by Theorem 5.2.3 with some particular references. We have Theorem 6.6.1. In a Cartan space Cn = (M,K(x,p)) the set of all N-linear connection DF(N) = (Htjk,Cijk), metrical with respect to the fundamental tensor glj of Cn is given by
where (Hljk,Cijk) = CF(N) is the canonical metrical connection and is an arbitrary 0-homogeneous d-tensor field and Zrk is an arbitrary (-1)-homogeneous tensor field. In this case we can remark that the mappings DF(N) —> DF(N) determined by (6.1) form an Abelian group, isomorphic to the additive groups of the pairs of d-tensor field (lD,1rsjXrsk,Qi3jZsrk). Theorem 5.2.4 has a particular case for Cartan spaces. Theorem 6.6.2. Let Cn = (M,K(x,p)) be a Cartan space and N its canonical nonlinear connection. There exists an unique metrical N-linear connection DT(N) = (iPjk^Ci^) with respect to the fundamental tensor gtj, having a priori given torsions fields: T'jk (= —T'kj), ^-homogeneous and.Sjk (= —S^), (-Y)-homogeneous. The coefficients of have the expressions H1* = ff'i* + \gls(9shThjk
- gjhThsk +
2
(6.2)
gis(gSh where (Hljk,Cjk)
gSh
+
h 9khT ]S)
gShn
are the coefficients of the canonical metrical connection CF(N).
If we take in particular case from (6.2) as follows 0-homogeneous; Sjk
&k
6k*
kk
rr is — 1-homogeneous
Ch.6. Cartan spaces
151
we get, for crk,rk arbitrary, the set of all metrical semisymmetric connections of Cartan space Cn. Let us consider the following N-linear connections of Cn : 1) CF(N) = {Hljk, – c a n o n i c a l metrical connection 2) BT(N) = {&Njk, -Berwald connection 3) RCF(N) =
-Rund-Chern connection
4 HT{N) = i&Njk,
)-Hashiguchi connection
These connections are determined only by the fundamental function K, of the Cartan space Cn. Every connection l)-4) can be defined by a specific system of independent axioms, [88]. b\
cf
We denote by |, | or |, | etc. the h- and v-covariant derivatives with respect to Cr,BT,... etc. Proposition 6.6.1. The properties of metrizability of the connection CT(iV)HT(N) are given by the following table
cr(N)
gij° = o
BT{N)
gijb
c
gijJC — Q
Ifc
=
\k
giif = -2Ciik
-2Ck^c |0
RCT{N) HT{N)
k
%
lj
The calculation of g \ or g
H
I*
g>if = -2Cijk
|0
is similar with Finslerian case (cf. Ch.2).
jfc
Let us consider a transformation of N-linear connection : DT{N) = DT(N) = defined by Vu'^;
"
l
]k —
n
jk
'
a
]k
k
where o jk,T~j are d-tensors 0-homogeneous and -1-homogeneous, respectively. The following particular transformations are remarkable:
: CT(iV) —> RCT(N)
h{-P\k, -Cjk)
: CT(N) —> BT(N)
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152
Now, it follows easily. Proposition 6.6.2. The following diagram holds good RCV(N)
CV(N)
•
HV(N)
The existence of this commutative diagram shows us that the N-linear connections CT(iV), BV(N), RCT(N) and HT(N) are important in the geometry of Cartan spaces.
6.7
Some special Cartan spaces
The Berwald connection BT(N) - (d'Njk,0) the torsion: T3k
of the Cartan spaces has d-tensors of
= 0, Si* = 0, CiJk = 0, P'jk = 0, and Rijk = SkNij - 5jNik
The d-tensors of curvature of BV(N) are:
(7.1)
hkh s/kh
= 6^,-6,Or>ijh =
i r>s r>i + O jkti
sh
-
B*jhB>.k
dhB{jk
= 0
where (7.2)
Bl k = &Nk
are the coefficients of the -connection. The Bianchi identities of the Berwald connection BT(N) 4.12.2. Now, let us give the following definition:
are given by Theorem
Ch.6. Cartan spaces
153
Definition 6.7.1. A Cartan space Cn is called Berwald-Cartan space if the coefficients B'jk of Berwald connection BT are functions of position alone: Bijk(x,p) = Bijh(x).
(*)
The Berwald-Cartan spaces can be characterized by: Theorem 6.7.1. A Cartan space is a Berwald-Cartan space if and only if the following tensor equation holds: Ctjkc = 0.
(7.3)
The proof is similar with the one of Theorem 2.3.1. From the formula (7.1) we obtain: Corollary 6.7.1. A Cartan space is a Berwald-Cartan space if and only if the b
.
d-tensor of curvature P/k
vanishes.
Definition 6.7.2. A Cartan space is called a Landsberg-Cartan space if its Berwald connection is h-metrical, i.e.
Theorem 6.7.2. A Cartan space is a Landsberg-Cartan space if and only if the following tensor equation holds: (7.4)
Cj>, = 0 .
As in the case of Landsberg-Finsler space we can prove: Theorem 6.7.3. A Cartan space is a Landsberg-Cartan space if and only if the d-curvature tensor Pijkh of the canonical metrical connection CT(N) vanishes identically. Corollary 6.7.2. If a Cartan space is a Berwald-Cartan space, then it is a LandsbergCartan space. Remark. We will study again these spaces, in next chapter using the theory of Cduality of Finsler spaces and Cartan spaces. Also, it will be introduced the Cartan space of scalar and constant curvature. Definition 6.7.3. A Cartan space C" = (M,K(x,p)) is called locally MinkowskiCartan space if in every point x € M there is a coordinate system (U, x') such that
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154
on 7T* 1(U) c T*M the fundamental function K(x,p) depends only on the momenta (Pi)Exactly as in the case of Finsler spaces we can proof the following important result: Theorem 6.7.4. A Cartan space Cn — (M, K(x,p)) is a locally Minkowski-Cartan space if and only if the d-tensor of curvature Rjlkh of the canonical metrical connection CY vanishes and C^k\h = 0.
Examples. 1° The Cartan spaces Cn = (M,K), Minkowski-Cartan space. 2° The Cartan spaces Cn = (M,K), Moór)
where K is given by (1.5), is a locally with the fundamental function (Berwald-
K = {pip2-Pn}K is a locally Minkowski-Cartan space.
6.8
Parallelism in Cartan space. Horizontal and vertical paths
In a Cartan space Cn = (M, K{x,p)) endowed with the canonical metrical connection CT(iV) = (H'jk,Cijk), the notion of parallelism of vector fields along a curve 7 : / —> T*M can be studied using the associate Hamilton space Hn — (Af, K2(x,p)), (cf.§5.8, Ch.5). _ _ Let 7 : / —> T*M be a parametrized curve expressed in a local chart of T*M
by
(8.1)
x* = xl(t),
Pl
= Pi(t),
d-y The tangent vector field — is (8 2)
-
where
Tt
t e I, rank||pi(i)|| - 1.
Ch.6. Cartan spaces
155
The curve 7 is called horizontal if I ^7 —\ Vfltt/ characterized by the equations: dt ~ dt Taking into account that N^ = pmHmij
= 0. So, an horizontal curve 7 is
- Njl A f — 0 dt ~ °'
it follows that:
Proposition 6.8.1. The horizontal curves 7 in the Cartan space Cn are characterized by the equations:
<->
*-.<(.).£ = $ - * * - „ £ - 0 .
The first consequence is follows: Theorem 6.8.1. The geodesies of the Cartan space C" are the horizontal curves. Proof. The geodesies of the space Cn are given by the Hamilton-Jacobi equations dF2 dxi '
d^__ dF^_ dPi _ dt ~ dpi ' dt ~
T t o e f o r e , we have & = f -
Njl%
. - ( g
+
Nt^
. -H, . 0. <,.e.d.
Corollary 6.8.1. The geodesies of the space Cn are characterized by the equations: p
~df
=
a
For a vector field X e Af(T*M), which is locally expressed by: (8.7)
X = X'(x,p)8i + Xi(x,p)di,
we have: Theorem 6.8.2. The vector field X, given by (8.7) is parallel along the curve 7, with respect to the canonical metrical connection CY of the Cartan space Cn, if and only if its coordinate X1, Xi are solutions of the differential equations (11.8), Ch.4, where ujlj(x(t),p(t)) is the 1-forms connection of CY. In particular, Theorem 4.11.2 can be applied:
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156
Theorem 6.8.3. The Cartan space Cn = (M,K), endowed with the canonical metrical connection CT is with the absolute parallelism of vectors, if and only if the d-tensors of curvature Rj'khjPj'k11 and Sijkh vanish. Theorem 4.11.3 can be particularized, too: Theorem 6.8.4. The curve 7 given by (8.1) is autoparallel with respect to the canonical metrical connection CT of the Cartan space C" if, and only if, the functions x'(t), Pi(t), t e / are the solutions of the system of differential equations ,
d2x{
,
(8 8)
-
^
+
dxs u\
^T dT
d Spt
= 0
Sps
' dt^~^-°-
As is known, the horizontal autoparallel curves of the Cartan space C" are the horizontal paths. Theorem 6.8.5. The horizontal paths of the connection CT of a Cartan space are characterized by the system of differential equations
Now, taking into account Theorem 4.11.5, we get: Theorem 6.8.6. The vertical paths in the point XQ € M with respect of a Cartan space Cn are characterized by the system of differential equations dt*
~l
v u
~ "' ; <2*
dt
6.9 The almost Kählerian model of a Cartan space To a Cartan space Cn = (M, K(x,p)) we can associate some important geometrical object fields on the manifold T*M. Namely, the N-lift T of the fundamental tensor gli, the almost complex structure F , etc. IfN is the canonical nonlinear connection of C , thus (G, F ) determine an almost Hermitian structure, which is derivated only from the fundamental function K of the Cartan space. This structure gives us the so-called [99] geometrical model on T*M of the Cartan space C". Let (6,,d) be the adapted basis to the distribution N and V, N being the canonical nonlinear connection of the space Cn. Its dual basis is (dx',6pi). The N-lift of the fundamental tensor field g'j of the space Cn is defined by (9.1)
G(x,p) = gij(x,p)dxi
We obtainas usual
® dxj + gij(x,p)SPi
® 6Pj.
Ch.6. Cartan spaces
157
Theorem 6.9.1. The following properties hold: 1° G is a Riemannian structure globally defined on T*M. 2° G is determined only by the fundamental function K of the Cartan space Cn. 3° The distributions N and V are orthogonal. Now, considering the covector fields fy — g^ in every point u G we can define the F(f*M)-\mear mapping F : XiT^M) —> X{T*M), defined in (6.7), Ch.4, by (9.2)
W(6i) = -du
By means of Theorem 4.6.2, we obtain Theorem 6.9.2. We have the followings: 1° F is globally defined on 2° F is the tensor field of type (1,1): (9.3)
F = -gijfr (8 dxj + gij8i ® 6pj.
3° F is an almost complex structure on T*M: (9.4)
FoF
= -/.
4° F depends only on the fundamental function K of the Cartan space Cn. Theorem 4.6.4 lead to Theorem 6.9.3. 1° The pair (G,F) is an almost Hermitian structure on T*M. 2° The almost Hermitian structure (G, F ) depends only on the fundamental function K of the Cartans space Cn. 3° The associate almost symplectic structure to the structure (G, F ) is the canonical symplectic structure 6 = SpjAdx1 = dpiAdx*. Corollary 6.9.1. The space {T*M, G, F ) is almost Kählerian and it is determined only by the Cartan space Cn. Finally, we remark:
158
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Theorem 6.9.4. The N-linear connection D, determined by the canonical metrical connection CT of the Cartan space Cn is an almost Kählerian one, i. e.
Due to the last property, we call the space (T*M, G, IF) the almost Kählerian model of the Cartan space Cn. It is extremely useful in the applications in mechanics, theoretical physics, etc.
Chapter 7 The duality between Lagrange and Hamilton spaces In this chapter we develop the concept of £-duality between Lagrange and Hamilton spaces (particularly between Finsler and Cartan spaces) investigated in [66], [67], [97] and a new technique in the study of the geometry of these spaces is elaborated. We will apply this technique to study the geometry of Kropina spaces (especially the geometric objects derived from the Cartan Connection) via the geometry of Randers spaces. These spaces are already used in many applications.
7.1
The Lagrange-Hamilton C- duality
Let L be a regular Lagrangian on a domain D c TM and let H be a regular Hamiltonian on a domain D* C T*M. Hence, the matrices with entries (11)
9ab(x,y) := dadbL{x,y)
and (1.2)
g*ab(x,p):=dad»H(x,p)
are everywhere nondegenerate on D and respectively D * b, c,... e , Note: The metric tensors (1.1) and (1.2) used in this chapter are those of Chapter 3 and Chapter 5 multiplied by a factor 2. This notation allows us to simplify a certain number of equations and to preserve the classical Legendre duality encounterd in Mechanics (see[3]). 159
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160
If L 6 ^F(D) is a differentiable map, we can consider the fiber derivative ofL, locally given by (1-3)
which will be called the Legendre transformation. It is easily seen that L is a regular Lagrangian if and only if i s a local diffeomorphism [3]. In the same manner if H £ T{D*) the fiber derivative is given locally by (1.4)
iP(x,p) = (xi,daH(x,p))
which is a local diffeomorphism if and only if H is regular. Let us consider a regular Lagrangian L. Then tp is a diffeomorphism between the open sets U C D and U* C T*M. We can define in this case the function H:U*—>$1: (1.5)
H(x,p)=paya-L(x,y),
where y = (ya) is the solution of the equations (1.5')
Pa
= daL(x,y).
Also, if H is a regular Hamiltonian on M, ip is a diffeomorphism between same open sets U* C D* and U C TM and we can consider the function L : U —> IR : (1.6)
L(x,y)=Paya-H(x,p),
where p = (pa) is the solution of the equations (1.6')
Va = daH(x,p).
It is easily verified that H and L given by (1.5) and (1.6) are regular. The Hamiltonian given by (1.5) will be called the Legendre transformation of the Lagrangian L (also L given by (1.6) will be called the Legendre transformation of the Hamiltonian H. Examples: 1. If L is m-homogeneous m ^ , regular Lagrangian, then locally, y), pa = daL{x,y).
Ch.7. The duality between Lagrange & Hamilton spaces 2. If L(x,y) = -aij(x)y1yi Hamiltonian
161
+ 6^' + c then its Legendre transformation is the H(x,p) = -al:>{x)pipj - blpi + d
where b1 := atjbj and d :— 6j6* - c. In the following, we will restrict our attention to the diffeomorphisms (1.7)
(where tp is the Legendre transformation associated to the Hamiltonian given in (1.6)). We remark that U and U* are open sets in TM and respectively T*M and generally are not domains of charts. The following relations can be checked directly (1.8)
(1.9)
Ifi Otp — l y . ,
diH(x,p) = -diL(x,y);
i()O(p=lU
didaL(x,y) =
-didbH(x,p)9:b(x,p)
9ab(x,y)g*bc(x,p)=Sca
(1.10)
where pa = daL{x, y), ya = daH{x,p). Using the diffeomorphism ip (or ) we can pull-back or push-forward the geometric structures from U to U* or from U* to U. A) if / 6 ^{U) we consider the pull-back off by ^ (or push-forward by )
Also, if / € :F(f/*), we get /° € (1-12)
f ;= f o p =
We have the following properties: (i) (Ay + Mfl)* = A/* + « T , (/)• = / V ; V A,/i € R, V /, g (ii) (A/ + /i 5 ) 0 = A,0 +M5°, (/ff)° = / V ; VA,/x 6 H , V / , f l
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(iii) ( / T = /, (9°r =9, / € F(U),ge
F{U*),
(iv) (9abr = 9*a», (gab)* = 5a6B) If X e X(U) the push-forward of X by v (or pull-back by ) is X* € (1.13)
X* := TV o X o y r 1 = T^""1 o A" o ip.
(
is the tangent map of) Also, if X 6
X° := Tip o X o ip-1 = Tip'1 o X o tp
The following relations are easily checked:
(i) (fX + gY)* = f*X* + g'Y*, Vf,ge
J ( f / ) , V I , Y 6 X(U),
(ii) (fX + gY)° = f(>X0 + g0Y0, \ff,g € T(U%VX,Y
€
(iv) (X*)° = X, {Y°Y = Y, X e C) If 6 e X*(U) the push-forward of 9 by ^ (or pull-back by ) is S' G A"([/*) (1.15)
(9* = ( I » * oOoip'1 = (Tip'1)* o6oip.
and if 6 € /f*(f/*) we can consider (1.16)
(9° = [Tip)* o 6 o i p ~ l = (Ttp-1)*
o9o
where (Tv)* denotes the cotangent map of
(AT ® T)* = K*® T", (K1 ® T')° = A"0 ® T'°.
Ch.7. The duality between Lagrange & Hamilton spaces
163
Let V be a linear connection on U. We define a linear connection V* on U* as follows:
°r, x, Y e x(W). Also, if V is a linear connection on U* we get a linear connection V° on U (1.19)
V°xy ••= (Vx-r)°, X,Ye
X(U).
It is easily checked, using the above examples, that V* and connections on U* and U. For the torsion and curvature tensors of V* we have
are indeed linear
(1.20)
T*(X, Y) = [T(X°, Y°)]\ VX, Y e X(U'),
(1.21)
R*{X, Y)Z = [R(X°, Y°)Z0}*, VI, Y,Z e X(U*).
Generally, if K € V(U) and K* e V(U*) is its push-forward by ip, then (1.22)
VK* =
Definition 7.1.1. We will say that f and f* (or f and ) X and X* (or X and ) , K and K* (or K and ) , V and V* (or and ) are dual by the Legendre transformation or are -dual. In the next section we will look for geometric objects on U and U* which are -dual. These geometric objects will be obtained easily each one from the other.
7.2
—dual nonlinear connections
Definition 7.2.1. Let HTU and HTU* be two nonlinear connections on the open sets U and U*. We say that HTU and HTU* are -dual if (2.1)
T
Let N = (N*) and N = (JV,-O) be the coefficients of two nonlinear connections on U and U*.
The Geometry of Hamilton & Lagrange Spaces
164
Theorem 7.2.1. The following statements are equivalent: (i) N and N are -dual; (ii )
Nr = -Nib9*ab - d{d"H or Nfa = -N?9ba + d,daL\
(iii) 8iV*a = - A / J " ; (iv) 8iP°a = ATP,; f^Af and N are £-duak=* Ty(HTU) V i £ 1, n. We must have
= HTU* <=> T
On the other hand, T
a> = 5{ and N°a =-N*gab ba
(or, equivalent, iV?* = -Nibg* Now we have
+dAL
a
- did H) and we have proved that (i)<=> (ii).
= (ft/)* - A?wr. Using these relations we get the proof. The -dual of the nonlinear connection N will be denoted by N*. (Similarly, the -dual of A7 will be denoted by ) Corollary 7.2.1. If N — (N?) and N* = {Nia) are two tions we have
-dual nonlinear connec-
frY = 61, (Sya)* = g*ab6*pb.
(These properties are characteristic for two -dual nonlinear connections, too.) Proposition 7.2.1. The following equalities hold good: (dx'Y = da:'; ab
(ft,)' = g^fr
d-p = g* (dbfY; ft/* = (difY + d^
Ch.7. The duality between Lagrange & Hamilton spaces
165
Corollary 7.2.2. Let N = (N?),N* = (Nia) be two £-dual nonlinear connections. The following assertions hold: (i) IfX = X% + Xada then X* = XUS* + g*abXa*^> and
{xHy = {x*)H, (xvy = (x*)v.
(ii) If to = cjiadx1 A 8ya then u* = (uiaYgtabdxi
A S*pb.
(iii) IfK = KtfbSi ®da® dxj
Examples. 1. The -dual of the metric tensor gat, has the components g*ab. 2. If If Cabc = l-da9bc then C*abc = ~d ~d»g*bc. If Cabc = \gadddgbc
then C'J* =
-^glfig"*.
3. If 8 is the Kronecker delta, with components 8la then the components of its dual S* are as follows: Let c(t) = (x(t),y(t)), t e / C R b e a differentiable curve on U. The tangent vector can be written as follows: (2.2)
jLS,
We say that c is a horizontal curve if —T- = 0. at A similar definition holds for differentiable curves on U*. Proposition 7.2.2. (N, ) is a pair of -dual nonlinear connections if and only if the -dual of every horizontal curve is also a horizontal curve.
The Geometry of Hamilton & Lagrange Spaces
166
t e I C Mtwo £-dual curves,
Proof. Let c(t) = (x(t),y(t)), c*(t) = (x(t),p(t)), therefore ya(t) = daH{x{t),p{t)). We have: at
at
at
bad
. at
a
* ^ + (did H + N?)^- = 0 at at
Let suppose that N and N are
-dual nonlinear connections. Using Theorem
8*pa
6ya
7.2.1, (ii), and the above relation we get -—- = 0. Conversely, from —r- = 0, at at —r-^ = 0 we obtain easily that N and A^ are at
-dual.
Example. For a Lagrange manifold the geodesies are extremals of the action integral of L and coincide with the integral curves of the semispray XL - y% - 2Gada,
(2.3) where
Ga=l-gab(ykdbdkL-dbL).
(2.4)
This semispray generates a notable nonlinear connection, called canonical, whose coefficients are given by N° = djGa
(2.5)
(see Section 3.3, Ch. 3). Using (ii) from Theorem 7.2.1, we get the coefficients A^a of its connection:
-dual nonlinear
and after a straightforward computation we obtain (2-6)
Ni3 = \
^
We remark that i s expressed here only using the Hamiltonian. This is the canonical nonlinear connection of the Hamilton manifold (M, H) obtained by R.Miron in [97]. We also remark that the canonical nonlinear connection (2.3) is symmetrical, that means (2.7)
ry = Ntj - Njt = 0.
Ch.7. The duality between Lagrange & Hamilton spaces Taking the C -dual of (2.7) we get the "symmetry" condition Nbj9ln - Nbi9bj = d^L
(2.8)
167 f o r
- OidjL
((2.8) can be also checked directly and thus (2.7) may be obtained as a consequence of (2.8)). Now, let us fix the nonlinear connection given by (2.5) and (2.6) on U and respectively U*. The canonical two form 6 = 6paA dxa is just the canonical symplectic form of T*M. The Hamilton vector field XH can be obtained from the condition: iXliw = -dH <=» ixAdx* A 8*pi) = 6'Hdx1 + diH6*pi. Consequently, XH = WHS* - 6*Hd\
(2.9)
The integral curves of XH are solutions of Hamilton-Jacobi equations
(equivalents with ^
= #H,
^
=
-&H).
The -dual of XH is just Xi, the Lagrange vector field from (2.3). In adapted frames (2.3) one reads XL = yl6i + (y*Nj - 2Ga)da and we remark that Xi is horizontal iff G° is 2-homogeneous. An integral curve of XL verifies the Euler-Lagrange equations:
which are -dual of (2.10). The -dual of the canonical one form u> = Pidx% is the canonical 1-form of the Lagrange manifold (2.12) and the (2.13)
u = biLdxi -dual of w is the canonical 2-form of (M, L) eL=gia6yaAdx\
The Geometry of Hamilton & Lagrange Spaces
168
Proposition 7.2.3. (i) If N and N are -dual then (Rajk)
= (R'ajk) = 9abR jk,
(ii) [%,%] = 0, (iii) (daNib)° = ga'(8i9cb -
gMdcNf).
Proof. (0 [s;,st] = [SjM = (Rbjkdby = Rajkir(ii) We use the symmetry of the tensor d^k in all indices, (iii) [96,6;} = 9 6 (iVj a )S a . On the other hand, and then we will get (iii). Proposition 7.2.4. Let N and N betwo C -dual nonlinear connections. (i) Nij = Njt ^ (ii )
N}gki - N?gkj - d^L -
d°Nl3 = d'Na ^=> gihdkN^ - gjhdkKh
Then
djjL,
= Sj9*k ~ 6i9*jk,
(iii) g*ihdhNjk - g*hd»Ntk = 5i9*k - 8i9*k *=> 4 w * = drf. Proof, (i) follows from Theorem 7.2.1, (ii), and (ii), (iii) are direct consequences of (iii) of Proposition 7.2.3.
7.3
-dual d— connections
Let (N,N*) be a pair of -dual nonlinear connections. Then -dual of the almost product structure P = Si
Proof. Weuse V*P = V*P* = (VP)*.
is a d-connection..
Theorem 7.3.1. Let CV(N) - (Lijk,Llc,Cijc,Cabc) be ad-connection on U, and CT(N*) = [H)k, H£c, V',-0, Vabc) be a d-connection on U*. Then CT{N) and CT{N*) are -dual if and only if the following relations hold:
Ch. 7. The duality between Lagrange & Hamilton spaces
=L
(i) W (ii)
169
b
(H ak)0=9
(iii) (VV)° = f» (iv) (K6C)° = 0 V Remark. We see that H]k is obtained very simple from U]k and Vlja as the -dual of Cijh that is \ri a .?m a V j —O j .
On the other hand, and therefore V* = C*abc *=* C*abc = -\gldddg*bc
^^
Cabc -
\gaddd9bc
Corollary 7.3.1.
(ii) Let K* be a d*-tensor on U*, the -dual of the d-tensor K on U, K* *k and K* * c its h- and v-covariant derivative with respect to V*, T := K\k, T'~K\C.
Then
A consequence of (1.20) and (1.21) is the following result: Proposition 7.3.2. Let and and curvature tensors of V* are
be two -duald-connections. Then, the torsion -dual of torsion and curvature tensors of V.
Remark. The proposition above states that the torsion and curvature tensors of V* can be obtained from those of V by lowering or raising vertical indices, using Soft-
As we have seen (Theorem 7.3.1), the a N*-connection.
-dual of a N-connection generally is not
Proposition 7 3 3 . Let CT{N) = {L)k, Cabc) be a N-connection on U and CT*{N*) = {H1jk,Hb1k,Vija,Vabc) its C-dual. Then
The Geometry of Hamilton & Lagrange Spaces
170
(3-1)
H
(3.2)
vv =
*
=
t
9
;
h
L
^
Conversely, if the coefficients of V* (the C -dual of ) verify (3.1) and (3.2) then is a N-connection. Proof. Let F* = -g*ada ® dxl + s* ia ^ ® <Spa be the £-dual of the almost complex structures F = -d^dx1 + 6t®8y\ Then the d-connection V is a N-connection if and only if V F = 0 <^> V*F* = 0 <^=> (3.1) and (3.2). On the tangent bundle we have the metrical structure (3.3)
G = gijdx1 ® eta-7 + gab&ya ® Syb.
The -dual of this metric tensor is (3.4)
G* = g*jdxx ® dxj + gtab6pa ® 6pb.
Therefore we are in the position to apply Theorem 3.10.1 and Theorem 7.3.1 and so we will get the canonical d-connection of the Lagrange and Hamilton manifolds (here restricted to U and U*). Theorem 7.3.2. The -dual of the canonical N-connection of a Lagrange manifold is just the canonical N*-connection of its associated Hamilton manifold. (Only in this case Vbc = C*bc.) Proof. Using Theorem 3.10.1 and Theorem 7.3.1, (i) we get {Hxjk)° = -g^iSjghk + 8kgjh ~ Shgjk), and using Theorem 7.2.1 we obtain
Also, making use of Theorem 7.3.1, (iv) and Theorem 3.10.1, we have dc Va abc= =-k- dk(9V + 9V M - aV) - aV 6c=) =-\9J9 -\9*Jb9tcd = =C\ C\bc d ( 9 V + 9V
Ch.7. The duality between Lagrange & Hamilton spaces
171
and from (ii) and (ii) of Theorem 7.3.1 we obtain f
zri
\ri
a
\r ia
bk
Theorem 7.3.3. Let V be the U. Then we have:
-dual of a N-connection CT{N) = {D]k,Cabc) on
(i) H]k = H)k iff V* is h-metrical ( ^ ** = 0, <4 *k = 0);
(ii) y y = V/c iff V* is v-metrical {g'j *c = 0,5*,, *c = 0). Proof. (i) The h-metrical condition for V* and (3.1) can be rewritten in the following forms: S)y#i* + 9ihH% = Stg'j (<«=> gtj *k = 0) 9*cbHcak + glM
= tig'*
(<=> 9lb *k = 0)
HH'jk - I f from the last equality we get the first two and from the last condition and the first we get Hjk — H?jk. By a similar argument we can prove (ii). Consequently, we can conclude finally: Theorem 7.3.4. The class of N-connections which is preserved by the class of metrical N-connections.
-duality is only
Let 0L be the canonical symplectic form of (M, L) given by (2.13). The following result is a consequence of the above theorems:
Theorem 7.3.5. (i) If' V is a N-connection on U C TM, then V0 L = 0 4=^- VG = 0. (ii) If V* is a N*-connection on U* c T*M then VF* = 0 <=> VG* = 0.
The Geometry of Hamilton & Lagrange Spaces
172
Proof. (i) Let V* be the £ -dual of Then VG = 0 *=> V*G* <^> V 0 = 0 <=> V0 L = 0. (ii) We use a similar argument. Let us stand out some problems connected with the deflection tensor field. The h-deflection tensor field of a d-connection V, on the tangent bundle can be defined as follows:
D : X(U) —> X(U), D(X) = V"C
where C is the Liouville vector field. Locally we have: C = yada, D = DaA ® dx\ D\ = ^ = L£y» - Nf.
,» ,, [ '
The deflection tensor field of a connection on the cotangent bundle, D{X) = 1 V^C where C is the Liouville vector field on T*M, C = pada, has the local form D = And" ® dx\ Dai - p.,,,- = 7Via - Hbaj>b.
(3.6)
Also we can consider the v-deflection tensor field d(X) = VVXC, d = dabdb ® 6ya, da" = »fo = <56° + C c a y c
(3.7)
and its correspondent for cotangent bundle d{X) = VVXC, d = dabdb ® 5pa, d\ = Pb\\a = 5ab - Vbcapc.
(3.8)
Using the -duality we see that generally, the -duals of D and d are different by D and respectively d. We have D*(X) = V ' / C
(3.9)
where C* is the -dual of the Liouville vector field,
and locally D' = ^ 9 ° ® da;', D*ai = ^ ( ^ 6 i ) ' or
(3.H) The (3.12)
Dr = ylM = Slyl-Hliyl. -dual of rf is d = d * 0 ^ ® i
s
where
^ a t = 3*6c5*ae(4c)* = 2/6**°.
Ch. 7. The duality between Lagrange & Hamilton spaces
173
The following result holds: Proposition
= -F2(x,y)
7.3.4. C* = C if and only if L(x,y)
+ u(x) where F is
1-homogeneous and u is a scalar field. In this case D = D* and d = d*. Proof. C* = C <=> g*abyb* = pa <=> yadadbL = dbL <=>• daL is l-homogeneous <=> L(x,y) = -F2(x,y)
+ u(x), and F l-homogeneous.
Remark. As we have seen from the last two sections there exists many geometric objects (nonlinear connections, linear connections, metrical structures and so on) which can be transfered by using -duality from U to U* and also from U* to U. Now let suppose we have a regular Hamiltonian defined on a domain D* C T*M. The Legendre transformation ip : U* —> U is a diffeomorphism between some open subsets U*, U of D* and TM. Taking the Lagrangian L(x,y) — paya - H(x,y), ya* _ Qaj{(Xjp) w e c a n construct a Lagrange geometry restricted to U and then we pull-back by ip the geometric objects on U, to U*. These will depend only by Hamiltonian, therefore we will be able to extend them on the whole domain D*.
7.4
The Finsler-Cartan
-duality
In this section we will give an idea for the study of the geometry of a Cartan space using the -duality and the geometry of its associated Finsler space. Let H be a 2-homogeneous Hamiltonian on a domain of T*M, ip : U* —> U the Legendre transformation and L(x,y) = ptf - H(x,p),
(4.1)
y" = &H{x,p)
its associate Lagrangian. We remark, using the 2-homogeneous property of H, that L(x,y) = H(x,p).
(4-2)
Proposition 7.4.1. The Lagrangian given by (4.1) is a 2—homogenous Lagrangian. Proof. Let us put f(x,p)
:= y " = $H{x,p),
9i{x,y)
:= p? =
diL{xty).
We know that /* is 1-homogeneous, then 9i{x,Xy) =ffi(x,A/ J (a;,p)) = g^xj^x,
Xp)) = Xp° = \gi{x,y)
The Geometry of Hamilton & Lagrange Spaces
174
and thus 9i is 1-homogeneous so, L is 2-homogeneous. Therefore, using the theory made in the previous sections we may carry some geometry of Finsler spaces on 2-homogeneous Hamilton manifolds. Remark. For 2-homogeneous Hamiltonian we have (4.3)
p° =
Vi
or
ft
= y\
Among the nonlinear connections of a Finsler space one has the most interest. It is the Cartan nonlinear connection N] = 7 j 0 -
(4-6)
C^oo
where \
dkg,h - a h g j)c ), C*jfc = |ff
Theorem 7.4.1. The -dual of the Cartan nonlinear connection (4.6) is (4-7)
^ - 7 i
Q
and
- ^
l
-dk9ii.
K*,, =
Proof. We have (iVy-)° = -N*gkj + didjL and making use of (4.2) we get (4.8)
JVy = -(TiirJ/T +
^
^
where A? := ft On the other hand, 9ir)* = (di9*r + dr9*j ~ % £ ) So, we will have
tf + ytT^i9*jr -
U°r(dagijy)
175
Ch.7. The duality between Lagrange & Hamilton spaces and hasryrys)*
= -Pkilksy*s + ytsy*r(dsg*ar =
-AbT{db9asy)
-Pki:ks+y*sy*rdsg*ar.
Then (4.8) becomes
But we can write: y*rdi9*r
= gtrkPkdl9*r
= -
= -gUd>(9*hhPk) = -g-hdtdhH
=
-Ahl9hj
and substituting it in the above equality we get (4.7). Among Finsler connections, the Cartan connection is without doubt very important. The following result is also well known [88]. Theorem 7.4.2. On a Finsler space there exists only one Finsler connection which verifies the following axioms (Matsumoto's axioms) Ci) 9ij\k = 0 (h-metrical);
C2) g.. . = 0 (v-metrical);
C3) D\ = -Ni + yiF)k = 0; C4) Tjk = 0; C6) S*jk = 0. The coefficients of this Finsler connection are: (4.9)
(4-10)
F}k = \gih{6i9hk Clik =
+ 6k9jh - 8h9jk) \gihdh9jk
and the nonlinear connection is given by (4.6). The Finsler connection given by (4.9), (4.6), (4.10) is the Cartan connection of the Finsler space Fn = (M, F). Using the results of the previous sections we can state the -dual of Theorem 7.4.2 for the Cartan space C" = (M, F). Theorem 7.4.3. On a Cartan space (M, ) there exists only one N-connection CT{N) = (H'jk, Vi>k) which satisfies the following axioms:
The Geometry of Hamilton & Lagrange Spaces
176 C[) s"'||* = 0;
Q ) g*ij\\k = 0;
CJ) Dlk = -Nik + PjHi
= 0;
k
CD T > = 0; CD SS = 0. It is the -dual of the Cartan connection above. That is: (4.H)
H)k = \9^{5]g*hk +
5lg*h-5lg]k)
\
(4.12)
and the nonlinear connection is given by (4.7). Proof. The connection given by (4.11), (4.12) and (4.7), verifies CX)-C\). For this connection Tjk = T"jk, 5,J'* = SV* and Dik = D*ik (Proposition 7.3.4). This connection is unique. Indeed, if there exists another one, taking the -dual of it we will get two Finsler connections (restricted to an open set) which satisfy Matsumoto's axioms. The linear connection of Theorem 7.4.3 is just the Cartan connection of the Cartan space Cn = (M,F). We remark that conditions Cj^-Q) are all the -duals GfCi)-C 5 ). Consequently, all properties of the Cartan connection from Finsler spaces can be transfered on the Cartan spaces only by using the -duality. Remark. When we look for a -dual of a d-tensor field we must pay attention to the vertical indices; in this section (and sometimes in the other sections) for the sake of simplicity we have omitted to use indices a, b, c, d, e,f to stand out the vertical part. Let c"(t) = (x(t),p(t)), t e l e R a differentiable curve on D*. c will be called h-path (with respect to Cartan connection CF*) if it is horizontal and dx
U
Theorem 7.4.4. The -duals of h-paths of CT*{N*) = (tf^.V?*) are h-paths of ( = (Fjk,Ciik). Proof. It follows from Proposition 7.2.2 and Theorem 7.3.1, (i). Corollary 7.4.1. Let c*(t) = (x(t),p(t)) be an integral curve of the Hamilton vector field (2.9). then c*(t) is an h-path of CT\
Ch. 7. The duality between Lagrange & Hamilton spaces
177
Proof. c*(t) = (x(t),p(t)) is an integral curve of XH iff its dual c(t) = (x(t),y(t)) is an integral curve of XL, and therefore an h-path so its -dual c* will be also an h-path. The next results will give us an interesting field where the -dual theory can be applied. As we know a Randers space is a Finsler space where the metric has the following form F(x,y) = a + P
(4.13)
(Randers metric) and a Kropina space is a Finsler space with the fundamental function F(x,y) = a2//3
(4.14)
(Kropina metric) where a = Jai:j(x)ylyi is a Riemannian metric and 0 = bi(x)y% is a differential 1-form. We can also consider Cartan spaces having the metric functions of the following forms F(x,p) = \Ja'ipiPj + b'pi or
(4.15)
(4-16) and we will again call these spaces Randers and, respectively, Kropina spaces on the cotangent bundle T*M. Theorem 7.4.5. Let (M, F) be a Randers space and b = length of bj. Then
(CL^W)1!2
the Riemannian
(i) If b2 = 1, the -dual of(M, F) is a Kropina space on T*M with
(ii) If b2 ^ 1, the -dual of (M, F) is a Kropina space on T*M with (4.18)
H(x,p) =
(in (4.18) " - " corresponds to b2 < 1 and "+" corresponds to b2 > ).
The Geometry of Hamilton & Lagrange Spaces
178
Proof. W e p u t a2 = y{y{, b' = aljbjt $ = ky', /3* = br'pt, p{ = aijpj, a*2 = p{p{ = aijPiPj. We have (4.19)
F = a + (3,
ft
= ^
=
Contracting in (4.19) by p' and bl we get:
(4.20)
a*2 = F ((— ++ /?*V /?*V /3* /3* = F F [^ [^ + + b2
Therefore, (4.21)
/?'= a
F2 (i) If b2 = 1, from (4.21) we obtain /?* = — and using (4.20), we get a a*2
(ii) If b2 ^ 1, from (4.20) and (4.21) we have:
±a« = - + (!'; 0*a = Fa and by substitution
From this last relation we obtain (4.18). Theorem 7.4.6. The -dual of a Kropina space is a Randers space on T*M with the Hamiltonian (4.22)
1 / r . - • \ H(x,p) = - (jtiipiPj ± b'pij a» = ~a^,
where
¥ = \b\
(Here "+" correspondsto/3 > 0 and "-" to (5 <
)
Ch.7. The duality between Lagrange & Hamilton spaces
179
Proof. We use the same notations as in the proof of Theorem 7.4.5. We have pi = FdiF
(4.23)
F =—(2yi-Fbi).
Contracting by p1 and then by 61 we get: a*2 = 4 - ( 2 F - /?*), (3* = ~{2f3 - Fb2).
(4.24)
Using these relations, after a simple computation, we obtain (4.22). We must have b2 ^ 0 for regularity of a y '. But, the regularity condition for the Kropina metric leads to b2 ^ 0. Remark. Using the Theorem 7.4.6, we can derive the geometric properties of Kropina spaces, very simply, from those of Randers spaces, by using £-duality. We will explain this more precisely in the next section.
7.5
Berwald connection for Cartan spaces. Landsberg and Berwald spaces. Locally Minkowski spaces.
Berwald connection BF = (Nij,dkNij,Q) of Cartan space (here Ntj are given by (4.6)) is not the -dual of Berwald connection of its associated Finsler space like Cartan connection. There exist some important distinctions here, which are consequences of the nonexistence of a spray and thus, the nonlinear connection cannot be obtained as a partial derivative of a spray. Theorem 7.5.1. On Cartan space (M, ) there exists only one Finsler connection with the following properties: D#\
Bl)
JT* IT*
h
n
D*\
n
n
n
9ihd Njk - g*hd Nlk = 5*g*k - 5*g*k,
Proof. It is easily checked that the connection BT — (Nij,dkNij,Q) with N,j given by (4.6) satisfies all B]) — Bl). Indeed, let us take the local diffeomorphism i> : U* -> U and consider N = (Nj), the £ - d u a l of N = (JVy). N = {Nj) is Cartan nonlinear connection of the associated Finsler space (M, F). We have = 0.
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B%) and B£) are obvious and B*z) is equivalent to diNf = djNf (see (2.8)). Let us provethe uniqueness of the Finsler connection which satisfies B{) — B$). If BT = (Nij, Hljk, VJk) is another connection, then it must have the following form:
Taking the lift of this connection on U*, we get a N-connection (d'Nij,dk N^, 0,0), and ((dk Nij)0, dkNj,0, gkh dhSij), is the £ -dual of this connection, where (JVj) is the - d u a l o f (Nij). This d-connection provides a Finsler connection of U
which has the following properties:
*|, = 0,
TV = 0,
DJ = O, Pljk = 0,
CV = 0.
These conditions are sufficient to assure the uniqueness of Finsler connection of U. Now, we can easily prove the uniqueness. Let us put G/k := &Njk. Berwald connection for Cartan spaces has the following, generally nonvanishing curvature tensors: Hh'jk (5.1)
= II {6*k Gh'}+ Q^jQh)
(h-curvature),
(K])
G/A:'1 =8hGj1k
(/w-curvature)
and a torsion tensor Rijk - S*Nki - 6kNji. Proposition 7.5.1. The following relations hold good: (i) Hjik
= -VRw,
(ii) piHh'fl =
-Rhjk.
Proposition 7.5.2. Let (M, )be a Cartan space and Hh'jk, Hhjk the h-curvature tensors of BT and of Berwald connection of its (locally) associated Finsler space, respectively. Then we have in U* (5.2)
Hhijk = -(Hihjky
-
2VlhsRsjk,
where (Hihjk)* means that the value ofHi^jk is calculated in (x,p), y' =
d'H(x,p).
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181
Let us denote by "*", the h-covariant derivative with respect to Berwald connection
T h e o r e m 7 . 5 . 2 . L e t CY a n d respectively. Then (5.3)
b e t h e C a r t a n a n d B e r w a l d c o n n e c t i o n s of ( M , ) ,
(i) Gj\ = H)k - V | , o ,
(ii) 9*ij % = -2Vk%.
Proof. (i) Let us restrict our considerations to the open sets U*, U such that Legendre transformation ip : U* —> U is a diffeomorphism. If we consider the £ - d u a l of Cartan connection CT, CT = (Nlj,Fjk,Cl jjt),we have in U PSjk=Csjk{0 ([88],pagell4). Taking the - d u a l of this relation, we get P.ik where Vsjkl]0 = Vsjkllhph,
= Vsjkll0 *=^ Hksj - 8kNJS = VSJkl]0, ph =
g""pT.
(ii) We have 9*ij I = W j )
+ Gh\g*hj + Ghh9tlh =
Definition 7.5.1. A Cartan space is called a Landsberg space if Vy'*||o = 0. It is called a Berwald space if V»knh = 0 (V^k = g*ilV^k = - \ frg*>k, V^^ = V'jk\]hph). Using the -duality between Cartan and Finsler spaces, we can easily prove: Proposition 7.5.3. A Cartan space is a Landsberg (Berwald) space if and only if every associated Finsler space is locally Landsberg (Berwald) space. The following theorems characterize Cartan spaces which are Landsberg and Berwald spaces. Theorem 7.5.3. A Cartan space is a Landsberg space if and only if one of the following conditions holds: (a) H)k = G/k,
(b) i Y / = 0,
(c) Pij" = 0.
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Proof. Using the £ - d u a l i t y , we get Vij*||o = 0 <=> (c) «=» (b). Then (a) (b) follows from (5.3). Theorem 7.5.4. A Carton space is a Berwald space if and only if one of the following conditions is true:
(a) GA" = 0, (b) BY is a linear connection (that is G/k are functions of position only), (c) Hjk are functions of position only. Proof. Obviously (a) <=> (b). Now let us prove V«fc||fc = 0 * = * Gj\h = 0. The associated Finsler space (U, F) is Berwald space because of ( = 0 -dual of V^*||h = ), therefore the coefficients Fjk of Cartan connection are functions of position only. From
we obtain also that Hljk are functions of position only. Now, using (5.3) and again Vljk\\o = 0, we get (a). Conversely, we obtain G/k' 1 = 0, and (5.3) yields )k - ^ ( V H O ) = 0.
Taking the -dual of this equation, we obtain dhFijk-dh(Ciim)
= 0 or Fjikh-Ciim.h
= 0,
where F/kh is the hv-curvature tensor of the Rund connection of (U, F). From this relation we obtain Cyt^ = 0, following the same way as in [88], page 161, and by -dualization we get V^k\\h = 0. Finally, from the above considerations we can easily prove (a) <=$• (c). Definition 7.5.2. A Cartan space (M, ) is called locally Minkowski space if^there exists a covering of coordinate neighborhoods in which g*xi depends on pk only. Proposition 7.5.4. A Cartan space (M, ) is a locally Minkowski space if and only if every locally associated Finsler space is locally Minkowski space. The following result characterizes the Cartan spaces which are locally Minkowski spaces.
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183
Theorem 7.5.5. A Cartan space is locally Minkowski space if and only if one of the following conditions holds:
Proof. If (M, ) is locally Minkowski space, then iVy = 0 and S*glk = 0. Therefore (i) and (ii) hold. If (ii) is true, using the -duality we can easily prove that (M, ) is locally Minkowski space (see also [130]). If (i) is true, G V / = 0 yields that (M, ) is Berwald space and thus Vi>k\\h = 0. On the other hand, Hh'jk = 0 and from (ii) and Proposition 7.5.1, it follows Rhjk = 0. In the same time G/* = H'jk holds and therefore, we get Rhjk = Hhljk = 0 from (5.1). Definition 7.5.3. a) Cartan space (M, ) is said to be of scalar curvature if there exists a scalar function K = K(x,p) such that (5.4)
Hhl]kPyXhXk
= K(gl]9;k
-
g ^
for every (z,p) e D* and X = (X1) e TXM. b) A Cartan space (M, ) of scalar curvature is said to be of constant curvature K if the scalar function from a) is constant From Proposition 7.5.1, (ii) and (5.4) we easily obtain that (M, ) is of constant curvature K if and only if Rijkp> = K F2h*k, where h*k = g*k — jsPiPk is the angular metric tensor of Cartan spaces. Theorem 7.5.6 (i) A Cartan space is of scalar curvature K(x, p) if and only if every associated Finsler space is of scalar curvature K(x, y), y = ), (ii) A Cartan space is of constant curvature K if and only if every associated Finsler space is of constant curvature K. Proof. Contracting (7.2) by p\p>, Xh, Xk, we get
HhljkPyxhxk =
-{xihjkyPyxhxk
or (5.5)
HhllkVip>XhXk
=
-(HihjkyyXhXky.
(Here the £ - d u a l of X = Xi(x)6* is X° = Xi(x)8i.) But a Finsler space is said to be of scalar curvature K(X, y) if HihjkyiyjXhXk = K{gijghk- gikghj)ylVJXhXk and if K(x,y) = const., it is said to be of constant K (see [88], page 167).
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Now, using (5.5), we obtain the proof. Remark. We can get some similar results as in Proposition 7.5.3, Proposition 7.5.4 and Theorem 7.5.6 for a Finsler space, considering Cartan spaces, locally associated to it. Therefore, some nice results in Finsler space can be obtained as the -dual of those from Cartan spaces.
7.6
Applications of the
-duality
In this section, we shall give some applications of the -duality between Finsler and Cartan spaces. In terms of the Cartan connection a Landsberg space is a Finsler space such that the hv-curvature tensor Phlik = 0 [90]. A Cartan space is called Landsberg if P*h%jk = 0. Using the -duality it is clear that a Finsler space is a Landsberg space iff its - d u a l is a Landsberg one. In [151] (see also [90], [77]) was proved that a Randers space is a Landsberg space iff b^ — 0 (here ";" stands for covariant derivative with respect to Levi-Cività connection of the Riemannian manifold (M, )). For Kropina spaces we have a "dual" of the above result: Theorem 7.6.1. A Kropina space is a Landsberg space if and only if (6.1)
bi;k = bifk - bkfi + ctofibj,
P := o*/i-
Proof. The Randers metric F(x,p) = \Ja%ip{Pj ± blpi is a Landsberg metric iff
b\\K = 0 (here "||" stands for covariant derivative with respect to the Levi-Cività connection of the Riemannian manifold ( M , ) . Hence, we have the following equivalent statements: The Kropina space is Landsberg <=$• its - d u a l is Landsberg
,cf = °-
Ch.7. The duality between Lagrange & Hamilton spaces
But { jk' }} = {{jk ' } - {8ifj + 8}}k - ajkaisfs},
185
where fk := dk(logb) and
u }> { iu } a r e t h e coefficients of the Levi-Cività connection of ay-, respectively JK JK fly (a«j = pGy therefore a nd a^ are conformal metrics). So we get {
which also was obtained in [90], [77] by a different argument. We can obtain other properties of Kropina spaces from those of Randers by using the -duality. Theorem 7.6.2 i) Kropina space is a Berwald space if and only if Vkb{ = bifk - bkf{ +
(6.2)
aikpbj.
ii) Kropina space is locally Minkowski space if and only if the condition (6.2) above holds and also Rihjk
(6.3)
= ] J {ofck/y + ayfhk +
fmfmaikahj},
where fk := dk(logb), / ' := atkfk, fa = V,/,- + / , / , . V* stands for the covariant derivative with respect to Levi-Cività connection of (M, ) and Rihjk is the Riemannian curvature tensor. First of all we need the following Lemma 7.6.1 Let (M, ) be Cartan space with Randers metric F(x,p) = Then
(6.4) (i) (ii)
(M, ) (M, )
is a Berwald space if and only if Vkbt = 0, is Minkowski space if and only if Vkbi — 0,
Rirjk — 0,
where V* stands for Levi-Cività connection of (M , ) and ® o R is Riemannian curvature tensor. Proof. The proof of this Lemma follows step by step the ideas of Kikuchi [77]. For example, here, to obtain that (M, ) is Berwald space on the condition V6j =
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0, we use Theorem 7.4.3 and prove that the Cartan connection of this space is CT = ( {
JK
} pi, { .,} JK
,Vjk),
where { ., } are the coefficients of the Levi-Civita JK
connection of a^. Proof of the Theorem 7.6.2. We take the and we get the Hamiltonian (4.18). We have cnj = — ai: = e2ffay,
-dual
of the Kropina metric (4.14)
a := log 2 - log b.
Therefore, Riemannian manifolds (M, ) and (M, ) are conformal and the coefficients of Levi-Cività connections are related as follows:
The condition (6.4) is written as (6.2). Also, for the conformal metrics, we have
For our position at — - / ; , a^ — —Vifj - fifj = —hi and using the above equality, we get (6.3). Finally, we apply the results of the previous section. Remark. (i) The results of Theorem 7.6.2 were obtained by Kikuchi [77] in a different form and also by Matsumoto [90] in this form, by using of totally different ideas. (ii) A Finsler space with a Kropina metric is Berwald space if and only if (6.2) is true. Indeed, it is easily checked that the Cartan space with Randers metric (4.15) is a Berwald space if and only if Vkbi = 0 (see [151] for Finsler spaces with Randers metric). Therefore, we follow the same idea as in Theorem 7.6.2. Let us now give another example of using of -duality (see also [63]). If (M, F) is Finsler space with
F{x, y) = {a^xWr (m-th root metric [11], [152]), its function Mai(aO) where \ + ^ = 1.
+ ... + -dual
an(x)(yn)m}^m
is Cartan space having fundamental
Ch.7. The duality between Lagrange & Hamilton spaces
187
I n p a r t i c u l a r , if ax{x) = ... = an{x) = em
and its - d u a l is
The geodesies of (M, F), parametrized by the arclength, are just the ecological equations (see [11], [12], [18], [152]). The -duals of these equations have a simpler form: ^ = *H, % = -8lH, at at (Hamilton-Jacobi equations), where H = \ F2. The solutions of (6.5) are h-paths of (M, ) [66]. (6.5)
Chapter 8 Symplectic transformations of the differential geometry of T* M It is well-known that symplectic transformations preserve the form of the HamiltonJacobi equations. However, the natural metric tensor (kinetic energy matrix) is not generally invariant nor is its associated differential geometry. In this chapter we address precisely the question of how the geometry of the cotangent bundle changes under symplectic transformation. As a special case, we also consider the homogeneous contact transformations. The geometry of spaces admitting contact transformations was initiated and developed by Eisenhart [53], Eisenhart and Knebelman [55], where the first contact frame was introduced. Muto [122] and Doyle [52] introduced independently, the second contact frame and the geometry of the homogeneous contact manifolds was intensively studied by Yano and Muto [173, 174]. The chapter is based on [14].
8.1
Connection-pairs on cotangent bundle
Let M be a n-dimensional C°° -differentiable manifold and : T*M —> M the cotangent bundle. As we have seen (Chapter 4) a nonlinear connection on T*M is a supplementary distribution HT*M of the vertical distribution VT'M = Ker TT* . ( is the tangent map of ) . It is often more convenient to think of a nonlinear connection as an almost product structure F on T*M such that VT'M = Ker (I+F) (Section 4.6). I / e f (T*M) and F is a connection on T*M, the push-forward of F by / generally fails to be a connection. Because of this, we will now define a new geometrical structure which nevertheless is an extension of the above definition for connections. Definition 8.1.1. A connection-pair 0 on T*M is an almost product structure on 189
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T*M such that Ker (/ -
(f>) the oblique bundle.
Remark. If 0 is a connection-pair on T*M, then a unique connection F can be associated to it, such that Ker (I -V) = Ker (/ - $ ) , therefore <£ = F on Ker (I -
h=±(i+r)t
(i.i)
v=±(i-r)
and by /i pr , w those induced by
(1.2)
/*'=i(/+0),
t«=i (/-*).
The local expression of F is given by T{di) = % + 2Tijdi, F(^) = -&
(1.3)
and the local vector fields: (1.4)
Si := h{di) =\(di
+ V{di)) = & +
provide us with a frame for HT*M at (x, p). We also obtain: (1.5) On the other hand, (1.6) Indeed, from ^(a { ) = a)fr + bij5j we get
and now (1.6) follows easily. The local vector fields
V^
Ch.8. Symplectic transformations of the differential geometry ofT*M
191
form a basis for WT*M at (x,p) and 0(£) = -S\
(1.8) cf>.
Therefore, {Si,8') is a frame for TT*M at (x, p), adapted to the connection-pair, The dual of this adapted frame is (Sx',Spi) where
(1.9)
6x* = dx'' + UjiSPj,
Spi = dpi -
Tjidxj.
Using the notation above we have the following local expression of associated connection T : (1.10)
<j> = 5i®&xi-8i®5pi,
T = 5i®dxi
and its
-8i®8pi.
With respect to natural frame, <j> has the local form (1.11) From (1.4) and (1.7) we get: Proposition 8.1.1. The adapted basis (<5i, 8l) and its dual {5x\ Spi) transform under a change of coordinates on T*M as follows: (1.12)
Sv = dvx%,
8V = dixH\
(1.13)
8x{' = dixHx\
Sp^ = dex'dp,.
Proposition 8.1.2. If a change of coordinates is performed on T*M, then the coefficients of the connection-pair 4> obey the following rules of transformation
Remarks. 1. In spite of being an object on T*M, IT^ follows the same rule of transformation as a tensor of type (1,1) on M, therefore II lJ are the components of a d-tensor field.
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2. If M is paracompact, there exists a connection-pair on T*M if and only if, on the domain of each chart on T*M there exists 2n2 -differentiate functions Fy and IT-7 satisfying (1.16) and (1.17) with respect to the transformation of coordinates on T*M. 3. Explicit examples of connection-pair on T*M. Let g = (gij) a Riemannian metric on M and { ., } the Christoffel symbols of g. We can define on every domain JK
of a chart
(1.16)
IV, = {*}„*, lfl 13
JL
These are the local components of a connection pair [146]. More generally, if (M, H) is a Hamilton manifold, we can take F^ as the coefficients of the canonical nonlinear connection and II1-' =
Proposition 8.1.3. We have the brackets: (1.17)
M>]=
(1.18)
[#,*i]= (<5injfe + IP1
+(&rik + uj%tk)6k, (1.19)
[6\ V] = R'ijk6k + (n'r
IF'5% k )6 k ,
where (1-20)
- ^n j * + (nir6jrTS Let us put
ujr6Trs)usk.
i[0,0] = fl + i?,
where [<^, <j>] denotes the Nijenhuis bracket of (j> and R, R' are given by R(X, Y) = w[h'X, h'Y],
R'(X, Y) = h'[wX, wY}.
We call R the curvature and R' the cocurvature of the connection-pair (p. R and R' are obstructions to the integrability ofHT*M and WT*M, respectively.
Ch.8. Symplectic transformations of the differential geometry ofT*M
193
Locally we have: (1.22)
R = Rijk8k ® Sxi
R' = RHjk6k ® Spt ® <5pj.
HT*M and WT*M are integrable iff ^ is integrable, or equivalently, R — R! = 0. Definition 8.1.2. A connection-pair
no = u .
Let w = pidx1 be the canonical one form of T*M and 6 = dw the canonical symplectic 2-form. The Definition 8.1.2 above is invariant because of: Proposition 8.1.4. A connection-pair <j> is symmetric if and only if
(1.23)
Proof. 6 has the local expression 6 = dp; A d z ' . We obviously have
From
P6(6i, &i) = -9(6U Sj) ^ we get
9(8i, 6j) = 0,
I F - w% + n M iP r (r jr - r rs ) - o, r sr = r rs .
We obtain therefore, I F = 11^ and Fy- = F^. Corollary 8.1.1. The following statements are equivalent (i) (f> is symmetric. (ii) WT*M and HT*M are Lagrangian (every subbundle is both isotropic and coisotropic with respect to ) . Proof. If 0 is symmetric, using the proposition above we get 0(6i, 6j) = 0, 6(6', P) = 0 and on account of dim WT*M — = dim HT'M = \ dim TT*M it follows that HT*M and VT*M are Lagrangian. Now, if conversely HT*M and VT*M are Lagrangian, using again the proposition above, we get that <> / is symmetric.
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194
Remark. A connection-pair
= 5plAdxi,
ff
6" = 8piA8xi.
4> is symmetric iff 6 = 0' = 0". Its associated connection F is symmetric (that is Fy = or equivalently T*9 = -6) iff 0 = ff. Let C = Pid1 be the Liouville vector field, globally defined on T*M. We denote by T*M the slit cotangent bundle, that is, the cotangent bundle with zero section removed. Definition 8.1.3. A connection-pair on T*M is called homogeneous if the Lie derivative of 4> with respect to C vanishes, that is (1.25)
Lc<£ = 0.
The following, characterize the property of homogeneity for connection-pairs in terms of homogeneity of its connectors. Proposition 8.1.5. A connection-pair 4> is homogeneous iff I\j and geneous, respectively, -1 - homogeneous, with respect to p. Proof. From L C 0 = 0 and (1.5) we get [C,6k]-4>[C,6k] = 0. But, and from (1.6)
-d11 + 2Uhs6s.
Therefore,
frr^fr + (rtH-Pi&Tudn^s, = o
and thus Fy are 1- homogeneous. We also must have: but using the 1 - homogeneity of Fy above, we get
[c, sk\ = \pj\ sk\ = (-uks - Pldinks)6s and thus
-uks
- pi&n1" = o,
that is, II*8 is - 1 - h o m o g e n e o u s relative t o p .
are 1-homo-
Ch.8. Symplectic transformations of the differential geometry ofT*M
8.2
195
Special Linear Connections on T*M
Let $ be a fixed symmetrical connection-pair on T*M and TTM = HTM © WTM, the splitting generated by it. HT*M is the horizontal bundle and WT*M is the oblique bundle. Every vector field X £ X(T*M) has two components with respect to the above splitting (2.1)
X =X
+X
where XH' = h'{X) is the horizontal component and Xw = w(X) is the oblique component of X. We can also introduce some special tensor fields, called -tensor fields as objects in the algebra spanned by {1, Si, 51} over the ring of T(T*M) of smooth real valued functions on T*M. For instance (2.2)
K = Kfk6i ® Sj ® Sxk
is a (2,2) ^-tensor field. For a change of coordinates given on T*M the components of a -tensor are transformed in exactly the same way as a tensor on M, in spite of Pi dependence, thus K is a d-tensor field. Definition 8.2.1. Let V be a linear connection of T*M and <j> a connection-pair. We say that V is a
V(/> = 0 and Vw = 0.
It can easily be proved that V
Proposition 8.2.1. Under a change of coordinates on T*M the coefficients of a $ connection V change as follows: (2.5)
H^,k, = dixi'd3,xjdk<xkHijk + dhx1'dj,dklxh,
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V{/ = dix{'djXj'dk>xkVklj +
(2.6)
dkxi'dhxj'Sh(dklxh).
Remarks. 1. (2.6) is equivalent to
Vlk'/ = dtf'd,x>''dvxkVkij +
(2.7)
Il'udhxi'dk>xtdedlx*.
2. A -connection can be characterized by a couple of coefficients (//]fc, Vjk) which obey the transformation law of (2.5) and (2.6), if a change of coordinates on T*M, is performed. A <j> -connection on T*M induces two types of covariant derivative: (a) the h-covariant derivative V$Y:=VXH>Y
(2.8)
V X, Y £ X{T*M)
(b) the w-covariant derivative V%Y:=VxwY
(2.9)
VX,Y
£X(T*M).
IfK is the -tensor field of (2.2), then the local expressions of its h- and w-covariant derivative have the following form:
VfkK = K£,\k6i ® 6j 0 Sxf ® 5Pi,, where, (2.10)
Kj)'\k = fikKj}' + HtkKj)' + HtkKjy ~ HjkK"? ~ Hj'kKft,
(2.11J
Kjf\
- d Kj:j, + Ve Kjj, + Vt Kjf
- Vj Kej, - Vjt
Kjt.
Let C = pid' the Liouville vector field on T* M. Definition8.2.2. A – connection V on T*M is of Cartan type if V H C = 0 and VWC= I.
Ch.8. Symplectic transformations of the differential geometry ofT*M Proposition 8.2.2. A (2-12)
(2.13)
-connection Pilj
= 0^
197
is of Cartan type iff E%Vk - r y = 0
= s{ <=• vppk + n"r« = o.
ftp
Remarks. 1. If (2.12) is verified, we say that V is h-deflection free and if (2.13) is true then V will be called v-deflection free. 2. When IP-7 = 0, <j> is just the connection F which arises as usual. We will denote this -connection by o . Locally, we have
o
(2.14)
o
o
V a * =H)&>
V6
o
VfS
Theorem 8.2.1. Let
o
V induces a V X Y = w{VxYw)
-connection + h'{VxYH'),
V on T*M, given by: X, Y €
X{T'M).
The local connectors of V are the following:
(2.16)
H* = Hfc
V?j - V? - ff'HJi.
Proof. V from (2.14) is clearly a linear connection. Let us find the local form of this connection.
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We have
= h'(vfsk - n>sHkjssk) = {vf - nishkjs)6k,
Therefore, V is a -connection and also (2.16) are verified. Let V be a (j> -connection and (2.17)
T(X,Y) = VXY-VYX-[X,Y],
its torsion. Locally, with respect ot the frame ($,<£'), we have
(2.18)
(5j,Si)
=T%Sk+fljk8k,
?(#*)
^ %^«fc
where
p.ki =
ski:> = v£ - vkij -
Ch.8. Symplectic transformations of the differential geometry ofT*M
199
o
Proposition 8.2.3. The torsion connectors of 'V and V are related as follows: o
T ij = T tj — RJUH pki (2-20)
=p
+
Tijk = Tijk, uuRjimUmk
+
dTjtntk),
=P'kj-U'eR]ek,
P'kj
sklj
ki _ (6jUik
1
,
= skij + (uiesirtk - we6irek).
However, V has an extra torsion tensor Si:>k = R'i%k which does not occur when o
Uij = 0. It is clear that V is h-deflection free iff V is h-deflectionfree. The following result gives the relations between v-deflection free tensors: o
Proposition 8.2.4. Assume that V is h-deflection free. Then V is v-deflection O
free iff V is v-deflection free. Proof.
?Pi - nkeHiePj
= - n w r « <=^> vikpj = o
ft!* - tf (here | denote the v-covariant derivative induced by The curvature tensor of a 4> -connection V,
).
R(X, Y)Z = VxVyZ - VyVxZ - V[XtY\Z has three essential components. We have: R(Sh,Sk)Sj (221)
=RlmSu
Ri^SJSj
=PlhjkSu
j^Sj
=S/kh8u
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200
where (2.22) #jkh /**
= U {&kH)k + HRHU) (h,k)
fc
- Rhkt(UimH'jm
= 6h(H'jk) - 6k(V?) + H°kVsih M
hm
e
u Rkmrw
+ V}e),
HlkV°h
+ a r f c r a n r a ')^ - (9T*, + uhmRkms)v}is, h
U where
U {• • •} indicates interchange of i and k for the terms in the brackets and
subtraction. Let us consider the diagonal lift metric tensor on T*M G = glJ(x,p)Sxi
(2.23)
® 8x> + gij(x,p)5Pi
where gtj is a symmetric nondegenerate
® 6Pj,
d-tensorfield.
Theorem 8.2.2. Let 4> be a fixed connection pair on T*M. Then there exists only one
=0,
(ii) gij\k
=0,
(iv)S/*
=0.
The coefficients of this <j> - connection are the following: (2.24)
H* = i
km
9
(6i9mj
(2.25) Vt = - \ gim(6j9mk
+ 6j9im - 6m9ij) - 1 gkm{A\m9sj + Skgjm - 6mg^k) - i
sj
9im(g
B?k
+ Asjmgsi + + g°kB^
where (2.26)
A)k := Rwll";
Bkj := Uhr6jTri -
Proof. (i) is written in the following from: (2-27)
6k9lj = H%
WT6krrt.
+
A^gm), gmsBk'),
Ch.8. Symplectic transformations of the differential geometry ofT*M
201
and by using the same technique used to find the Christoffel symbols for Riemannian manifolds, and the first of (2.19), we get (2.24); similary for (ii), but using the last O
of (2.19). In particular, the – - connection A*. = \ gkm(6i9mj
(2.28)
Vf = - \
(2.29)
V has the coefficients + 5]9im -
mk
6m9lj),
+ dk9>m - dm9>k)
9m(&g
and verifies (i) - (iv); this connection is metrical with respect to G = gij{x,p)dxi
(2.30)
o
o
Theorem 8.2.3. Let V be the G - metrical connection above, and V its induced •$ — connection (2.15). Then V is G — metrical. Proof. We must show only that V is v-metrical. By virtue of (2.16) we have: 9ij\k
=Skgij
= dk9ij - UksSs9lJ - ( V f - nk°Hi)gej = dk9lJ - Vik9ji ok
os
- Vfgit
- {Vf - UksH%
- Uks(6s9ij - H\sgt3 -
H(js9u)
- n f c % | =o. o
Remark. This -connection, induced by V, is the appropriate one for studing the geometry of T*M endowed with the metric tensor (2.23). Eisenhart [53] and also Yano-Davies [172] used a similar connection.
8.3
The homogeneous case
We specialize, here, the results of previous sections in the particular case when (3-1)
g*{x,p)=±&
and H is a real smooth function on a domain D* c T*M, 2-homogeneous in Pi and such that the tensor (g^(x,p)) is everywhere nondegenerate on D*.
(3-2)
fy=7°~
The Geometry of Hamilton & Lagrange Spaces
202 where we have put, as usual,
1% •=lfjPkPj,
Pl-=9ijPj-
Note that F is deflection free, [97]. However, the geometry of Cartan manifolds as given, is dramatically changed under a diffeomorphism which is not fiber-preserving. In this case, the geometrical approach described in the previous section is the correct one to use. o
Theorem 8.3.1. Let <j> be a homogeneous connection-pair on T*M, such that PiUlj — 0. If V is the cj> - connection given by (2.24), (2.25) and g'3are those of (3.1), then (i) V is h-deflection free iff
(3.3)
f i, = Ti3 + ~ pm(Rimenes9sj +
Rjmentsgsi).
(ii) If V is h-deflection free, then VtPk = 0. (iii) If V is h-deflection free, then V is also v - deflection free. Proof. (i) By definition V is h-deflection free iff Fy = H^pk- From (3.1) we see that Vijk = dlgjk
(3.4)
is completely symmetric and because gik is 0 - homogeneous we get Pi^y* = Pldjg'k = vifrgij = 0, and also (3.5)
ptygtj = -p'gagjmd'g1"1 = -pt^g^m
= 0.
Taking into account that PiUlj = 0 and using the above identities, from (2.24) transvecting by Pk, we get: ri> = 7° - \pmrmsds9lJ
- \pm{Rim0sgSJ
+
Rjmeues9si).
Ch.8. Symplectic transformations of the differential geometry ofT*M
203
Transvecting again by p>, we obtain Fi0 = f°0 and thus ds9i0°so - \ pm(Rimene°gsj +
r y = 1ij-\
RjmiUlsgsi),
that is, (3.3) holds true. To prove (ii) we need the following: Lemma.
If V is h-deflection free, then Rijtp1 = 0.
Proof. Because V is h - metrical and h - deflection free and H = g^PiPj we get 8kH = 0, that is, diH = -Tij&H.
(3.6) Therefore,
RsktP1 = [S.TU - 8kvsl)pl = [(dsrke - dkrs() + (rjTkt +Tsidi(rktdlH)
- TufrpttfrH)
~
- VsiVkldidtH + TkiTsldidlH
= -dsdkH + dkdsH - Tkld,frH + T^frH
- Tti&dkH + Tkid%H
= 0. Let us now prove (ii). We have
- \WsdsdmH-
\ W'Tufr^H - gj
\ WsdsdmH = i
Thus, Pksg
WarTu
rkifrr.t)]dlH
n
and by using the above lemma we get, pk6jgmk - pk8mg'k = 1 (n^<5 m r rs - Umr8jrrs)ps + (Ujsgme
The Geometry of Hamilton & Lagrange Spaces
204
On the other hand,
and
Pk9'kBTj = \ ( i r » w r j - rpr 6mrrs)P*. From the last four equalities, by again using (3.5) and p;IP 3 = 0, we finally get,
Vfpk = 0. (iii) From the last of equation (2.19) we get
= Vppj + (IF'fiT* -
tf'PTJpj
= -UksTsi
and thus V is v - deflection free. Remarks. (1) The condition pillt-' = 0 was used by Yano-Davies [172] and YanoMuto [174] and it holds when WT*M is the image of the vertical subbundle through a -regular homogeneous contact transformation (see the next section). In fact, in this case the Liouville vector field C = ptd' belongs to WT*M. (2) Equation (3.3 ) shows how to select the connection Y such that (2.24), (2.24) hold and the connection is of Cartan type.
8.4 f-related connection-pairs Let (f> be a connection-pair on T*M and WT*M ->• T*M, HT'M -> T'M, the oblique and horizontal bundles. We denote by F its associated connection (nonlinear). Definition 8.4.1. / € Diff (T*M) is called F - regular if the restriction of the tangent map (TT/)» to HT*M (TT/)» : HT'M
-¥ TM
is a diffeomorphism. If / has the local expression f(x,p) = (i(i,p),p(x,p)),then it is F -regular iff (7r/),(Ji), i e l , n are linearly independent, that is, the matrix with entries (4.1)
0* := 6txk = d{xk + Tijdjxk
has maximal rank. Theorem 8.4.1. Let (j> be a connection-pair on T*M and f E Diff (T*M). The following statements are equivalent
Ch.8. Symplectic transformations of the differential geometry ofT*M
205
(i)
Theorem 8.4.2. The coefficients of two f— related connection-pairs <j> and
efry^p,, (d% - d%xeTek)Ukh = nij0$ -
Proof. From
=(Sixk)d,
P
Therefore, that is (4.2). Now, let us prove (4.3). We have
WT'M := Ker(/
The Geometry of Hamilton & Lagrange Spaces
206 But,
=
Vx'T^P.
f,(Sl)eWT*Mffi <j%* - ^ 5 , 1 1 ' * + <^lpsff* = o.
By using (4.2) we get (4.3). Corollary 8.4.1. If {8i,8'), I f are the adapted frames at (x, p) and {x,p) induced by $ and <> / respectively, then: (4.4)
/,(*) =*??*,
(4.5)
h{8i)=Sip,
where ey^d^-d^T^.
(4.6)
The regularity of ^j follows from (4.5) but is also a consequence of P r o p o s i t i o n 8 . 4 . 1 . ( i ) f i s - regular I 6 f i s - regular,
iff f~x i s then
(ii) ((Ji35*)(Jfci>) = <5?; and (iii) ( * p t - d*xhThk)(dkPj
- Wx*Ysj) = S>.
Proof. We have
and (i), (ii) follow. To prove (iii) we use the following equalities:
regular.
Ch.8. Symplectic transformations of the differential geometry ofT*M
207
Generally, if F is a connection on T*M the push-forward of T is not a connection. Some consequences of Theorem 8.4.2 are the following. Proposition 8.4.2. Let V be a connection on T*M and f e Diff (T*M) F - regular. Then = / . I 1 / " 1 is a connection-pair; the coefficients of the connection Y associated to
(4.7)
U1' =
0ixjFsk-Ppk)dkxj.
Proposition 8.4.3. The push-forward of a connection-pair cf> by a V -regular diffeomorphism is a connection if and only if (4.8)
IF = dhffihX1.
Corollary 8.4.2. The push-forward of a connection T by a V -regular diffeomorphism is also a connection iff f is fiber preserving (that is, locally, f(x,p) = {x{x),p{x,p)). Now we will study when the push-forward of symmetric connection-pair by f is also symmetric. Theorem 8.4.3. Let
By using (4.4) and (4.5) these conditions are equivalent to f*e(8i,Sj) = 0 and /*0(<5\
The Geometry of Hamilton & Lagrange Spaces
208
Corollary 8.4.3. <j> is a symmetric connection pair iff (4.9)
Wfc** = Wfctf, frpiPx1
(4.10)
= 6hPi6kx'.
Proof. We have: f*6 = dpt A dxi = (Skpi6xk + S'piSps) A (ShxiSxh + F&Spr). By using Theorem 8.4.3 we get the equalities above. Note: F is symmetric iff (4.9) is verified. Theorem 8.4.4. Let f be a V-regular iff (j> is symmetric.
symplectomorphism.
Then
is symmetric
Proof.
-Lagrangian
To summarise the results above we can state the following: Theorem 8.4.5. Let f G Diff {T*M), T-regular, such that /*0(<5\ &i) = 6). Then each pair of the next statements implies the third: (i) f is a symplectomorphism (ii) 4> is symmetric (iii) 4> is symmetric. Remark. 1. The condition f*6(6', 8j) = <5} is equivalent to (4.11)
frp&i*
- S'x'Sjp, = 5)
and it is obviously verified when f is a symplectomorphism. 2. If f is a -regular symplectomorphism then e WT*M) = },(HT*M) Proposition 8.4.4. Let f be a V-regular
®
f,(WT*M).
symplectomorphism
ofT*M.
Ch.8. Symplectic transformations of the differential geometry ofT*M
209
Then
(4-12)
*# = 4, *$ = <£•
Proof. It follows from using (4.4) and (4.5). Note: Under the conditions of the above result and Proposition 8.4.1 we have: (4.13)
6) = dp + Tjkdh? = PPj - rtjPx',
and its reciprocal, (4.14)
6) = d% - T^x* = djx' + Tks8sx\
Proposition 8.4.5. I € Diff(T*M) is a T -regular symplectomorphism and i (6x ,8pi) is the dual of (?<,?*) then (4.15)
M6xi) = %S3>;
Proof. and first equality (4.15) follows. A similar proof holds for the second.
f.
Let us now study the connection between curvature tensors R' and R' of F and
Proposition 8.4.6. Let be a connection-pair and <j) its push-forward by a – regular symplectomorphism. If R and R are the curvature tensors of F and F then: (4.16) (4.17)
The Geometry of Hamilton & Lagrange Spaces
210 Proof. From (1.17) we get:
Msit
/.[Mil
=io?5k,ef5t} = ek6k(e™)5m - e$.{e?)$m + eke°[RksJm + Rksenem5m)
and the relations above follow immediately. Corollary 8.4.4.
(4.18)
8.5
If
= 0 then
RiJkuke = erUO - ojUelj.
f-related >-connections
Let us now investigate the behaviour of geometrical objects described in Section 8.2 under symplectomorphisms. If <j) is a connection-pair on T*M and / : T*M —» T*M is a -regular symplectomorphism, then the symmetry of the connection-pair (j> = f*4>f~l is preserved and also
where We can construct a new geometry on T*M, generated by f, by pushing forward all geometrical objects described in Section 8.2, there by extending to a more general setting, the results of [55], [146], [147], [175]. For instance, if K is the tensor field, locally given by (2.2), then we can consider its push-forward: TC = TF-^Si'
In particular, the push-forward of G from (2.23) has the following local form: (5.2)
G = gij8xl®'5x:>+
gij5pi®5pj,
Ch.8. Symplectic transformations of the differential geometry ofT*M
211
where
r o / = eiei9kh.
(5.3)
If V is a linear connection on T*M, we define its push-forward by f as follows: (5.4)
VrF:=/,(Vxn
JC = f,(X),
Y = U{Y).
V is clearly a linear connection on T*M. Proposition 8.5.1. (i) V is a > -connection iff V is a 4> -connection. (ii) V is G-metrical iff V is —metrical. Proof. (i)
V0 = o ^v Y ^(F)=^(V r F)) On the other hand, because f is symplectomorphism and thus V0 = 0 <=> W = 0. (ii) This follows from
Proposition 8.5.2. The coefficients of V are related to those of V by the following relations: (5.5)
% =
9^9^,
(5.6)
Tl = ei^e^v^'
Similar theorems to those of Section 8.2 (Theorems 8.2.2 and 8.2.3) hold when V is replaced by V and G by G.
The Geometry of Hamilton & Lagrange Spaces
212
8.6
The geometry of a homogeneous contact transformation
In this section we will restrict our considerations to the slit tangent bundle T*M (the cotangent bundle with zero section removed) instead of T*M. Let u) be the canonical one form of T*M, locally given by ui = pidx1.
(6.1)
Definition 8.6.1. Adiffeomorphism / : T*M —> T*M is called a homogeneous contact transformation (h.c.t.) if w is invariant underf, that is
(6.2)
/•« = w.
Proposition 8.6.1. Iff is a h.c.t. then /»(C) = C. Proof. We use the property of the Liouville vector field C = Pidl, as the only one such that icdu — u> where "i" denotes the interior product of C and da;.We have
for every X G X{T*M). Note: The set of h.c.t. is clearly a subgroup of the group of symplectomorphisms of T*M. Corollary 8.6.1. If f{x,p) = (x(x,p),p(x,p)) is the local expression of a h.c.t. then x — x(x,p) and p = p(x,p) are homogeneous of degree 0 and 1 with respect to
Proof.
f,(O
=c
<=^ Pid'xkdk + Pid'pkdk = pkdk { k = 0, Pld% = pk. Pid x
Remarks. (1) See also [53] for another proof of this result.
Ch.8. Symplectic transformations of the differential geometry ofT*M
213
(2) A h.c.t. is a symplectomorphism, therefore we must have: (6.3)
%•** = &&,
dipk = -dkPudixk
= -dkx\
d% = dkx\
If x = x(x,p), p = p(x,p) are homogeneous of degree 0 and 1 with respect to p , Eq. (6.3) are also sufficient conditions for f(x,p) = (x,p) to be a h.c.t. In [53] it is proved that f is a h.c.t. then (6 .4)
dixkdjpk-djxkdipk djxkd'Pk-dixkdjpk
= 0, =6),
dixkdipk~djxkdipk
= 0,
which in fact results from (6.3). (3) If /o e Diff (M) then the cotangent map induced by i s a h.c.t. In fact, if x = i(a;) is the local form of /Q then, f(x,p) = (x(x),p(x,p)),
pk = Pidkxl.
In this case f is called an extended point transformation. It can easily proved that every fibre preserving map which is also a h.c.t. is an extended point transformation (see also [146]). (4) The reason to use the word "contact" in the name of this transformation is given by the property of preserving the tangency of some special submanifolds of T*M. (See [53], [146].) Proposition 8.6.2. Let 4> be a connection-pair, Y its associate connection andf a -regular h.c.t. (i)
Let H : T*M —> M be a 2-homogeneous regular Hamiltonian and H the pushforward of H by f, (6.5)
H = Hof-1.
The Geometry of Hamilton & Lagrange Spaces
214
77 is also 2-homogeneous Hamiltonian, but the matrix with entries gij = WWTI
(6.6)
c
c
may not be regular. Assume also that f is T -regular, where T is given by (3.2). Using the homogeneity property of f we get ft = P*#ft = PkifrVi - dkx°Ysi) = fl?Pfc. Therefore, Pi = ekiPk and pk = 0\pi.
(6.7)
Let G be the metric tensor (5.2). The push-forward of G is given by (2.23), where glj = d'&H.
(6.8) We have
gii{x,p)pipj
=pipj9ik93hgkh{x,p)
=
pkphghh(x,p).
Therefore, (6.9)
77=
-t'pipj.
Of course, we also have but g'j 7^ g1* may happen. In fact, the metric tensor induced by f is g*J and not glj, in general. The tensor i s 0-homogeneous with respect to p{ and nondegenerate, but it may lack the property which assures that following Section 8.5, the geometry, as in Section 8.3, can be derived from it. Therefore, it is from
Proposition 8.6.3. (i) WE = O^Ho
f~\
Ch.8. Symplectic transformations of the differential geometry ofT*M
215
Proof, (i) WE = dsH~5kx3 + daH~5kps. But 5SH = 0 = » dsH = -Yst&H of homogeneity of ftj,
= -Tsepe
and by using (3.2) we get, because
d.H = - 7 s ° 0 = - i W .
(6.10) Therefore,
- (7,° -
\
where we also have used (4.13). Therefore, we get (i). Now, W&H = 6in{dmdkHWpk
+ dmdkHWxh)
+
Using equation. (6.9) we obtain dmdkH = -dmrkePe
-
and this equality transforms into (ii) after a straightforward calculation. Note: The relation (ii) above is just (3.19) combined with (3.20) of [54], if we start with a Riemannian metric glJ = 7<J'(a;). Remark. <^ = rfi •<=> pk{dmd\dmW - (9^j. - eihdhTek)dex:i) we see that this equality is equivalent to pWei = A}kd^xk
and
3
P
(dii
k
^
for some functions A\. But from these equalities we get
(see also [54], (3.26)). Also note,
tfi
=tfi
holds d{kjPi
= 0. By using (6.3)
= 0
The Geometry of Hamilton & Lagrange Spaces
216
As a consequence of the discussion above and results of previous sections, we have the following summary: c
o
o
(a) If we start with a Cartan manifold (M,H), we get the triple (F, V,G) given by (3.2), (2.28), (2.29) and (2.30) V is a f-connection, G -metrical, of Cartan type and the torsion tensors T\^ S^ vanish. C
(b) Taking a -regular h.c.t. we get a new triple (
6iH = 0
and
H'f\3' = H ]'}< = 2fj
where | denotes the v - covariant derivative with respect to V. (c) I fJ = g1*, then H is a regular Hamiltonian and Theorem 8.3.1 is valid (baring all the coefficients). A simple consequence, for the deflection-free case, is : (6.12)
r = f
iff
(Rtmtnts9aj
+ Rjmtnes9si)pm
= o.
The relation (6.11) can be also written by using Proposition 8.4.6 and (5.3) in terms of similar objects derived from H. (Rimi and gsj are contact transformations of Rimi and ). When (6.12) is verified, by virtue of Theorem 8.2.1, Proposition 8.2.2 and Theoc
o
o
o
c
o
rem 8.2.3 we can pass to the triple (P, V , G) when V is a F - connection, G - metrical, h- and v-deflection free, but generally fails to have vanishing torsion tensor TJ fc ,Sj*. Therefore, it does not coincide with the Cartan linear connection for the Hamilton manifold {M,H) [66], [97]. c
If f is an extended point transformation, then n = 0, F = F and the pushforward of the geometry of Cartan manifold (M, H) is just the geometry of (M, H) so this geometry is invariant.
8.7
Examples
We now construct a connection-pair on T*IR2 which is horizontally flat, but with complicated I n fact, we construct a homogeneous contact transformation be2 tween (T*1R ,H), where IF' = 0 and F
Ch.8. Symplectic transformations of the differential geometry ofT*M
217
Ti:i = 0. Here, M = \ (P? + P22) is the Euclidean Hamiltonian in T*IR2, spanned by
(Q\Q?,PUP2),
and^ = ^ .
Select, once and for all, a Finsler metric function A = X{ql,q2,Pi,P2) metric f^q^Pi) = e ' 2 ^ ^ • (fr&X2) and set
where,
1 2 1
1 2 1
tf
1 2
and the
W = A,
is defined in terms of C°° functions / i , / 2 and A, some constant. Noting that Pidti? - Pidq* = 0, we have the possibility of constructing the desired contact transformation (Q, Pi) •(q',Pi) and its inverse, locally. But, two side conditions will be necessary for this. Firstly, det, = det ( f ^ ) ^ 0, must hold in some chart (U, h). Then Pi = - A W Q1 - A f^1 df,
p2 = - A W
Q1 - A f?92 ^
has a unique solution in {U,h),
Of course, Pi
so that the transformation is determined by fi and djfi. It is also required that the transformation by - regular. In this case, F^- = 0, so this condition is merely det (
o ^) = I where,
We have supposed that 4>{ql,q2) is known and defined in a chart (U, h).
218
The Geometry of Hamilton & Lagrange Spaces
Secondly, we must now select fi(q\q2) so that the F -regularity condition holds, locally. Set IR = - (P2 A) d\ + (pi A) d?, to denote this linear operator and assume A is independent of ql,q2. Thus, A is a Minkowski metric function in the chart (U,h). Note that Q1 = - l/det / ]R(/ 2 ) and Q2 = - l/det / IR(/ 1 ). Proposition 8.7.1. Under the condition det/ ^ 0 in (U,h) – regularity holds for A Minkowski <{=» di(£n det / )[(]R/2)(IRoS 2 )(/ 1 ) - (IR/j) • (IRo9 2 )(/ 2 )] -d 2 (&i det/) x[(JR/2)(IRo «,))(/!) - (R/ 1 )(IRo3 1 )(/ 2 )] / [ ( E o ^ t / , ) ] • p o f t ) ( / 2 ) ] - [(IRo in (tf,/.). Corollary 8.7.1. In addition, assume / 2 = c-q2, where c> 0 is a sufficiently small constant. Then F -regularity holds in (V,h) «=>• Hess (.A) ^ 0 in (V,h) (Hessian determinant) and J\ = \j^ 2-disk).
— (c • q2)2 . Here, V C B C U
(B is interior of closed
Proof. A short calculation shows that the condition of the proposition reduces to the non-zero Hessian condition. An easy continuity argument shows that /1 above is well-defined in some closed 2-disk in (U,h). Merely note m < 4>{ql ,q2) < M holds in any closed disk B C (U, h) and take the radius r — £ em so that c-q2 < e^ in this, B (radius = r). Now choose a smaller chart V in the interior of This completes the proof. Also note that by linear adjustment, we can always suppose that $ (center of ) = 0 in IR2. We can now state the Theorem 8.7.1. If has a non-degenerate critical point x in (U,h) of IR2, then - regularity holds in some neighborhood of x. Consequently, (T*IR2, H) is homogeneous contact equivalent to (T*1R2, H) where H = l^PiPj — 5 e~2tt>(dl&\2)piPj. Moreover, I F is by (4.7) not zero generally and is completely determined by W — 0, V^ = 0 and this transformation. Similar results are possible even if <j> has no nondegenerate critical points. For example, if <j> = utq', u>i constants, the conclusion of the theorem above holds. It can be reformulated as Theorem 8.7.2. Any 2-dimensional constant Wagner space is the Legendre-dual of the homogeneous contact transformation of the flat Cartan space (T*JR2,H) with non-trivial oblique distribution II. Similar reformulations can be made of the main theorem on F -regularity, as well, using the known result that Wagner spaces with vanishing h -curvature must have local metric functions of the form e* • A, [11]. These have been found to be of fundamental importance in the ecology and evolution of colonial marine invertebrates (ibid.).
Chapter 9 The dual bundle of a k-osculator bundle The cotangent bundle T*M, dual of the tangent bundle carries some canonical geometric object field as: the Liouville vector field, a symplectic structure and a Poisson structure. They allow to construct a theory of Hamiltonian systems, and, via Legendre transformation, to transport this theory in that of Lagrangian systems on the tangent bundle. Therefore, the Lagrange spaces Ln = (M, L(x, y)) appear as dual of Hamilton spaces Hn = {M,H(x,p)), (cf. Ch.7). In the theory of Lagrange spaces of order k, where the fundamental functions are Lagrangians which depend on point and higher order accelerations, we do not have a dual theory based on a good notion of higher order Hamiltonian, which depend on point, higher order accelerations and momentum (of order 1, only). This is because we have not find yet a differentiable bundle which have a canonical symplectic or presymplectic structure and a canonical Poisson structure and which is basic for a good theory of the higher order Hamiltonian systems. The purpose of this chapter is to elliminate this incovenient. Starting from the fc-osculator bundle (Osck M,irk,M), identified with ^-tangent bundle (TkM, irk,M), we introduce anew differentiable bundle (T*kM,n*k,M) called dual bundle of kosculator bundle (or k-tangent bundle), where the total space T*kM is the fibered product: T*kM = Tk~1MxMT*M. We prove that on the manifold T*kM there exist a canonical Liouville 1-form, a canonical presymplectic structure and a canonical Poisson structure. Consequently we can develop a natural theory of Hamiltonian systems of order k, for the Hamilton functions which depend on point, accelerations of order 1,2,..., k - 1 and momenta. These properties are fundamental for introducing the notion of Hamilton space of order k. We will develop in the next chapters the geometry of second order Hamilton 219
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220
spaces and we remark that this is a natural extension of the geometry of Hamilton spaces, studied in the previous chapters. All this theory is based on the paper [110] of the first author.
9.1
The {T*kM,K*k,M) bundle
Let M be a real n-dimensional manifold and le tixk,M) be its k-osculator bundle. The canonical local coordinates of a point u e TkM are ( I ' I J / ' 1 ' 1 , . . . , ! / ' ' ' 1 ) , •Kk(u) = x and the point u will be denoted by u = (z,2/ (1) , . . . , ( * ) The changes of coordinates on TkM are given by [106]:
=
*»
x'(xl,...,xn),
det
-
For every point u e TkM, the natural basis
dxi
Ch.9. The dual bundle of the k-osculator bundle in Tu(TkM)
221
is transformed by (1.1), as follows: dx] dxi
d dxi
QyWl dx%
U
dxi
d d'l
(1.2) dy^
d
d
Qy(k)i
where (1.3)
dxi
For a 6 {0,1,..., k -1} we denote by TT£ : TkM locally expressed by
teby
the canonical submersion,
Every of these submersions determines on TkM a simple foliation, denoted by Ta+iThe sheafs of Fa+\ are embedding submanifolds of TkM of dimension (k - a)n on which {yia+1]i,--.,y{k)i) are the local coordinates and (x\yW, ...,2/ (a)l ) are the transverse coordinates. Every foliation .F a+ i determines a tangent distribution Va+l = TFa+l = Kerdirka, (a = 0,..., k - 1), where dvr* is the differential of the mapping TT*. Therefore, we have a number of k distributions V*i,..., V* which are integrable, of local dimension kn, (k - 1)n,...,n, respectively, and having the propertythat V u e TkM,
V I ( u ) D V2(u) D - - - D
Vk{u).
The manifold TkM carries some others natural geometrical object fields [106], as
the Liouville vector fields T, ...,F. T belongs to the distribution Vk and is given by 1 k m- & t 9 3 loy Wt (1 F = y (fc)., ... , F belongs to Vx and has the form V = y >n „ , , + ? d ky Mi
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Moreover, there is a tangent structure J defined on TkM. It is given by r\
7—
-®dy>
,(*-!)•
We have the properties k-l
*
k-\
fc-2
2
1
1
jr = r , J r = r ,..., JT = r, jr = o. However, it does not exist a canonical symplectic or presymplectic structure over the manifold TkM. We introduce the following differentiable bundle: Definition 9.1.1. We call the dual of the k-osculator bundle (TkM,irkM) differentiable bundle (T*k M, ir*k, M) whose total space is TtkM = Tk~1M
(1.4)
the
xMT*M.
The previous fibered product has a differentiable structure given by that of the (k - 1)-osculator bundle Tk~1M and the cotangent bundle T*M. Of course, for k = 1, we have T*lM = T*M. We will see that over the manifold T*kM there exist a natural presymplectic structure and a natural Poisson structure. Sometime we denote (T**M,TT**,M) by T*kM. A point u e T*kM will be denoted by u = {x,y(l\...,y{k~l\p). The canonical projection TT** : T*kM —> M is defined by -K*k{x,y^\...,y^k~l\p) = x. Of course, we take the projections on the factors of the fibered products (1.4): TT£I :
TkM
rpk-l
M, 7T* : T*M
as being n^Li[x, ?/(1), ...,y{k~l\p) = (i, yw,...,y<-k the following commutative diagram:
J)
M
) and n*(x,p) = x.lt results
T"kM
T'M
M where •n* (x,yw,..,yik
1)
,p) = (x,p).
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Ch.9. The dual bundle of the k-osculator bundle
This diagram implies the existence of some natural geometrical object fields over the manifold T*kM. Let {xt,yWl,...,y(k~Vlt,Pi), (i = l,...,n), be the coordinates of a point u = (ar,!/ (1) ,...,y ( *- 1) ,p) € T*kM in a local chart (ir*-1 (U),
xi = xi(x\...,xn),
det
—
(1.5)
_
dxj
where the following relation holds:
(1.5)'
It follows that T**M is a real differentiable manifold of dimension (k + 1)n. It is the same dimension with that of the manifold TkM. With respect to (1.4) the natural basis
d
d
dpi
224
The Geometry of Hamilton & Lagrange Spaces
in Tu(T*kM) is transformed as follows: dxj d dxi dxJ
_ dxi
Qxi
(1.6)
d
\-
dxi
Qy{k + l)j d
H
h
Qy(k-l)i
3 ~~ i' a ~~
'
the conditions (1.5)' being satisfied. Consequently, the Jacobian matrix of the transformation of coordinate (1.5) is given by 0
0
as* (1.6)'
o
o
0
0
Jfc(u) =
m
°°
9a;'
It follows: (1.6)"
dx'
det Jk(u)= |det(^j(u
it—I
Theorem 9.1.1. We have: a. Ifk is an odd number, the manifold T*kM is orientable.
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225
b. If k is an even number, the manifold T*kM is orientable if and only if the manifold M is orientable. Following the property of the bundle (TtkM,K*kk_l,Tk~1 M) it is not difficult to prove: Theorem 9.1.2. If the differentiable manifold M is paracompact, then the differentiable manifold T*kM is paracompact, too. Let us introduce the following differential forms: (1.7)
u}
(1.7)'
e = du = dpiAdxi.
Then, we have: Theorem 9.1.3. 1° The forms ui and 8 are global defined on the manifold T*kM. 2° 9 is closed, i.e. dO = 0. 3° 8 is a presymplectic structure of rank 2n on the manifold T*kM. Proof. 1° The forms ui and 8 are invariant with respect to (1.5). 2° d8 = d2u = 0. 3° 8 is a 2-form of rank In and 2n < a + n = d
q.e.d.
Now, let us consider the system of Poisson brackets: (1.8)
{ }a : (/,g) e T(T*kM) x T{TkM)
—• {/,g} a €
F(T'kM),
defined by
We obtain: Theorem 9.1.4. Every bracket { } Q , (a = 0,..., k — 1) defines a canonical Poisson structure on the manifold T*kM. Proof. We prove that the Poisson bracket {/,g} a , (a = 1,...,A; - 1) is invariant under the transformations of coordinates (1.5) on T*kM. Indeed, by means of (1.6)
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we have
_ dy(a)m
df
dy(a+l)m
df
)
^ I
r-.
df
df
m / _\ •
nw/L
i\
dpm df
I
Qy(a)i
But the first formulae, by means of (1.5)', can be written as follows: dxm df dx' dy(a)m
df
dxi
dy{x)m df dx{ dy<-a+l'>m
dy(k~l)r
And also, taking into account the identities: dy^m W
dxi 8x>
dpm we obtain:
dy{a)m d~x* =°' fo"* = 1 . 2 . - , * - l ,
„ , , . = 0,(a = l,...,kl),
, , #Pm 9 l '
„ = 0,
^d^dl__dj_dg_ dyW dPi~ dyW dpi
{
'""
;
'
Consequently {/,}« = {/Ts} (Q) , (o = 0 , . . . , * - l ) . It is clear that: {f,9}Q = -{g,f}a;
{f,g}a is fl-linear in every argument
holds. Finally, we prove that the Jacobi identities hold, i.e.: {{/, }a, h}a + {{9, h}a, f}a + {{h, f}a, g}a = 0, (1.8)"
227
Ch.9. The dual bundle of the k-osculator bundle
Finally, by a direct calculus, it is not difficult to prove the Jacobi identities (1.8)" for every Poisson bracket {/, g}a. q.e.d. For more clarity, all these considerations will be applied in the case k = 2, in order to study the geometry of Hamilton spaces of order 2.
9.2
The dual of the 2—osculator bundle
The theory from the previous section can be particularized to the case k = 2. We obtain in this way the dual of the 2-tangent bundle (T 2 M,TT 2 , M) given by (T*2M,TT*2,M), where T*2M = TlM xM T*M can be identified with T*2M = TM xMT*M. Notice that the last one is exactly the Whitney sum of vector bundles TM 0 T*M. We have T*2M = T*2M = TM © T*M,
(2.1)
where {TM, 7r,M) is the tangent bundle of the manifold M and (T*M, ir*, M) its cotangent bundle. A point u € T*2M can be written in the form u — (x,y,p), having the local coordinates {x^y^pi). The projections ir*2(u) = Tr*2(x,y,p) = x, 2 2 2 •K\ : T* M —> TM are defined by TT2(U) = n (x,y,p) = (x,y) and W : T* M —> 2 T*M is given by W*(u) = ff*(x,y,p) = (x,p). Let us denote n (u) = w(u). We get the following commuttive diagram: T*2M
TM
.1/
A change of local coordinates is given by 7fi
x
(2.2)
• /
—
i
r'tr'
vn\
\
.
i
rift I
— x \x , ...,x ;, uei ( :
y* — — i dxj Pi = -^=ri
UJu
a^
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228
The dimension of the manifold T*2M is 3n. For every point u e T*2M the natural basis d_ dxi
d_
,'dPi
of the tangent space TUT*2 transformes, with respect to (2.1), as follows: dxi 'dx1
1
'dx (2.3)
d
dF>u
pj
d
dx* dyi
d
dx> 'dx1
dp
dx1
d
The Jacobian matrix of the change of coordinates (2.1) is given by drJ
(2.4)
0
\
dxi
J(u) =
0
2r(«)
^-(
Theorem 9.2.1. The differentiable manifold T*2M is orientable if and only if the base manifold M is orientable. The nul section 0 : M —> Tr2M of the projection TT*2 is defined by 0 : (x) 6 M —• (x, 0,0) € T*2M we denote by T**M = T*2M \ {0}. Let us consider the tangent bundle of the differentiable manifold T*2M. It is given by the triade (TT*2M,T*2,T*2M), where r*2 is the canonical projection. Taking into account the kernel of the differential dr*2 of the mapping r*2 we get the vertical subbundle VT*2M. This leads to the vertical distribution V : u 6 T*2M —>• V(u) C TUT*2M. The local dimension of the vertical distribution V is 2n and V is locally generated by the vector fields < usuat edby denote (2.5)
d
J , Vu
d
d
f.
*'
d
dp.
Ch.9. The dual bundle of the k-osculator bundle
229
It follows that the vertical distribution V is integrable. By means of the relation (2.3), we can consider the following subdistributions of V: Wx : u e T*2M —> W^u) c TUT*2M
(2.6)
locally generated by the vector fields { 4 , u e T*2M} It is an integrable distribution of local dimension n Let us consider also the subdistribution (2.6)'
W2 : u G T*2M —> W2(u) C TUT*2M
locally generated by the vector fields {&• , u G T*2M}. Of course, W2 is also an integrable distribution of local dimension n Clearly, we have Proposition 9.2.1. The vertical distribution V has the property (2.7)
V{u) = Wx{u) © W2{u), Vu € Tt2M.
Now, some important geometrical object fields can be introduced: (i) the Liouville vector field on T*2M: (2.8)
C(u) = i/<3Si|ti, V u € T ' 2 M ,
(ii) the Hamilton vector field on T*2M: (2.9)
C*(u) = Pi dl\u , Vu e Tt2M,
(iii) the scalar field (2.10)
We remark that CeWl and C* G W2. Also, let us consider the following forms (2.11) uj=pidxl, (Liouville 1-form) (2.12) 6 = dw = d Then, Theorem 9.1.3 leads to the following result: Theorem 9.2.2. 1° The differential forms w and 6 are globally defined on the manifold T*2M.
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2° The 2-form 6 is closed and rank 0 is 2n. 3° 9 is a presymplectic structure on T*2M. The Poisson brackets { } 0 , { }i can be defined on the manifold T*2M by: ,
df dg dxl dpi
df dg dpi dx1
dy* dpi
dpi dy*
(2.13)
Therefore, Theorem 9.1.4 can be particularized in: Theorem 9.2.3. Every bracket { }0 and { }i defines a canonical Poisson structure on the manifold T*2M. Now, it is not difficult to prove that the following T(T*2M)-linear
mapping
J : X(T*2M) — • X{Tt2M) defined by (2.14)
J{di) = %, J0i) = 0, J{ff) = 0, u e T*2M
has geometrical meaning. It is not difficult to prove: Theorem 9.2.4. The following properties hold: 1°. J is a tensor field of type (1,1) on the manifold T*2 M. 2°. J is a tangent structure on T*2M, i.e. J o J = 0. 3°. J is an integrable structure. 4°. KerJ = WX@ W2, imJ = Wv We are going to use these object fields to construct the geometry of the manifold T*2M.
Ch.9. The dual bundle of the k-osculator bundle
9.3
231
Dual semisprays on T*2M
An important notion suggested by the geometrical theory of the Lagrange spaces of order 2 is that of dual semisprays. Definition 9.3.1. A dual semispray on T*2M is a vector field S on T*2M with the property: JS = C.
(3.1)
Taking into account (2.6) and (2.11) it follows: Proposition 9.3.1. A dual semispray S on T*2M can be represented locally by
The system of functions (V(x,y,p), fi{x,y,p)) is called the coefficients of the dual semispray S. However, they are not any arbitrary functions. In fact, {£*} and {/J are important geometrical object fields. Theorem 9.3.1. With respect to the transformation law (2.2) on T*2M, the functions {C} and {/,•} transform as follows: 2f = — 2 ^ + — yj
(3.3)
~
dx3 ,
dpi ,
Conversely, if on every domain of local chart on T*2M are given the systems of functions {£'} and {/i} (i = l,...,n) such that, with respect to (2.2), the formula (3.3) and (3.3)' hold, then S given by (3.2) is a dual semispray on T*2M. The proof is not difficult. It is similar with the proof given by semisprays on the osculator bundle T2M. Two immediate properties are the following: Proposition 9.3.2. The integral curves of the dual semispray S, from (3.2), are given by the solution curves of the system of differential equations: /„ .\
(3.4)
dxl
_
, d?xl
=
^ _
,,,
s dpi
= 2ax ,,, p) ,
J. =
.,
/i(w).
.
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Proposition 9.3.3. Every dual semispray S on the manifold T*2M with the coefficients (£', fi) determines a bundle morphism (3.5)
£ : (i, y, p) e Tt2M -> {x, y™, y™) € T™M,
defined by (3-5)'
j = x\ y{l)i = y\ y{2)i =
?(x,y,P).
Moreover, it is a local diffeomorphism if and only if rank II d'•J£J"ll = n. We shall see in Chapter 10 that the bundle morphism £, defined in (3.5)', is uniquely determined by the Legendre transformation between a Lagrange space of order two, Z/ 2)n = {M,L(x,y(l\y^), and a Hamilton space of order two, # ( 2 ) n = (M,H(x,y,p)). Consequently, if the bundle morphism £, defined in (3.5)' is apriori given, the dual spray S, denoted by (3.2)'
S( = y'di + 2Cdi + 0
is characterized only by the coefficients We have, also:
fi(x,y,p).
Proposition 9.3.4. The formula: (3.6)
w(5{) = tp,
holds. An important problem is the existence of the dual semisprays on T*2M. Theorem 9.3.2. If the base manifold M is paracompact, then on T*2M there exist dual semisprays S^ with apriori given bundle morphism £. Proof. Assuming that the manifold M is paracompact by means of Theorem 9.1.2, it follows that the manifold T*2M is paracompact, too. We shall see (Ch.10) that a bundle morphism £, defined in (3.5)'exists. Now, let 7i ; (z), b e a Riemannian metric on M and Yjk(x) i t s Christoffel symbols. Setting (3-7)
fj = lU^Piv"
we can prove that the rule of transformations of the systems of functions respect to the transformation of local coordinates (2.2) are given by (3.3)'.
with
Ch.9. The dual bundle of the k-osculator bundle
233
Indeed, we have
[
~i dxr dxs _ ~dti dx^ ~
lrs
'
dxi d& '
d2?
s ljh
The contractions with yh and p, lead to
which is exactly (3.3)'. Therefore f rom (3.7) are the coefficients of a dual semispray. As a consequence, we have: Theorem 9.3.3. The following systems of functions N*i
(3.9)
&
are geometrical object fields on having the following rules of transformations, with respect to the changing of local coordinates (2.2): ~t &?_ _ N. .<&_ _ <&_ s J dx> dx° dx^ (3.10)
~ _ dx^ dx^_ 'j~ dx' d&
dV dx>dxi'
sr+Pr
These properties can be proved by a direct computation starting from the formulae (3.3)'.
Remarks. 1°. We will see that the system of functions (Nlj,Nij) a nonlinear connection on T*2M.
gives us the coefficients of
2°. With respect to (2.2), the system of functions
is transformed like a skew symmetric, covariant d-tensor field.
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234
9.4
Homogeneity
The notion of homogeneity for the functionsf(x, y, p), defined on the manifold T*2M can be defined with respect to y\ as well as, with respect to Pi respectively. Indeed, any homothety
K • (x,y,p) —> (x,ay,p), a € H +
is preserved by the transformation of local coordinates (2.2). Let Hy be the group of transformation on T*2M,
Hy = {ha : (x,y,p) —> (x,ay,p) | a € IR + }. The orbit of a point u0 = (zo,2/o,Po) b
y
is given by
x* = x'o, yl = atfo, Pi = p°t, Va e B + . The tangent vector in the point u0 -
i
s the Liouville vector field C(u0) =
Definition 9.4.1. A function / : T*2M —> IR differentiable on T* 2 M\{0} = T* and continuous on the null section of the projection TT*2 : T*2M —> M is called homogeneous of degree r £ 2 w i t h respect t o y \ if foha = aTf, V a e R + .
(4.1) It follows[106]:
Theorem 9.4.1. A function f e F(T*2M) differentiable on T*2M and continuous of the null section is r-homogeneous with respect to yl if and only if (4.1)'
Ccf = rf,
where £ c is the Lie derivation with respect to the Liouville vector field C. We notice that (4.1) can be written in the form: (4.1)"
y
^ = rf.
The entire theory of homogeneity, with respect to y\ exposed in the book [106], can be applied. However, in our case it is important to define the notion of homogeneity with respect to variables pj. Let Hp be the group of homotheties Hp = {tia : (x, y,p) e T'2M —> {x, y,ap) e T*2M | a e JR+}.
Ch.9. The dual bundle of the k-osculator bundle
235
The orbit of a point u0 = (x0, yo,Po) by Hp is given by x{ = 1*0, Vi = V{o, Pi = ap°i, Va € R + . Its tangent vector in the point u0 = h[(u) is the Hamilton vector field C*(u0). A function / : T*2M —> R differentiable on T*2M and continuous on the null section is called homogeneous of degree r, r , %) with respect to the variables p, if f°h'a = arf, VaeJR+.
(4.2) In other words:
f(x,y,ap) = arf(x,y,p).
It follows Theorem 9.4.2. A function f e F(T*2M), differentiable on T*2M and continuous on the null section is r-homogeneous with respect to Pi if and only if we have Ccf = rf.
(4.2)' Of course, (4.2)' is given by
(4.2)"
*J£ = r/.
A vector field X eX(T*2M)
is r-homogeneous with respect to Pi if
X o h'a = a1""1 A'B* oX, V a € H + . We have: Theorem 9.4.3. A vector field X e X{T*2M) is r-homogeneous with respect to Pi if and only if Cc-X = (r
(4.3)
Corollary 9.4.1.
d d d The vector fields -^-^'^r^'Trare 1, 1,0-homogeneous dxl dyl dpi
spect to pi, respectively. Corollary 9.4.2. A vector field (4.3)'
X
= Xt~ + Xl^- + Xi^, dx' dy' Opt
with re-
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236
is r—homogeneous with respect to pit if and only if (0).
X' is r - l homogeneous, X1 is r — l homogeneous, (2)
Xi is r homogeneous. It results: Proposition 9.4.1. If f I f X 6 X{T*2M) are r- and with respect to pi, respectively, then fX is r + s-homogeneous. In particular:
s-homogeneous
1°. The Hamilton vector field C* is 1-homogeneous. 2°. A dual semispray
is 1-homogeneous with respect to pi if and only if its coefficients £' are 0-homogeneous and fi are 1-homogeneous with respect to pt. Proposition 9.4.2. If X s I f is r-homogeneous and f is s-homogeneous with respect to pi, then Xf is r + s - l - homogeneous with respect to p\. Corollary 9.4.3. If r € T(T*2M) entiable on T*2M, then
df
1°. —— are (r dpi d2f 2°. -—-— is (r dpidpj A q-form
is r-homogeneous with respect to Pi and differ-
1)-homogeneous. 2)-homogeneous. is called s-homogeneous with respect to uoti*
if
=asuj, V a e IR + .
Corollary 9.4.4. If the functions f, !F(T*2M) are r - and s - homogeneous with respect to pl, respectively, then the functions given by the Poisson brackets {f,g}o and {f,g}i are (r + s - 1)-homogeneous, respectively.
Ch.9. The dual bundle of the k-osculator bundle
237
The following result holds: Theorem 9.4.4. A q-form u> € Aq(T*2M) is s-homogeneous with respect to pt if and only if Cc'W = su-
(4.5) It follows:
Proposition 9.4.3. The 1-forms dx',dy\dpi with respect to Pi, respectively.
(i = 1,..., n) are0, 0, 1 homogeneous
In the next section we will apply these considerations for study the notion of the Hamilton spaces of order 2. Finally, we remark: Proposition 9.4.4. A dual semispray S^ is2-homo geneous with respect to yx if and only if the coefficients £* are 2-homogeneous and fi are 1-homogeneous with respect toy*. A dual semispray S$ which is 2-homogeneous with respect to yl is called a dual spray.
9.5
Nonlinear connections
We extend the classical definition [97] of the nonlinear connection on the total space of the dual bundle (T* 2 M,7r' 2 ,M). Definition 9.5.1. A nonlinear connection on the manifold T*2M is a regular distribution N on T*2M supplementary to the vertical distribution V, i.e. (5.1)
TUT*2M = N(u) © V(u), V u 6 T*2M.
Taking into account Proposition 9.2.7 it follows that the distribution N has the property: (5.2)
TuT*2M =
N(u)@W1(u)®W2(u).
Therefore, the main geometrical objects on T*2M will be reported to the direct sum (5.2) of vector spaces. We denote by (5.3)
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a local adapted basis to N, Wi,W2. Clearly, we have _6_ = _d_ _
d_
N3
d__
The systems of functions (N^i(x,y,p),Nij(x,y,p)) ear connection N.
are the coefficients of the nonlin-
6 d d 5x dy dpi
With respect to the coordinate transformations (2.2), -—are transl >-r—,-— % formed by the rule: .
,.
(5.4)
5 Sx'
r=
dxj
5
d
dxl 6x]
dy'
r —^\
: =
dxj
d
d
ox' dyJ
dpi
r —^\
dxi
= ~^r
d
—^r'
dxJ dpj
It is not difficult to prove the following property Theorem 9.5.1. The coefficients ofa nonlinear connection N on T*2M obey the rule of transformations (3.10) with respect to a changing of local coordinates (2.2). Conversely, if the systems of functions (N'j,Nij) are given on the every domain of local chart of the manifold T*2M such that the first two equations (3.10) hold, then (Nlj,Nij) are the coefficients of a nonlinear connection on T*2M. It is convenient then to use the basis (5.3), if the coefficients N'j and TVy are determined only by the coefficients /; of the semispray S ? . It is not difficult to prove the following theorem: Theorem 9.5.2. If S^ is a dual semispray with the coefficients fi, then the systems of functions (5.5)
Nlj = —3-, Nij = —A
are the coefficients of a nonlinear connection. Conversely: Theorem 9.5.3. If (N'j, N^) are coefficients of a nonlinear connection N, the following systems of functions
are the coefficients of a dual semispray S^, where & are apriori given. Taking into account Theorems 9.3.2 and 9.5.2, we can affirm:
then
Ch.9. The dual bundle of the k-osculator bundle
239
Theorem 9.5.4. If the base manifold M is paracompact, then there exist nonlinear connections on the manifold T*2M. From now on we denote the basis (5.3) by: (6i,di,6r).
(5.3)'
The dual basis of the adapted basis (5.3) is given by {5x\5y\5Pi)
(5.6) where (5.6)'
8x* = dx\ Syl = dyl + N'jdx',
8pt = dpt - Njidx*.
With respect to (2.2), the covector fields (5.6) are transformed by the rules:
(5.6)'
6ff = f § ^ , 5tf = d£.5y>, 6Pi = ^SPj.
Also, we remark that the differential of a function / e !F(T*2M) can be written in the form
9.6
Distinguished vector and covector fields
Let N be a nonlinear connection. Then, it gives rise to the direct decomposition (5.2). Let h , be the projectors defined by the distributions N , They have the following properties: h + w\ + w2 = I, h2 = h, w\ = w\, Wj = w2, (6-1)
h o w\ = W\ o h = 0, h o w2 = w2 o h — 0, wx o w2 = w2 ° w\ — 0. €f
(6.2)
we denote: XH = hX, XWl = WlX, Xw* = w2X.
Therefore we have the unique decomposition: (6.3)
X = XH +
Each of the components XH,Xm,Xw*
XWl+Xw*.
is called a d-vector field on T*HI.
The Geometry of Hamilton & Lagrange Spaces
240 In the adapted basis (5.3) we get (6.3)'
XH = Xm'8i,
XWi = X^%
XWi = X\2)d\
By means of (5.4) we have
(6.4)
^ = H^ (O)J ; *(1)' = ||*(1)j; ^
=
%x^-
But, these are the classical rules of the transformations of the local coordinates of vector and covector fields on the base manifold M [97]. Therefore X (0)i ,X (1) * are called d-vector fields and Xp' is called a d-covector field. For instance, the Liouville vector field C and Hamilton vector field C* have the properties: C = C^1 = y'di, CH = 0, CW2 = 0,
c* = ctW> = Pid\ c*H = o, c*Wi = o. A dual semispray % from (4.4), in the adapted basis (5.2), has the decomposition (6.5)
5^ = Sf + S?1 + Sp = y'Si + 2A^ + h{d\
where: (6.6)
2k' = 2f + NW,
hi = fi-
NjiV\
k' being a d-vector field and hi a d-covector field. Assuming that the nonlinear connection N provides from a dual semispray S$ with the coefficients fi, we get (6.6)'
IT; = frfj, Na = djfi-
It follows that the vector k' and covector hi are given by (6.6)"
2fc* = 2f + {frtfyi,
hi = fi-
(difM.
A similar theory can be done for distinguished 1-forms. With respect to the direct decomposition (5.2) a 1-form u E X*(T*2M) can be uniquely written in the form: (6.7)
«=
where (6.7)'
UJH = UJ o h, wWl = w o u i , u}w* = ui o w2.
Ch.9. The dual bundle of the k-osculator bundle
241
In the adapted cobasis (5.6) and (5.6)', we have CJ =Wi 6x{ + o W + WSJ*
(6.8) The quantities u>H,UWI,UJW2 (0) (1)
are called d-1-forms. (2)*
The coefficients w*, un and u are transformed by (2.2) as follows: (o) (0)
d& (£) (i)
dx'W
(2)i
dxi(2)i
(2) 1
(1)
Hence Wj and Wi are called d-covector fields and u will be called d-vector field. If the nonlinear connection N is apriori given, then some remarkable d-1-forms can be associated in a natural way. Namely, let us consider: CJ = OJH =
pidx1,
a = aWl = Pi5y\
(6.9)
P = PW*= y'6Pi. We will use these d-forms for studying the Hamilton geometry of order 2 on T*2M. Proposition 9.6.1. The following properties hold: If S% is a dual semispray, as in (6.5), and nonlinear connection N is determined by Sf, as in (6.6)', then we have (6.10)
«(S€) = piV\ a(S() = 2Vik\
Now, let us consider a function f on T*2M. Its differential can be written in the form (5.7). Therefore
(6.11)
df = (df)H where
(df)" = 5ifdx\
(df)w> = dJ6y\
[df)w> =
Let us consider a smooth parametrized curve 7 : / c H —> T*2M such that Im7 C (•n*2)"1 (U). It can be analytical represented by: (6.12)
xl = xl(t),
yl = i/*(t), Pi = Pi(t),
t e I.
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242
The tangent vector —, in a point of the curve 7, can be written in the form: (Jit
where ( 6 14\
dt
3
dt
dt dt dy
n
dt
dt
( dy \
The curve in (6.3) is called horizontal if — = \-r\ in every point of the curve 7. dt \dt) Proposition 9.6.2. An horizontal curve on T*2M is characterized by the following system of differential equations:
(6.15)
x ^ x ' i t ) , 5-£ = o,S-?j- = o , t e i .
Clearly, the system of differential equations (6.15) has local solutions, if the initial points x%0 = xl(t0), yl0,p° on T*2M are given, t0 e I.
9.7 Lie brackets. Exterior differentials In applications, the Lie brackets of the vector fields {^,4,9*}, from the adapted basis to the direct decomposition (5.2), are important. Proposition 9.7.1. The Lie brackets of the vector fields of the adapted basis are given by [Sj,Sh}
=
R'jhdi+Rijhd*,
= B{ jhdi+B (i)
(7.1)
M"l
l
ijhd
(2)
,
= Bfdi+B^d1, (i)
J
(2)
J
Ch.9. The dual bundle of the k-osculator bundle
243
where IV
(7.2)
Jh
= SHN'J - 5jN\,
Bj ih = d^j, Bi
R ijh = 63Nhl -
ShNjt,
B M = -dhNji,
.h = QhNi^
Bh
.. =
-dhNj._
The proof of this relations can be done by a direct calculus. Now we can establish: Proposition 9.7.2. The exterior differentials of the 1-forms (6xl, Sy1, 6pi), which determine the adapted cobasis (5.6)', are given by
( 7 3N
didx1)
= 0,
d{Syl)
= I i R{ jmdxm+
d(6Pi) = l\ll
Bl jmSym+
Bim j
ijmdxm+ B i]mSym+ B
Indeed, from (5.6)' we deduce
d(6y') = dNijAdxi,
d(SPi) =
-dNjtAdxj.
Using (6.11) for dNlj and dNji we have the formula (7.3). Now, the exterior differentials of the u>,a,/3, from (6.9), can be easily determined. Let us consider the following coefficients from (7.1): (7.4)
Bl jh - dhN'j; - B{ jh = dlNhj.
(1)
(2)
By means of (3.10) it follows: P r o p o s i t i o n 9.7.3. The coefficients Bl ih,~ B' h (i)
(2)
ave the same rule
mation with respect to the local changing of coordinates rs
di
dk~
dx"
3k
on T
his is
oftransfor-
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244
We will see that these coefficients are the horizontal coefficients of an N-linear connection. We obtain also: Proposition 9.7.4. The coefficients: B{ / = frfiTj, B ijh = -d.Nji
(7.6) are d-tensor fields.
9.8
The almost product structure P. The almost contact structure F.
Assuming that a nonlinear connection N is given, we define a T(T*2M) -linear mapping P : X(T*2M) —» X{T*2M), by defined
= XH, W{XWl) = -Xw\ (8.1)
F(XW*) -
-Xw\
V i e X{T*2M).
We have also, (8.2)
P o P = /, { !P = I-2(w1+w2) rankP = 3n.
= 2h-I,
Theorem 9.8.1. A nonlinear connection N on T*2M is characterized by the existence of an almost product structure P on T*2M whose eigenspaces corresponding to the eigenvalue -l coincide with the linear spaces of the vertical distribution V on E. Proof. If N is given, then we have the direct sum (5. 1). Denoting by h and v the supplementary projectors determined by (5.1) we have P = / — 2v with the properties P ( X H ) = XH, P(X V ) = -Xv. So, the imposed condition is verified. Conversely, if P 2 = / and W{XV) = -X v ,then let v = | ( / - P ) and h = ^ ( / + P ) We verify easy that h + v = I. So N = Kerv. It follows N ® V = TT*2M. q.e.d.
Ch.9. The dual bundle of the k-osculator bundle
245
Proposition 9.8.1. The almost product structure P is integrable if, and only if, the horizontal distribution N is integrable. Proof. The Nijenhuis tensor of the structure F (8.3)
NTP{X, Y) = P 2 [X, Y] + [PX, PY] - P[PX,
Y] - P[X, TPY]
gives us for X = XH, Y = YH NTP{XH,YH)
=
NP(XH,YV)
=
NP(XV,YV)
= [XV,YV] + [XV,YV}
[XH,YV}-[XH,YH}-1P[XH,YV}
Therefore NP = 0 if and only if [XH,YH]V = 0. But [XH,YH]V = 0, VX,Y £ X(T*2M) allows to say that the horizontal distribution N is integrable. The nonlinear connection N being fixed we have the direct decomposition (5.1), (5.2) and the correspondingadapted basis (5.4). Let us consider the P(T*2M) -linear mapping: F : determined by (8.4)
F(Si) = -di, TF{di) = Su W(&) - 0.
Then, we deduce: Theorem 9.8.2. The mapping F has the following properties: 1°. It is globally defined on T^M. 2°. F is a tensor field of type (1,1). 3°. KerF = W2, ImF = N®WV 4°. rank||F|| = 2n. 5°. F 3 + F = 0.
The Geometry of Hamilton & Lagrange Spaces
246 Proof.
1°. Taking into account (5.4) we have-r—rF I-^—I = — ^— ^ - r implies F I —rT = d
-^• 5x3
., Also
and
d x * [ d \ w
=
dxj 6
' di [W') MS*
. d x > - ( d \ and w
^
,~ ( n , 0>lead t 0 F
[to) =
d
W)
\ =
\dpi
2°. F isT{T^M) -linear mapping from X(f**M) to d_ ) = 0 implies W is trivial and W(N @ W © W ) = N ® F (— ]Wl x 2
3°.
mpli
4°. Evidently, by means of 3°. 5°. F2{XH) = F{-XWl) = ~XH- F3(XH) = Xw> and F(XH) = -XWK So (F 3 + F)XH = 0, MX" E N and (F 3 + F){Xm) = 0, (F 3 + F)(XW>) = 0. We can say that F is a natural almost contact structure determined by the nonlinear connection N. The Nijenhuis tensor of the structure F is given by: Afr(X, Y) = F2[X, Y] + [FX, FY\ - W[WX, Y\ - F[X, FY] and the normality condition of F reads as follows: (8.5)
Mv{X, Y) + f^diSpdiX, Y) = 0, \fX, Y £
X{f^M).
Of course, in the adapted basis, using the formula (7.3) we can obtainthe explicit form of the equation (8.5).
9.9
The Riemannian structures on T*2M
Let us consider a Riemannian structure G on the manifold T*2M. The following problem is arises: Can the Riemannian structure G determine a nonlinear connection N on T*2 Ml Also, can G determine a dual semispray S^ on In order to determine a nonlinear connection on T*2M by means of G it is sufficient to determine a distribution N orthogonal to the vertical distribution V. The solution is immediate. Namely, it is important to determine the coefficients N'j and Na of N.
Ch. 9. The dual bundle of the k-osculator bundle
247
In the natural basis, G is given locally by
G = g ijdx' ® dxj + g ijdx* ® dyj +
gjdx*
(9.1) (33)..
g x'
where the matrix
is positively defined.
Let {#,}, (i = 1 , ..., n), be the adapted basis of N:
The following conditions of orthogonality between N and V:
give us the following system of equations for determining the coefficients N*j and
(9.4)
where, the matrix
(9.4)' \
g
3
m
g
is nonsingular. Therefore the system (9.4) has an unique solution.
(a/3)
Whether, take into account the rule of transformation of the coefficients g y from G we can prove that the solution ( A ^ ' j , ^ ) of (9.4) has the rule of transformation (3.10), by means of the transformations of local coordinates on T*2M. Consequently, we have: Theorem 9.9.1. A Riemannian structure G on T*2M determines uniquely a nonlinear connection N, if the distribution of N is orthogonal to the vertical distribution V. The coefficients Nlj, Nij of N are given by the system of equations (9.4). Remarking that ft = N^yi are the coefficients of a dual spray S(, we have:
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248
Theorem 9.9.2. A Riemannian structure G on T*2M determines a dual semispray S( with the coefficients Si = tfyV, Nij being determined by the system (9.4). Let F be the natural almost contact structure determined by the previous nonlinear connection N. The following problem arises: When will the pair (G, IF) is a Riemannian almost contact structure? Of course, it is necessary to have: G(JFX, Y) = -G(X, FY),
VX, Y €
X{T*M).
Consequently, we get: Theorem 9.9.3. The pair ( G , F ) is a Riemannian almost contact structure if and only if in the adapted basis determined by N and V the tensor G has the form G = gtjdx* <S> dx3 + g^Sy* ® 5yj + tij5p{ ® Spj.
(9.5)
Corollary 9.9.1. With respect to the Riemannian structure (9.5) the distributions N,Wi,W2 are orthogonal respectively. Remarks. 1° The form (9.5) will be used to define a lift to T*2M of a metric structure given only by a nonsingular and symmetric d-tensor field <jy. Namely, we have (9.5)'
G = gijdx1 ® dxj + g^y* ® 6yj + gij6p{ ® 6pj.
These problemes will be studied in a next chapter. 2° Using the metric G on T*2M we can introduce a new almost contact structure F , defined by (9.6)
F(8i) = - g ^ ,
t(di)
= 0, F ( 5 y ^ ) = 6,
We will prove that (G, F ) is a Riemannian almost contact structure and its associated 2-form 6 is given by § — Spi Ada;'.
The pair (G, F ) will be studied in the Chapter 11 about the generalized Hamilton spaces of order 2.
Chapter 10 Linear connections on the manifold T*2M The main topics of this chapter is to show that there are the linear connection compatible to the direct decomposition (5.2) determined by a nonlinear connection N, on the total space of the dual bundle (T* 2 M,TT* 2 ,M). We are going to study the distinguished Tensor Algebra (or d-Tensor Algebra), N-linear connections, torsions and curvatures, structure equations, autoparallel curves, etc.
10.1
The d-Tensor Algebra
Let N be a nonlinear connection on T*2M. Then N determines the direct decomposition (5.2), Ch.9. With respect to (5.2), Ch.9, a vector field X and an one form u> can be uniquely written in the form (6.3) and (6.7), Ch.9, respectively, i.e. (1.1) Definition 10.1.1. A distinguished tensor field (briefly: d-tensor field) on T*2M of type (r, s) is a tensor field T of type (r, s) on T*2M with the property:
(1.2)
Tfo...,Z,X,..,X) = T(2,H,...tZ,">,Xa,...,Xw*), is
i s
for any (i,...,w) G X*{T*2M) and for any (X,...,X) 6 X{T*2M). For instance, every component XH,XWl and Xm of a vector field X e X{T*2M) is a d-vector field. 249
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250
Also, every component u>H,uWl and u>W2 of the 1-form u € X*{T*2M) is a d-l-form. In the adapted basis (5i, fy, d'J and cobasis (dx\ 8y\ 5pi) to the direct decomposition (5.2), a d-tensor field T of type (r, s) can be written in the form: ®---9>'®dxjl
T = Tjl;;£(x,y,p)Sh
(1.3)
It follows that the set (l,Si,di,dl\ the ring of functions T(T*2M).
®---®8pir.
generates the algebra of the d-tensor fields over
For example, if / 6 F(T*2M), then —- = Sif, - 4 = fyf are rf-1-covectors and ox' ay' TT- = d'f is a d-vector field. dpi Clearly, with respect to a local transformation a coordinates on T*2M, the coefficients T*l;jr3(x,y,p) of a d-tensor fields are transformed by the classical rule:
10.2
N-linear connections
The notion of N-linear connection will be defined in the known manner [97]: Definition 10.2.1. A linear connection D on T*2M is called an N-linear connection, if: (1) D preserves by parallelism distributionsN, W\ and W2. (2) The 2-tangent structure J is absolute parallel with respect to D. (3) The presymplectic structure 6 is absolute parallel with respect to D. Starting from this definition, any N-linear connection is characterized by the following: Theorem 10.2.1. A linear connection D is an N—linear connection on T*2M if and only if: (1) D preserves by parallelism every of distributions N (2) DX(JYH) = J(DXYH), € X{T*2M). (3) D0 = 0.
DX{JYW") = J{DXYW"),
, (a = 1,2),
Ch.10. Linear connections on the manifold T*2M
251
The proof is similar with the case given in the book [106]. We remark that DX(JYW~) = J(DXYW°) is trivial, because JYW° = 0, VF e X(T*2M) and that by means of the property (1), it follows J{DXYW°) = 0. We obtain also: Theorem 10.2.2. For any N—linear connectionD we have (2.1)
Dxh = DxWl
= Dxw2 = 0,
(2.2)
DXW = 0, DXW = 0.
Indeed, from (Dxh)Y = Dx(hY)-h{DxY) if Y = Y", and Y = Yw-,{a = 1,2) we obtain Dxh = 0. Similarly, we get Dxw\ = 0, Dxw2 = 0. Now, taking into account the expression (8.2), Ch.IX, of P it follows DX1P = 0. The last equality DXF = 0 can be proved in a similar way. Let us consider a vector field X € X{T*2M), written in the form (1.1). It follows, from the property of an N-linear connection that (2.3)
DXY = DXHY + DxWlY + Dxw2Y, VX, Y € X{T*2M).
We can introduce new operators of derivation in the d-tensor algebra, defined by: DHX = DXH , D^ = DxWl,
(2.4)
D^ =
Dxw2.
These operators are not the covariant derivations in the d-tensor algebra, since Dxf — XH f ^ Xf (etc.). However they have similar properties with the covariant derivatives. From (2.3) and (2.4) we deduce (2.5)
DXY = D%Y + D^Y
+ D^'Y, V I , Y e X{T*2M).
By means of Theorem 10.2.2, the action of the operator Dx on the d-vector fields i s ^ g s a m e as i ts a c t i o n o n the d-vectors YH. This property holds for the operators D%1 and D™2, too. Theorem 10.2.3. The operators D", D™1, D™2 have the following properties: 1) Every £>f, D^1, D™2 maps a vector field belonging to one of distributions N , in a vector field belonging to the same distribution. 2) D$f = XHf,
D%lf = XWlf,
3) D${fY) = XHfY
+ fD$Y; D^"fY
=
Xw"fY
4) D$(Y + Z) = D%Y + Dx'Z; D^a(Y + Z) = D^
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252
7) D$(JY) = JD^Y; D^"(JY) = JD^Y,
(Va = 1,2).
8) D"0 — 0, D^a6 = 0, (a = 1,2), 9 being the presymplectic structure from Theorem 6.7.1. 9) The operators D%, D^1, D^2 have the property of localization on the manifold Tt2M, i.e. {D%Y)\v = D% Y[(J etc. for any open set U C T*2M. The proof of the previous theorem can be done by the classical methods [106]. The operators D",D^l,D^2 will be called the operators of h-, and w2covariant derivation. The actions of these operators over the 1-form fields on T*2M, are given by
= (2.6)
(DTw){Y)
XHw(Y)-cu(D«Y),
= XWMY)
-
u(D^Y),
(D%>w)(Y) = Xw>u,(Y)-u>(D%>Y). Of course, the action of the previous operators can be extended to any tensor field, particularly to any d-tensor field on T*2M. Now, let us consider a parametrized smooth curve 7 : t c / —> 7(t) € T*2M, having the image in a domain of a local chart. Its tangent vector field 7 = — can be uniquely written in the form at
(2.7)
7= 7 " +
y'+7^
In the case when 7 is analytically given by the equation (6.12), Ch.9, then j H , j W l , j W 2 are given by (6.13), Ch.9. And we can define the horizontal curve. A vector field Y defined along the curve 7 has the covariant derivative D^Y = D?Y + DV The vector field Y(u(j)) is called parallel along the curve 7 if = 0. In particular, the curve 7 is autoparallel with respect to an N-linear connection D if D^ = 0. In a next section we will study these notions by means of adapted basis.
Ch.10. Linear connections on the manifold T*2M
10.3
253
Torsion and curvature
The torsion IT of an N-linear connection D is expressed, as usually, by Y(X,Y) =
(3.1)
DXY-DYX-[X,Y].
It can be characterized by the vector fields T(XH, YH), T(XH, Yw°), T(XW% YWb), (a, b = 1,2). Taking the h- and
(3.2)
-components we obtain the torsion d-tensors 2
T(XH,YH)
=
TT(XH,YW")
= hT(XH,Yw')
T{XW;YW>)
= hT(Xw°,Yw>) + J2
+
6=1
c=l
Since D preserves by parallelism the distributions H Wx, W2 and the distributions W\, W2 are integrable it follows Proposition 10.3.1. The following property of the torsion T holds: (3.3)
hT(Xw°,YW>>)
= 0, ( 0 , 6 = 1,2).
Now we can express, without difficulties, the torsion d-tensors by means of the formula (3.1). The curvature of D is given by (3.4)
m.{X,Y)Z = (DXDY - DYDX)Z - D[X>Y]Z, VX,Y,Ze
X{T*2M).
We will express H by means of the components (2.5), taking into account the decomposition (1.1) for the vector fields on T*2M. Proposition 10.3.2. For any vector fields X,Y,Z e X(T*2M) the following properties holds: (3.5)
J(R{X, Y)Z) = U{X, Y)JZ; DX6 = 0.
The previous properties have an important consequence:
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254
Corollary 10.3.1. 1° The essential components of the curvature tensor field JR. are JR.(X,Y)ZW* and R{X, 2° The vector field JR(X,Y)ZH
TR(X,Y)ZH,
is horizontal.
3° The vector field U(X,Y)ZWa
(a = 1,2) belongs to the distribution Wa.
4° The following properties hold wa[1R(X,Y)ZH}
= 0,
h[TR(X, Y)ZW°] = 0,
(3.6) wb\R(X, Y)ZW"} = 0, (a ? b, a, b = 1,2). Of course, we can express the d-tensors of curvature by means of the operators of h, covariant derivatives (2.5)". From (3.4) we get the following Ricci identities M(X,Y)ZH
\DX,DY]Z»
=
+
D[XtY]Z»
[Dx, DY]ZW«
= JR(X, Y)ZW° + D[XtY]Zw«
(3.7) {a = 1,2).
As a consequence, we obtain: Theorem 10.3.1. For any N -linear connection D there are the following [DX,DY]C
=
[DXiDy\C
= m.(X,Y)C*
identities
1R(X,Y)C-D[X,Y]C
(3.8) -
D[XtY]C*
where C is the Liouville vector field, and C* is the Hamilton vector field on the manifold T*2M. Using the previous considerations we can express the Bianchi identities of the Nlinear connection D, by means of the operators D$, D™1, D™3 taking into account the classical Bianchi identities
£ (XYZ)
{(DXT)(Y, Z) - ]R(X, Y)Z + T(T(X, Y), Z} = 0,
52 {(DxJR)(U,Y,Z)-JR(T(X,Y),Z)U} = 0,
(XYZ)
where
^2
{XYZ)
means the cyclic sum.
Ch.10. Linear connections on the manifold T*2M
10.4
255
The coefficients of an N-linear connection
An N-linear connection is characterized by its coefficientsin the adapted basis A
S
d
k
ox'
ay'
d
»
dpi
These coefficients obey particular rules of transformation with respect to the changing of local coordinates on the manifold T*2M. Taking into account Proposition 10.3.2, we can prove the following theorem: Theorem 10.4.1. 1° An N-linear connection D can be uniquely represented, in the adapted basis (5i,di,dl) in the following form: DSj8t = (4.1)
2° With respect to the coordinate transformation (1.1), Ch.6, the coefficients Hjk(x,y,p) obey the rule of transformation: H
(42)
3° The coefficients Cljk{x,y,p) and (2,1), respectively.
%E*~
=
^ %
and C{k(x,y,p)
are d-tensor fields of type (1,2)
Indeed, putting As A =Hk iiSk, Ds & =Hk udk, Ds & = - H{ (0)
(1)
(2)
k
kid
,
and taking into account Theorem 10.2.3, it follows Hk ij (0)
=
Hk ij (1)
=
Hk (2)
ij.
The statements 2° and 3° can be proved by a direct calculus, taking into account the rule of transformations (5.4)', Ch.9, for S^di and d*. The system of functions (4.3)
The Geometry of Hamilton & Lagrange Spaces
256
are called the coefficients of the N-linear connection D. The inverse statement of the previous theorem holds also. Theorem 10.4.2. If the systems of functions (4.3) are apriori given over every domain of local chart on the manifold T*2M, having the rule of transformation mentioned in the previous theorem, then there exists a unique N -linear connection D whose coefficients are just the systems of given functions. Corollary 10.4.1. The following formula hold:
(4.4)
Ds.dx* = -H'kjdxk,
Di/y* = -Wkj8yk,
DSj8Pi = Hk58Pk,
D6]dxi = -Cijdxk,
Dg.Sf^-Cyy",
Ddj6Pi k
,
DbJ8Pi = Ckj8Pk.
Indeed, the formula (4.1), the condition of duality between (6j, dj, &>) and (dx\ 8y\ leads to the formula (4.4).
10.5
The h -, covariant derivatives in local adapted basis
Let us consider a d-tensor field T, of type (r, s) in the adapted basis (Sj, dj, &>) and its dual (see (1.3)): T = T i l ; £ 5 h
(5.1)
For X = XH = X^i, applying (4.1), (4.4) and using the properties of the operator D% we deduce: fc o\ j,^ i *
n^T Y'mT^1*"*r A - & \ . . .
'
Jl
-A
•••Ja
|" I
*i
- ^ i ^ i f
where ji•• jj|m
(5.2)'
~ _
m
ji—ji "*" 31—U ™km + " ' " + -*ji...j,
/Ti«i...jr r j t
_
_ rp%i...ir
"•km~
Tjk
The operator "|m" is called h-covariant derivative with respect to DF(N). Now, putting X = X W l = X ^ , we obtain for the d-tensor field T from (5.1), the formula: v
/
-'t
J j •••Ja 1)71
*1
^^
r^lr
'
Ch.10. Linear connections on the manifold T*2M
257
where (5.3)'
The operator "| m " will be called "the
DT(N).
-covariant derivative" with respect to
Finally, taking X = XWi = X&, then D^T
has the following form
(5.4) where (5.4)' The operator " | m " will be called the – It is not difficult to prove:
covariant derivative.
Proposition 10.5.1. The following properties hold:
are d-tensor fields. The first two are of type (r, s + 1) and the last one is of type (r + l,s). Proposition 10.5.2.
The operators "\m", "\m", and "\m" have the properties:
1°. f\m = Smf, f\m = dmf, f\m = dmf, V / 6
HT*2M).
2°. They are distributive with respect to the addition of the d-tensor of the same type. 3°. They commute with the operation of contraction. 4°. They verify the Leibniz rule with respect to the tensor product. As an application let us consider "the (y)-deflection (5.5)
D'j - y%, d1; = y%, d* -
tensor fields" J l
Proposition 10.5.3. The (y)-deflection tensor field have the expression (5.5)'
D'j - -TV1, + ymHmij,
d*j = 6'j + y^Jj,
The Geometry of Hamilton &Lagrange Spaces
258
These equalities are easy to prove, if one notice y% =
Now, we consider the so called "(p)-deflection tensor fields": $„ = pi\j, Si3 = pi\J.
A ^ = p^,
(5.6)
Proposition 10.5.4. The (p)-deflection tensors are given by (5.6)'
A y = Ntj -
m PmH ij,
^ = -PmC^j,
6S = $3 -
A particular class of the N-connection with the coefficients DT(N) is given by the Berwald connectioin. Definition 10.5.1. An N-linear connection D with the coefficients (4.3) is called a Berwald connection if its coefficients are: H)k = d3N\,
(5.7)
C)k = 0,
This definition has a geometrical meaning if we take into account Proposition 9.7.4. The existence of the Berwald connection is an interesting example of N-linear connection. Remarking that the Berwald connection is uniquely determined by the nonlinear connection N and the fact that the nonlinear connection exists over a paracompact manifold T*2M (cf. Theorem 9.3.2), we can state: Theorem 10.5.1. If the base manifold M is paracompact, then on the manifold T*2M there exists the N -linear connections. Of course, the (y)-deflections and (p)-deflection tensor fields of the Berwald connection BT(N) = ( 5 , ^ , 0 , 0 )
(5.8) are very particular. We get (5.9) Ay = Nl3 -
m PmdiN j,
«y = 0,
Sj = 6S.
Hence, Dl0 = 0 if and only if the coefficients N'j are 1-homogeneous with respect toy1.
Ch.10. Linear connections on the manifold T*2M
10.6
259
Ricci identities. The local expressions of curvature and torsion.
In order to determine the local expressions of d-tensors of torsion and curvature of an N-linear connection we establish the Ricci identities applied to a d-vector field, using the covariant derivatives (4.5)', (4.6)' and (4.7)'. Theorem 10.6.1. For any N-linear hold: {
jk
connection D the following Ricci identities
- X\hThik - X\
Rh ik - Xf R hjk, (2)
(i)
(6.1)
i I \i\k
— Yh P
yi\k,
i \k
l
— yhp, i k
V* Oh
-r:
/-i hk _ yi\
yi
^•|A
r> hk _ yi\h
B
hjk,
rjk
.
and yi
A
\j\k
_ j~
A
yhc i yi\ dh. jk - A |
(6.2)
where the following tensors (6.3)
g { ,-fc = S.N'j - ^-i
- 8kNjU &jk, C
R
, B /k,
and (6.3)'
Tjk =
- &kj,
Sijk = Cjk - d
and (6.3)"
l p * jk = 8kN 3 -
(1)
are torsion d-tensors.
p
(2)
l
jk
= Wjk - d{Nkj
B
The Geometry of Hamilton & Lagrange Spaces
260
The d-tensors of curvature are given by
R' ik + Chis R ,jk,
(i)
(6.4)
C\s Ps
jk
(2)
™ B
Cish P* sj + C\s Bs
mjk,
f,
and nf. -
C™kCxmj,
(6.4)'
Proof. By a direct calculus we have
Interchanging j and k and subtracting, we get
And since the Lie brackets [Sk,Sj] are given by (7.1), Ch.9., the previous equalities give us the first identity (6.1), the coefficients being given by (6.3), (6.3)' and (6.4). The identities (6.1) and (6.2) can be proved in the same manner. q.e.d. Remark. Cf. Proposition 9.7.4, the d-torsions B ,-'*, B ijk are d-tensors. (1)
(2)
As usually, we extend the Ricci identities for any d-tensor field, given by (1.3).
Ch.10. Linear connections on the manifold T*2M
261
As a first application let us consider a Riemann d-metric #„, which is covariant constant, i.e.: = 0, 9ij\k = °> 9ijt = 0.
(6.5) Then we have:
Theorem 10.6.2. If the Riemann d-metric g,j verifies the condition (6.5), then the following d-tensors (6.5)'
Rijkh = 9jmRimkh,
Pijkh = 9jmPimkh,
Simkh, siJk» = 9jmsrkh,
Pijkh =
V
=
9jmPimkh,
9jmSimkh
are skew-symmetries in the first two indices (ij). Indeed, writing the Ricci identities for d-tensor &., and taking into account by the equations (6.5) we deduce 9imRjmkh + 9jmRimkh = 0, ...
And using (6.5)', we get R]ikh + Rijkh = 0, etc. The Ricci identities (6.1), (6.2) applied to the Liouville d-vector field y\ and to the Hamilton d-covector field Pi lead to the same fundamental identities. Theorem 10.6.3. identities:
Any canonical N-linear i
jk
(6.6)
- D\Thjk
jk ~ D\Chik *jk - D\Cjhk
connection D satisfies the following - d\ Rh - d\ Ph - dih Pk
- D^Cj1* -
(6.6)' H\k - dik\J =
jk
jk
- dih R - d'h B
h]
dihCkhj,
hjk
hjk
The Geometry of Hamilton & Lagrange Spaces
262 as well as
- 6lhRhjk -
(6.7)
A
I*
k\
h k j
_
j k - Sih
Ph
kh
r> ./
s
? Bhjk, r
and
(6.7)'
In the case of Berwald connection BF(N), the previous theory is a very simple one. Also, if the (y)-deflection tensors and (p)-deflection tensors have the following particular form = 0,
'j = 0,
(6.8) (
y = 0, % = 0,
^ = 5%
then, the fundamental identities from (6.6), (6.6)' and (6.7), (6.7)' are very important, especially for applications. Proposition 10.6.1. If the deflection tensors are given by (6.8), then the following identities hold:
W
k
jk
Jj" =B j i k
(6.8)'
f = C\k,
= 0
and PhRihjk = ~ R nk\ PhPfjk = - B o-fc, PhPihjk =Pk a (6.8)" phSihjk
= 0, phSikjk
=
-Ckjk;
By means of this analytical aparatus we will study the notion of parallelism on the manifold T*2M.
Ch.10. Linear connections on the manifold T M
10.7
263
Parallelism of the vector fields on the manifold T*2M
Let D be an N-linear connection with the coefficients DF(N) = (Hljk, C)k, C{k) in the adapted basis (5i,di,d%). Let us consider a smooth parametrized curve 7 : / —> T*2M having the image in a domain of a chart of T*2M. Thus, 7 has an analytical expression of the form: *< = * ' ( * ) , yl = y \ t ) ,
(7.1)
Pt
=Pi(t), t e l .
d'y
The tangent vector field 7 = -r 1 . by means of (6.13) and §6, Ch.9, can be written at as follows:
where 6y^_dy^ dt ~ dt
+N
{dx^
> dt'
5
Ii-^Ei_N dt ~ dt
3i
^_ dt
Let us denote nv
(7.4)
n Y
D^X = ¥—, DX = -—dt, VX e ^ ( ai ar
DX The quantity DX is the covariant differential of the vector X and —p- is the dt covariant differential along the curve 7. If X is written in the form X = XH + XWi + XWi =X 8i + X % and we put
then, after a straighforward calculus, we have (7.5)
DX = (dX
+X
u{\ 6i + IdX + x'wj ] ds + {dXi - X,u\) d\
The Geometry of Hamilton & Lagrange Spaces
264 where
ijj = H)kdxk + qkSyk + Cik6pk.
(7.6)
Here, wlj are called 1-forms of connection of D. Putting — = H\—
(7.6)'
+ C*k— + Cik—,
the covariant differential along the curve 7 is given by
~dT (7.7)
The theory of the parallelism of the vector fields along a curve 7 presented in Sect.2 of this chapter can be applied here. We obtain: Theorem 10.7.1.
0'
1».
The vector field X =X <J4 + X d; + Xtf
is parallel along the 0*
1'
parametrized curve 7, with respect to D, if, and only if, its coordinates X , X , Xt are solutions of the differential equations
DX The proof is immediate, by means of the expression (7.7) for ——•• CLL
A theorem of existence and uniqueness for the parallel vector fields along a given parametrized curve in T*2M can be formulated in a classical manner. The vector field X € X{T*2M) is called absolute parallel with respect to the canonical N-linear connection DT{N), if DX — 0 for any curve 7. It is equivalent to the fact that the following system of Pfaff equations (7.9) is integrable.
d X + X LJS{ = 0, dXi - Xsusi = 0, (a = 0,1)
Ch.10. Linear connections on the manifold T*2 M
265
The system (7.9) is equivalent to the system ai
a»
ai .
X \j =X I. =X \> = 0, (a = 0,1) (7.9)'
|
which must be integrable. Using the Ricci identities, the previous system is integrable if and only if the ai
coordinates (X ,Xt) of the vector X satisfy the following equations
x s^fc = o, x Sfc'/ = o, x sj* = o and (7.11) The manifold T*2M is called w;Yft absolute parallelism of vectors with respect to D, if any vector field on T*2M is absolute parallel. In this case the systems (7.10), (7.11) are verified for any vector field X. It follows: Theorem 10.7.2. The manifold Tt2 M is with absolute parallelism of vectors, with respect to the N-linear connection D if, and only if, all d-curvature tensors of D vanish. The curve 7 is autoparallel with respect to D if D^j = 0. By means of (7.2) and (7.7) we deduce dt
\dt2
dt dtj1
\dt dt
dt dt
(7.12) (dLSp1_Sp1^f \dt dt dt dt Theorem 10.7.3. A smooth parametrized curve (7.1) is an autoparallel curve with respect to the N-linear connection D if and only if the functions x'(t),yl(t),pi(t),
The Geometry of Hamilton & Lagrange Spaces
266 t£l,
verify the following system of differential equations dt2
+
±Sjl
(7 13)
dt dt d 5pi dt dt
dt dt~ ' +
^l^i
dt dt
= ()
Sps wl ^ Q dt dt
Of course, the theorem of existence and uniqueness for the autoparallel curve can be easily formulated. We recall that 7 is an horizontal curve if 7 = j H . The horizontal curves are characterized by
<7.U,
x-xW.S.o.fe.o.
Definition 10.7.1. An horizontal path of an N -linear connection D is a horizontal autoparallel curve with respect to D. Theorem 10.7.4. The horizontal paths of an N —linear connection D are characterized by the system of differential equations:
Indeed, the equations (7.14), (7.6)' and (7.13) imply (7.15). A parametrized curve 7 : / -> T*2M is -vertical in the point xQ € M if its tangent vector field 7 belongs to the distribution Wa, (a = 1,2). Evidently, a -vertical curve 7 in the point x0 £ M is represented by the equations of the form (7.16)
xi =
xlyi=yi(t),Pi=0,teI
and a W2-vertical curve 7 in the point x0 e M is analytically represented by the equations of the form (7.16)'
x^x'o,
yl = 0, Pi=pt(t),
tel.
We define a -path (a = 1,2) in the point x0 € M with respect to D to be a a -vertical curve 7 in the mentioned point, which is an autoparallel curve with respect to D. By means of (7.16), (7.16)' and (7.12) we can prove:
Ch.10. Linear connections on the manifold T M
267
Theorem 10.7.5. 1°. The-verticalpaths in the point x0 e M are characterized by the system of differential equations
2°. The w2-vertical paths in the point x0 e M are characterized by the system of differential equations
Remark. We assume that there exists the coefficients C'jk(x0, y, 0). In the case of the Berwald connection BT(N), tions of -paths appear in a very simple form.
10.8
(5.8), the previous characteriza-
Structure equations of an N-linear connection
For an N-linear connection D, with the coefficients DT(N) = (H'jk,C*jk,Ctk) adapted basis (5i,di,dl) we can prove:
in the
Lemma 10.8.1. 1°. Each of the following geometrical object fields d{dxl) - dxmAcjlm,
diSy1) - SymAwim,
d{5Pi) + 5pmAwm,,
are d-vector fields. However, the last one is a d-covector field, with respect to the index i. 2°.
The geometrical object field
is a d-tensor field, with respect to indices i and j . Using the previous Lemma we can prove, by a straightforward calculus, a fundamental result in the geometry of the Hamilton spaces of order 2.
The Geometry of Hamilton & Lagrange Spaces
268
Theorem 10.8.1. For any N -linear connection D, with the coefficients Dl\N) — — (Hljk,Clik,Cik), the following structure equations hold good: d(dx') - dxmAuJim = - ft ,
and dJj - u/",Au/m =
(8.2)
-Wj,
where 0 , ft and ftj are the 2-forms of torsion: 1
(0)'
ft
i
=
l r Rl jkdx:>f\dxk+ Pl jkdx3ASyk+ B
=
x
2 (1)
R
lk
(2) J
(1)
dx}A6pk+
HkdxjAdxk+ B ijkdxjA6yk+ Pk -Si
and where £llj is the 2-form of curvature: j = \RiikmdxkAdxm
+ PjikmdxkA8ym
+
+ Sjikm5ykASpm
+
PjikmdxkAdPm+
(8.4) i
l
-S3ikm8pkA8pm.
In the particular case of the Berwald connection we have C'jk = Ctkj — 0, P jk = 0, 5',-fc = Si>k = 0, S/km = 0, 5 / f c m = 0 and S^km = 0. i
Ch.10. Linear connections on the manifold T*2M
269
Remark. The previous theorem is extremely important in a theory of submanifolds embedding in the total space T*2M of the dual bundle T*2M = TM xMT*M, endowed with a regular Hamiltonian of order 2.
Chapter 11 Generalized Hamilton spaces of order 2 One of the most important structures on the total space of the dual bundle T*2M is the notion of generalized Hamilton metric of order two, g^{x,y,p), [110]. It is suggested by the generalized Hamilton metric, described in the section 1 of Ch. 5, which has notable applications in Relativistic Optics of order two. We define the concept of generalized Hamilton space as the pair QH^n = (M,gli{x,y,p)) and study a criteria of reducibility, the most general metrical connections, lift of a GH-metric, the almost contact geometrical model. We end this section with some example of remarkable GH^n -spaces.
11.1
The spaces
Definition 11.1.1. A generalized Hamilton space of order two is a pair
(M,glj(x,y,p)), where
1° g1* is a d-tensor field of type (2, 0), symmetric and nondegenerate on the manifold T*2M. 2° The quadratic form g^XiXj As usually 2)
i
has a constant signature on T*2M.
s called the fundamental tensor or metric tensor of the space
In the case when T*2M is a paracompact manifold then on T*2M there exist the metric tensors g%i{x,y,p) positively defined such that {M,g%i) is a generalized Hamilton space. Definition 11.1.2. A generalized Hamilton metric gl*(z, y,p) of order two (on short GH-metric) is called reducible to an Hamilton metric (H-metric) of order two if 271
The Geometry of Hamilton & Lagrange Spaces
272
there exists a function H(x, y,p) on T*2M such that 9'3
(1-1)
Let us consider the d-tensor field (1.2) z
We can prove: Proposition 11.1.1. A necessary condition for a generalized Hamilton metric glJ of order two to be reducible to a Hamilton metric of order two is that the d-tensors C'3k is totally symmetric. Theorem 11.1.1. Let g'3(x, y,p) be a 0-homogeneous GH—metric with respect to Pi- Then a necessary and sufficient condition that it to be reducible to an H-metric is that the d-tensor field Cljk is totally symmetric. Proof. If there exists a GH-metric g'3 reducible to a H-metric, i.e. (1.1) holds, then Cljk = -dkb%d1H
is totally symmetric (Proposition 11.1.1.).
Li
Conversely, assuming that gtJ'(x, y, p) is 0-homogeneous with respect to p,, taking into account the formula H(x,y,p) — gl3(x,y,p)piPj, and the fact that C y * = dkgl] is totally symmetric, it follows gij - -bl&H.
q.e.d.
Remark. Let Jij(x) be a Riemannian metric. Then it is not difficult to prove that the d-tensor field gij(x,y,p) = e-2'(x*»V(a;)1 a €
T{T2M),
is a GH-metric which is not reducible to an H-metric of order two, provided dodoes not vanishes . The covariant tensor field g^ is obtained from the equations (1.3.)
9 ^ = 61
Of course, g^ is a symmetric, nondegenerate and covariant of order two, d-tensor field. Theorem 11.1.2. The following d-tensor fields C'jk = 2 ^ " \pi9tk + dkgjs - dsgjkj , (1.4)
Ch.ll. Generalized Hamilton spaces of order 2
273
have the properties: g% = 0,
(1.5) and (1-6)
C)k = C'kj,
Indeed, the d-tensors C'jk, C{k have the properties (1.6). By a direct calculus we can prove (1.5) taking into account of (1.3). q.e.d. Remarks. P The tensors gij\k and glj\k are the mental tensor field gij, respectively.
-
-covariant derivatives of the funda-
2° The tensors Cljk and C\k are the w^ and –coefficients of a canonical metrical N-linear connection D, respectively. Some particular cases 1. Let gij{x,y) be the fundamental tensor field of a Finsler space Fn = (M,F) and let g%i{x,y) be its contravariant tensor field. Let us consider g'j(x,y,p) defined on T*2M by g1(x,y,p) = g^(x, y). The tensors C*k are given by the first formula (1.4) and by Cjk = 0. It follows Cljk = 0. The Gfl-metric glj{x,y) has the covariant metric gtj(x,y) reducible to a particular metric: gtj = -didjF2. We have: Theorem 11.1.3. The nonlinear connection N of the space GH^n — has the coefficients:
{
Nlj = dj(j*rs(x,y)yrys)
(M,g^(x,y))
— (Cartan nonlinear connection of Fn),
They are determined only on the fundamental function of the Finsler space Fn. Proof. The tensors i s exactly the Cartan non-linear connection and diN^ is its Berwald connection. A straightforward calculus shows that the rule of transformation of Nij is exactly (3.10), Ch. 6. q.e.d. 2. Let glj(x,p) be the fundamental tensor field of a Cartan space Cn = ij
[97]. It follows g with respect to Pi.
l
= -d cVH
2
(M,H(x,p)),
and therefore we obtain that H is 1-homogeneous
The Geometry of Hamilton & Lagrange Spaces
274
We consider the extension g'j(x,y,p) — gi:i(x,p) of the tensor glj to T*2M. The tensor C'jk vanishes and C{k is given by the second formula (1.4), where k k = SS. In this case, we can determine a nonlinear connection N depending only by the fundamental tensor gli [see Ch.6]. Indeed, let ^lik{x,p) the Christoffel symbols of gtj(x,p) and let us put 7 ° h = 7 > i , 7°o = 7°hjA Vh = 9hjVj = \dhH2. Theorem 11.1.4. The space GH (2)n = {M,gij{x,p)), determined by the Cartan space Cn has a nonlinear connection N with the following coefficients deduced only fromgtj : N; = (1-8)
To a generalized Hamilton space of order two GH^n — (M, glj) we associate the Hamilton absolute energy (1.9)
£(x,y,p)=
glJ{x,y,p)piPj
and consider the d-tensor field (1.9)'
j
dl
g"
The space GH^)n is called weakly regular if: (1.9)"
rank||0* ij | = n.
We can prove the following fact: The weakly regular GH^n-spaces on the fundamental tensor field g'j.
have a nonlinear connection N depending only
11.2 Metrical connections in –
spaces
If a nonlinear connection N, with the coefficients (TVj,^), is a priori given, let us consider the direct decomposition (see (5.2), Ch.7): (2.1)
TUT*2M = N(u) © Wx{u) ® W2{u), Vu e T*2M,
and the adapted basis to it, (6^,^,8'), (2.2)
where
S ^ d t i
Ch.11. Generalized Hamilton spaces of order 2 The dual adapted basis is (dxl,5y1,8pi)
275
where
Sy{ = dy{ + N}dxj, SPi = dPi - N^dx*
(2.3)
An iV-linear connection DT(N) = (Htjk,Cjk,CJik) determines the h-, w\-, covariant derivatives in the tensor algebra of d-tensor fields.
w2-
Definition 11.2.1. An N-linear connection DT(N) is called metrical with respect to GH-metric gij if (2.4)
ff'V
= 0, g \
= 0, 9ij\h = 0.
In the case when gij is positively defined we can introduce the lengths of a d-covector field Xi by (2.5) The following property is not difficult to prove: Theorem 11.2.1. An N-linear connection DT(N) is metrical with respect to GHmetric i f and only if along to any smooth curve 7 : / —> T*2M, and for any parallel d-covector field X, —— = 0, we have -~-^ dt dt
= 0.
The tensorial equations (2.4) imply: h
(2-6)
Now, using the same technique as in the case of Ch. 5, we can prove the following important result:
Theorem 11.2.2. 1. There exists a unique N-linear connection DT{N) = (Hjk,Cjk,C{ the properties:
) having
1°. The nonlinear connection N is a priori given. 2°. DT(N)
is metrical with respect to GH-metric g'i i.e. (2.6) are verified.
3°. The torsion tensors T1jk,S1jk and S^11 vanish. 2. The previous connection has the coefficients Cxjk and C^k given by (1.4) and H'jk are the generalized Christoffel symbols:
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276
H)k = -gis(Sjg3k
(2.7)
+ Sk9js -
Ssgjk).
The known Obata's operators, are given by I*
(2.8)
They are the supplementary projectors on the module of d-tensor fields TI{T*2M). Also, they are covariant constants with respect to any metrical connection DT(N). Exactly as in Ch.5, we can prove: (Htjk,Ctjk,Z% %k)),
Theorem 11.2.3. The set of all N-linear connections DF(N) = which are metrical with respect to g'3', is given by IT — (2.9)
where DT(N) = (Hjk, Cljk, C{k) is given by (1.4), (2.7) and X)k, Yjk, Z? are arbitrary d-tensor fields. We obtain: Corollary 11.2.1. The mapping DT(N) —*• DT(N) determined by (2.9) and the composition of these mappings is an Abelian group. Remark. It is important to determine the geometrical object fields invariant to the previous group of transformations of metrical connections [105]. From Theorem 11.2.3 we can deduce: Theorem 11.2.4. There exists a unique metrical connection DT(N) — (Hljk,C?jk.,C:ii ) with respect to GH-metric g'i, having the torsion d-tensor fields Tljk, Sljk, Sjk a priori given. The coefficients of DT(N) are given by the following formulas % (2.10)
= \9"(Sjg.k
+ SkgJS - 6sgjk) + \gis{gshThjk
T?ik = l-gis(dj93k + dk9js - dsgjk) + ~gis(9shShjk f
L
- dsgjh) ~ lgis(93llShik
- gjhThsk + -
h 9jhS 3k
+
- gjhShsk +
gkhThjs), h 9khS js),
ghhS^).
Ch.11. Generalized Hamilton spaces of order 2
277
We can introduce the notions of Rund connection, Berwald connection and Hashiguchi connection as in Chapter 2, and prove the existence of a commutative diagram from the mentioned chapter. Finally, if we denote Rhijk = ghsRs'jk etc., then applying the Ricci identitiesto tJ , and taking into account the equations (1.6), we get Theorem 11.2.5. The curvature tensor fields ghsRsljk, symmetric in the indices h, i.
11.3
ghsPsljk,
etc. are skew
The lift of a GH-metric
Let the nonlinear connection N be given, then the adapted basis (Si, di,d') and its dual basis (dxl, 5y\ Spi) can be determined. Therefore, a generalized Hamilton space of order two GH^n = (M, g1*) allows to introduce the N-lift: (3.1)
G = giidxi ® dxj + g^y* ® 6yj + gij5pt
defined in every point u € T*2M.
Theorem 11.3.1. 1°. The N-lift G is a nonsingular tensor field on the manifolds T*2M, symmetric, of type (0,2) depending only by the GH-metric g'j and by the nonlinear connection N. 2°. The pair (T*2M, G) is a (pseudo)-Riemannian 3°. The distributions N
,
space.
are orthogonal with respect to G, respectively.
Indeed, every term from (3.1) is defined on T*2M, because #„ is a d-tensor field, and dx',6yl,6pi have the rule of transformations (5.6)", Ch. 9. The determinant of G is equal to the determinant of matrix ||<7y||. Hence det ||G|| i= 0. Now it is clear that G is a (pseudo)-Riemannian metric. And it is evident that the distributions N , W2 are orthogonal with respect to G, respectively. The tensor G is of the form (3-2) G = G* + G Wl + Gw\ GH = gtjdx* ® dxj, GWl = g^Sy' ® Syj, Gm = gljSPi ® SPj. Here GH is the restriction of the metric G to the distribution H, GWx is its restriction to Wx and Gw* is its restriction to the distribution W2. Moreover GH, GWl GW2 are d-tensor fields.
The Geometry of Hamilton & Lagrange Spaces
278 It is not difficult to prove:
Theorem 11.3.2. The tensors G, G^, G^ 1 G^2 are covariant constant with respect to any metrical N-linear connection DT(N). Therefore, the equation DG = 0 , V X e X(T*2M)is equivalent to gijih = g..L = gij\h = 0. The same property holds for the d-tensors GH, GWl Gm. The geometry of the (pseudo)-Riemannian space (T* 2 M,G) can be studied by means of a metrical N-linear connection. Let IF be the natural almost contact structure determined by N and is given in the section 8, Ch.9. Theorem 11.3.3. The pair ( IF) is a Riemannian almost contact structure determined only by GH—metric
G(FX, Y) = -G(FY, X), VX, Y e X{T*2M)
is verified. The 2-form associated to the structure ( (3.4)
IF) is given by
8{X,Y) = G{WX,Y).
Since DXG = 0, D X IF = 0, we get that Dx0 = 0. In the local adapted basis 9 has the expression: (3.4)'
6 = gi
Theorem 11.3.4. The 2-form 6 determines an almost presymplectic structure on the manifold T*2M. It is not an integrable structure if the metric g%i depends on the moments pi. Indeed, 8 is a 2-form of rank 2n < 3n and for d'g^ ^ 0 the exterior diferential of 8 does not vanish. The last theorem suggests to consider another almost contact Riemannian structure on T*2M. In order to do this, let us consider some new geometrical object fields on T*2M : (3.5)
pl =
(3.6)
Bi =
jiPj, 9i]d\
Ch.11. Generalized Hamilton spaces of order 2 Let us define the
279
-linear mapping F : X{T*2M) —> X{T*2M)
given in the local adapted basis by (3.7)
F(<5,) = -Bit F(3,-) = 0, F(di) = 6it (i = 1,..., n).
Theorem 11.3.5. The mapping F has the following properties: 1°. F is globally defined on T*2M. 2°. F is a tensor field of type (1, 1) on 3°. KerF = Wu imW = N®WX. 4°. rank F = 2n. 5°.
The proof is completely similar with the one of exactly like Theorem 9.8.2. The mapping F will be called the (p)-almost contact structure determined by gtJ and by N. The Nijenhuis tensor of the (p)-almost contact structure is A/"F(X, Y) = ¥2[X, Y] + \FX, WY] - W[WX, Y) - W[X, WY],
and the condition of normality of F is as follows (3.8)
Afp(X, Y) + XXcJz/X*, Y), = 0, \/X, Y e X(T*2M).
The relation (8.3) can be explicitely written in adapted basis. Theorem 11.3.6. The pair (G, F ) is a Riemannian almost contact structure determined by gtJ and by N. Indeed, we have verified the property: G(WX,Y) = -G(JFY,X). The 2-form associated to (G, F ) is (3.9)
9(X, Y) = G(FX, Y).
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280
In the adapted basis 9 is given by (3.9)' 6 = 8piAdx\ As we know, if the torsion r^ = Nij - Nji of the nonlinear connection vanishes (Ch.9), then we have: 9 = 6 = dw = * Hence for r^ = 0, the 2-form 9 is canonical presymplectic structure. It does not depend on the nonlinear connection N (see, Theorem 9.1.3). Theorem 11.3.7. The associated 2-form 0 of the almost contact structure (G,IF) has the properties: 1°. 6 is globally defined on T*2M. 2°. rank||0|| = 2n. 3°. 6 depends by g'j and by N. 4°. 6 defines an almost presymplectic structure on T*2M. 5°. If the torsion Tij of the nonlinear connection N vanishes, then 6 is canonical presymplectic structure: (3.9)"
8 = du) = d
6°. 9 is covariant constant to any N-linear connection DT(N). The Riemannian almost contact space (T*2M, G , F ) will be called the geometrical model of the generalized Hamilton space Gi/ ( 2 ) " = (M,glj).
11.4
Examples of spaces GH^n
We shall consider a generalized Hamilton space of order two, GH ( 2 ) n — (M,gtj), whose fundamental tensor is as follows: (4.1)
9lj(x,y,p) =
e-2^x^1ij(x,y),
where jij(x,y) is the fundamental tensor of a Finsler space Fn = (M,F), 7 y is its contraviant tensor field and a : T*2M -> R is a smooth function. In the particular case where d'a = 0, and 7y(a;, y) = iij{x) is a Lorentz metric, this structure was used for a constructive axiomatic theory of General Relativity by R. Miron and R. Tavakol [121].
Ch.11. Generalized Hamilton spaces of order 2
281
If iij(x, y) is a locally Minkowski Finsler space and d V = 0, then (4.1) gives a class of a generalized P.L. Antonelli and H. Shimada metric [19]. In order to study the GH^n spaces with the metric (4.1) in the general case when the d-vector field d oes not vanish, we prove at the beginning that this metric is not reducible to an H-metric. Theorem 11.4.1. The generalized Hamilton space of order two with the metric (4.1) is not reducible to an Hamilton space of order two. Proof. Taking into account Proposition 11.1.1 is sufficiently to prove that if d%u ^ 0, then the tensor field
i
= -dkgij
s not totally symmetric.
q.e.d.
From the formula (4.1), we deduce
Consequently, Cijk is totally symmetric if and only if dka = 0. Let us consider the Cartan nonlinear connection of the Finsler space Fn, with the coefficients Ni(x, y). Thus, using Theorem 11.1.3, we can a priori take the nonlinear connection N with the coefficients (1.7) as the nonlinear connection of the considered space GH^n. Proposition 11.4.1. The nonlinear connection N, with the coefficients {N1 j , TVy) from the formulas (1.7) depend only on the GH-metric (4.1). Now we can determine the metrical connection DF(N) of the space GH^n Theorem 11.2.3. This metrical N-linear connection will be called canonical. It is not difficult to prove:
using
Theorem 11.4.2. The canonical metrical connection DT{N) of the space with fundamental tensor field (4.1), has the coefficients: H)k =hik
+S)8kc7 + SlSja -
is
7]k7
6s
o o where (F'jk,C'jk) is the Cartan metrical connection of the Finsler space Fn. Now, applying the theory from the previous chapters and using Theorem 11.4.2, we can develop the geometry of the spaces G// ( 2 ) n , with the metric (4.1). For instant, we can write the structure equations of the canonical connections (4.2), etc.
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282
Other important example suggested by the Relativistic Optics is given by the following GH-metric of order two: (4.3)
i 9 i{x,y,P)
= 7lJ(x,y) + ^ J
M
p< = j i j P j ,
where n2(x,y) > 1, on the manifold T*2M, and 7lJ(:r, y) is contravariant tensor of a fundamental tensor field 7y(i,y) of a Finsler space F n = (M, F(x,j/)). In this case we can prove that GH^n, with the metric (4.3), is not reducible to a Hamilton space of order two. Taking into account the nonlinear connection N with coefficients (1.7) we can determine, by means of Theorem 11.2.3, a canonical metrical connection DT{N) depending only by the considered space GH^n. As a final example, we can study by the previous methods "the AntonelliShimada metric" defined in the preferential charts of an atlas on the manifold T*2M by (4.4)
9ii(*,V,P) = e-2o{***)'riiip),
where
(4.5) H2(P) = {(Pi)m + • • • + (Pn)m}™, m > 3. Finally, we remark that the theory exposed in this chapter will be useful in the next chapters for study the geometry of Hamilton spaces of order two.
Chapter 12 Hamilton spaces of order 2 The theory of dual bundle ( M ) mentioned in the last three Chapters allows to study a natural extension to order two of the notion of Hamilton spaces studied in the Chapters 4,5,6. A Hamilton space of order two is a pair H^n = (M, H(x, y, p)) formed by a real, n-dimensional smooth manifold M and a regular Hamiltonian function H : {x,y,p)eT*2M -¥ H{x,y,p) e R. The geometry of the spaces H^n can be constructed step by step following the same ideas as in the classical case of the spaces H^n = (M, H(x,p)), by using the geometry of manifold T*2M endowed with an regular Hamiltonian H(x, y,p).
12.1
The spaces H^n
Let us consider again a differentiable manifold M, real and of dimension n and the dual bundle T*2M of the 2-osculator bundle T2M. Definition 12.1.1. A regular Hamilton of order two is a function H : T*2M -> R, differentiable on T*2M and continuous on the zero section of the projection •n*2 : T*2M —> M, whose Hessian, with entries (l.l)
gHx,v,p) = \
is nondegenerate. In other words, the following condition holds (1.1)'
Taa]4g^(x,y,p)\\=n, on
Moreover, since gli{x,y,p) being a d-tensor field, of type (2,0), the condition (1.1)´ has geometrical meaning. 283
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284
Of course, if the base manifold M is paracompact, then on T*2M there exist the regular Hamiltonians. The d-tensor glj is symmetric and contravariant. Its covariant d-tensor field will be denoted by gij(x,y,p) and it is given by the elements of the matrix Hff^ip1Hence we have: 9i393k = l>l
(1.2)
Definition 12.1.2. A Hamilton space of order two is apair H^n = (M, H(x, y,p)), where H is a regular Hamiltonian having the property that the tensor field gli{x, y,p) has a constant signature on the manifold T*2M. As usually, H is called the fundamental function and g^ fundamental tensor field of the Hamilton space or order two, H^n. In the case when the fundamental tensor field g%i is positively defined, then the condition (1.1) ´ is verified. Theorem 12.1.1. If the manifold M is paracompact then always exists a regular Hamiltonian H such that the pair (M, H) gives rise to a Hamilton space of order two. Proof. Let Fn = (M,F(x,y)) be a Finsler space having 7i3(x,y) as fundamental tensor. Then, the function defined on the manifold T*2M by
H(x,y,p) = af3{x,y)piPj,
a € R+
is a regular Hamiltonian or order two, and the pair # ( 2 ) n = (M, H(x, y,p)) is an Hamilton space of order two. Its fundamental tensor is gij(x,y,p) = aYj(x, y). Obviously, M being paracompact, a Finsler space Fn = = (M, F(x, y)) exists and therefore H(x, y, p) exists. q.e.d. One of the important d-tensor field derived from the fundamental function H of the space # ( 2 ) n is: (1.3)
Cy* = - ^ a V =
--dld>dhH.
Proposition 12.1.1. We have: 1° Clih is a totally symmetric d-tensor field. 2° Cijh vanishes, if and only if the fundamental tensor field glj does not depend on the momenta Pi.
Ch.12. Hamilton spaces of order 2
285
Other geometrical object fields which are entirely determined by means of the Hamiltonian of order two, H(x, y, p), are the w\- and w2- coefficients of a metrical connection, respectively. Theorem 12.1.2. The
d-tensorfields c
*jk = o 9" (djgSk + dkgjS - d3gjk) , ^ \
(1-4)
have the following properties: 1° They depend only on the fundamental function H. 2° They are symmetric in the indices jk. 3° The formula C(k =
(1.5)
gisCsjk
holds. 4° They are the coefficients of the toj- and w2- metrical connection. So we get: (1.6)
9 \k = 0' 9 \
=
0.
The proof is not difficult. The curvature d-tensor fields Shljk,Shljk and 5h y * expressed in formulae (6.4)", Ch.10, depend only on the fundamental function H. The wi and w 2 -vertical paths of the Hamilton space of order two are given by Theorem 10.7.5, respectively. Namely Theorem 12.1.3. 1° The wi -vertical paths of the space H^" in the point (x'o) € M are characterized by the system of differential equations * = 4 , Pi = 0, ^
+ C^Xo,
y,Q)^^
= 0.
2° The w2-paths of the space H^n in the point {xl0) e M are characterized by the system of differential equations
The horizontal paths of the Hamilton space of order two will be studied after a canonical nonlinear connection will be introduced.
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286
12.2
Canonical presymplectic structures and canonical Poisson structures
As we know on the total space T*2M of the dual of 2-osculator bundle there exist different remarkable canonical structures and object fields. Namely: (2.1)
(2.2)
w=
(2.3)
e
where 6 is a presymplectic structure on T*2M of rank 2n. There exist, also, the canonical Poisson structures { }o and { }i defined for any f,g E F(T*2M) by U 9i0
'
dx> dPl
dx> dpt'
if x =^L^_-^_^L U 9il ' dt dpt dy* dpi Each of these Poisson brackets are invariant with respect to changes of coordinates on the manifold T*2M they are R-linear with respect to each argument, skewsymmetric, satisfies the Jacobi identities and the mapping
is a derivation in algebra of the functions T(T*2M). Proposition 12.2.1. The following identities hold: 1°
{X\xi}a = {yW}a = {Pi,Pj}a = 0, (a = 0, 1),
(2.5)
{xi,yi}a
= {yi,pj}a = 0,
(a = 0,1),
2° For any H e T(T*2M) we have: air
{x\H}0 = ^-; OPi
(2.6)
{xi,H}1=0,
dH {y\H}a = Q,{y\Hh = ^ r
m
dH
f
m
dH
Ch.12. Hamilton spaces of order 2
287
Assuming that the manifold T*2M is endowed with a regular Hamiltonian H such that H^n = (M, H) is an Hamilton space of order two, we get an Hamiltonian system of order 1 given by the triple (T*2 M, H (x, y, p), θ) and we can treat it by the classical methods [cf. M. de Leon and Gotay [85]]. In this case, evidently only the Poisson structure { }o will be considered. Therefore we will study the induced canonical symplectic structures and the induced Poisson structures on the submanifolds S o and Ei of the manifold T*2M, where E o and £ i will be described below. Let us consider the bundle (T*2M,W*,T*M) and its canonical section, a0 : (x,p) e T*M -> (x,Q,p) 6 T*2M. Let us denote by E o = I m a 0 . It follows that S o is a submanifold of the manifold T*2M. Let us denote the restriction of to the submanifold £ 0 by 90, and let us remark that E o has the equation y{ = 0, where (xl,Pi) are the coordinates of the points {x,p) G Eo. Theorem 12.2.1. The pair (Y,o,6o) is a symplectic manifold. Proof. Indeed, 0O = dpiAdxi
(2.7)
is a closed 2-form and rank||0 o || = 2n = dimE o .
q.e.d.
In a point u = (x,p) e E o the tangent space T U E O has the natural basis I TT-^I-— I , (i = 1,..., n), and natural cobasis (dx\ dpi)u. \dx* dpiju Let us consider ^(EoJ-module X(E0) and ^"(Eoj-module X*(E0) vector fields to E o and of cotangent vector fields to S o , respectively. Then, the following ^(Eo)-linear mapping SBo defined by (2.8) has the property: (2.8)'
S
A glance at the formula (2.8)´, gives: Proposition 12.2.2.
The mapping Se0 is an
isomorphism.
of tangent
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288
Let us consider the space H(2)n = (M,H(x,y,p)) and denote Ho = H^o. Then the pair (M , ) is a classical Hamilton space (see Ch. 5) having the funda1 BH mental tensor field gtj(x,0,p) = 2 j The Proposition 12.2.2 shows that there exists a unique vector field XH0 € £ A'(Eo), such that (2.9)
o
In local basis, we get , ( (210)
dH0 d
X
dH0 d
Theorem 12.2.2. The integral curves of the vector field XHo are given by the " -canonicalequations": dxi
e>u\
(211)
-dH°
For two functions f g €
dpi
-
, let
be the corresponding vector fields
given by iXf90 = -df, iXg90 = -dg. Theorem 12.2.3. The following formula holds (Z.IZ)
[Jtgfo = U()(A.f,A.g).
Proof. Indeed, we have zi/v" v \ (A a\lv \ vo\Af,Ag) = \lXj"o)(-X-g)
(2.13)
(df
dg
=
c l v \( v \ At (v \ v f ^>9o\-^f)\^g) — ~"J\^-g) ~ ~-*-gJ
=
dg df"
Now, taking a canonical 2-form 6\ on the fibres of the bundle (T*2M,TT*2,M) we can obtain a similar relation for the Poisson structure { }i. Let Ei be the fibre ( T T * 2 ) - ^ ^ ) C T*2M in the point x0 € M. Then Ex is an immersed submanifold given by Ex = {{x,y,p) € T*2M \ x — x0}- In a point u = and natural cobasis by (dy',dpi).
Ch.12. Hamilton spaces of order 2
289
We can prove, without difficulties, that the following expressions give rise to geometrical object fields on the manifold E^ wi = pidy\ If
U 9n
'
X
=
i
Qt dPi
_
dtf dPi
Consequently, we get: Theorem 12.2.4. The following properties hold: 1° 9\ is canonical symplectic structure on the manifold Si. 2° { }i is a canonical Poisson structure The relationship between these two canonical structures can be deduced by the same techniques in the case of the pair of structures (0Q,{ }O). Indeed, the mapping S
8 l
••
defined by (2.15) has the properties (2.15)'
^
Proposition 12.2.3. Sgl is an isomorphism. That means that there exists an unique vector field XHl such that (2.16)
ix^O, =
-dHu
where Hi : Si ->• R is the regular Hamiltonian, Hi{y,p) - H(x0, y,p). Locally, XHi is given by d
dHl
and its integral curve are as follows (218) (2 18) '
a' x
^ ' dt ~ dPi'
dt
d
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290
These are called "the state:
-canonical equations" of the space
Theorem 12.2.5. The integral curve of the vector field canonical equations (2.18).
n
. Therefore, we can
are given by the
Finally, we can prove: Theorem 12.2.6. The following formula holds {f,g}i=01(Xf,Xg),
V/ )
The previous theory shows the intimate relations between symplectic structures 6a and the Poisson structure { }Q on the manifolds EQ, (a = 0,1).
12.3
Lagrange spaces of order two
We shall prove the existence of a natural diffeomorphism between the Hamilton space of order two, H^n = (M,H(x,y,p)) and the Lagrange spaces of order two £(2)n _ [M,L(x,y^\y^)). To this purpose, we shall briefly sketch the general theory of the space L<2)n (see, §1,2, Ch.6, of the book [106]). The fundamental function of the space L ^ n is a Lagrangian of order two, L : (x,y^l\y^) e T2M —> 6 R, which is regular and the fundamental tensor field d2L
(3.1)
has a constant signature. On the manifold T2M there exist two distribution V\ and V2. The distribution Vj is the vertical distribution of dimension 2n and V2 C V\, dimV2 = n. Clearly, dimM = n, T2M = 3n. A transformation of local coordinates on T2M : (xl, j / 1 ' 1 , j/ 2 ' 1 ) —> {x\y^%,y^%), (i,j, h, k,…= 1, 2,…, n) is given by the formula (1.1), Ch.6, for k = 2. Namely, x{ = xi{xl,...,xn),
det
(3.2)
+2
dx{
Ch.12. Hamilton spaces of order 2
291
where
dT dxi
dy^ dyMi
&f^ dyW
dy^ dxi
8y^
Of course, for every point u = (a;,2y(1),2/(2)) € T2M the natural basis of the tangent space TU(T2M) transforms as: d _ dx1 d Ijx* dxi ~~ dx 9 ?{ dxi 95>
d
dyil)j dx{
d C 1W dyW '~'
dyi2)j dx{ d
d
d
dy^i
By means of these formulae one can prove that the vector fields determine a local basis of the distribution V2, and d
d
d
d
is a local basis of the vertical distribution V\. These distributions are integrable and V2 C Vi, dimVi = 2n, dimV^ = n. Let us remark that there are two Liouville vector fields:
f - y(1)i
(3 4) {
~V
'
d
, f - v(1)i V
dyM*
d
dyW
I 2u f2) '
d
V
1
2
with the properties that F belongs to the distribution V2 and T belongs to the 1
2
distribution V\. The vector fields F and F are linear independent. There exists a 2-tangent structure J, on T2M, defined by
(3 5)
-
j
{£?)=
The following properties of the 2-tangent structure J hold: 1° J is a tensor field on T2M of type (1, 1). 2° J is an integrable structure. 3° Im J = Vi, Ker J = V2, ./(Vi) = V24° rank||J\\ = In.
The Geometry of Hamilton & Lagrange Spaces
292 5° JT = f, JT = 0. 6° J 3 = 0.
A 2-semispray on T2M is a vector field S on T2M with the property
JS = f.
(3.6) Locally S is given by
S^^^y^^W"^
(3.7)
where G* are the coefficients of S and they characterize the vector field S. A nonlinear connection N on the manifold T2M = Osc2M is a vector subbundle N(T2M) of the tangent bundle T{T2M) which, together with the vertical subbundle V{T2M), give the Whitney sum:
TT2M = NT2M 8 VT2M. Noticing that ./(./Vo) = Nit No = N, and that Nx is a subdistribution of the vertical distribution Vi, we obtain the direct decomposition of linear spaces TUT2M = NQ(u) © Nx(u) © ^ ( u ) , Vu e T2M.
(3.8)
An adapted basis to this direct decomposition is given by S
*
d
where
The systems of functions < jV \, N \ > give the coefficients of the nonlinear connection l(i) (2) J N. The adapted cobasis, which is the dual basis of (3.9),
where (3.10)'
&y
— dj/
+ MJ ldx^, <5j/ (1)
^ dy
+ M \dy"^+ (1)
71/ Ida;''
(2)
Ch.12. Hamilton spaces of order 2
293
The new coefficients M \,M 3\ of the nonlinear connection N that appear here are (1)
(2)
called the dual coefficients of N. They are related with the primary coefficients N l3i,N 3) by the formula:
(1)
(2)
(3.11)
M ) =N ) , M ) =N i+N^N (i)
3
(i)
3
(2)
3
(2)
3
(i)
m
(i)
™ 3
Conversely, the previous formulae, uniquely determine N, N as functions of M, M • (1) (2)
(1) (2)
R. Miron [106] and I. Bucataru [36], [37] showed that a 2-semispray with the coefficients G\ uniquely determines a nonlinear connection. The dual coefficients given by Bucataru are very simple: (2)
3
Studying the variational problem for the regular Lagrangian of order 2, L(x, y^, t/ 2) ) we can determine a canonical nonlinear connection of the Lagrange space of order two, \
12.4
Variational problem in the spaces
Let us consider a Lagrange space of order two, L (2)n = (M, L{x,y^\y^)). If X e €.X(E), let us denote the operator of Lie derivation with respect to X by £x1 2
Applying this operator with respect to the Liouville vector fields F, F, (2.4) we get two important scalar fields determined by the Lagrangian L: (4.1)
I(L) = &L, }{L) = C,L.
They are called the main invariants of the space Let c: : 1 ] —> M be a smooth parameterized curve and assume Im c C U C M, where U is a domain of a local chart on the manifold M. The curve c is represented by the equations xl = x'(t), t £ [0,1]. The extension c of c to the manifold T2M is: (4.2)
x* = x\t), yW' = ^(t),
yW = \ ^(t),
t € [0,1].
The integral of action of the Lagrangian L(x, y fl) , j/ (2) ) along the curve c is defined
by
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It is known that if I(c) does not depend by the parameterization of the curve c 1
2
then I(L) = 0, I(L) = L (Zermelo conditions). In this case rank||oy|| < n. So, the fundamental tensor of the space L^n is singular. Consequently, the functional I(c) depends on the parametrization of the curve c. Along the curve c, the following operators can be introduced: *~ dx> ~ dt dyW (4-4)
^ = -
^
+
+
2dl?
5
2 In the monograph [106], the following theorems are proved: Theorem 12.4.1. 1° For any differentiable function <j>(t), t € [0,1] we have
(4.5)
Ei {4>L) = <>/ k (L) + ^ k(L) + ^
k(L).
2° Ei(L), (a = 0,1, 2) are d-covector fields.
Theorem 12.4.2. The variational problem on the integral of action I(c) leads to the Euler-Lagrange equations ,Ae\ 2 , , , ^ dL d (4.6) EUL):=g--JtW^
dL
+
1 d2 dL - ^ 2 ^ - t = 0 n,
y
dx< 1 d2xi ^ = - , y (2)i ^ = - ^ .
n)i
dL Taking into account of the Theorem 12.4.1 it follows that — do not vanishes dt along with the integral curve of the Euler-Lagrange equations. Therefore, it introduce the notion of (Hamiltonian) energy, [106]. However we point out that in the case of the space L^n it will depend on the curve c : [0,1] —: M. Definition 12.4.1. Along the smooth curve c: [0,1] —> M the following functions (4.7)
£C(L) =
2
I(L)-\±I(L)-L,
£C(L) = - \
Ch.12. Hamilton spaces of order 2
295
are called the energy of order two and of order 1 of the Lagrange space Z/ 2) " = (M, L), respectively. In the monograph [106] it is proved: Theorem 12.4.3. 2
For any differentiable Lagrangian L(x,y^\y^)
the energy of
order two SC{L) is conserved along every solution curve c of the Euler-Lagrange o equations Ei (L) = 0. 2
1
Along the smooth curve c : [0,1] —> M, the energies £C{L), and SC(L) can be written in the form
(4.8)
e
.
(
t
)
|
!
i
«(!)
where
dL 1 d dL W ~2lt dy^' are the Jacobi-Ostrogradski
_ 1 dL ~~2 ^
Pi
momenta.
Theorem 12.4.4. Along a smooth curve c we have
(4.10)
Ox
«'
This property is useful in order to prove: Theorem 12.4.5. Along each solution curve c of the Euler-Lagrange o Ei (L) — 0, the following Hamilton equations hold: d£c(L) _ dx{
dpw
dt'
d£c{L)
dxj
~
d£c(L) = rfV d£c{L) dpi ~ dt2 ' dyW ~ 2
equations
dt dPi dt
From this reason £C(L) is called the Hamiltonian energy of the space Z/ 2 '". Some of the previous results hold even in the case when L is not a regular Lagrangian. If L is the fundamental function of a Lagrange space of order 2, Z/2'™ = (M, L) we can determine a canonical nonlinear connection N depending only on L.
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Indeed, we consider the Synge equation [106]:
where
\
J
(4 13) (4.13) Thus, the canonical semispray of Z/2)n is given as follows:
Theorem 12.4.6. Any Lagrange space Z/ 2 ' n = h L) as a canonical semispray, determined only by the fundamental function L. It is given by: (4 14) (414)
9 - v^— 5-1/
+ 2?/(2)i +^2/
d
_ o^./ (l) (2h 9 dOliy tf ;
,
where the coefficients G' are expressed in the formula (4.13). Consequently, we obtain [106]: Theorem 12.4.7. For any Lagrange space Z/ 2 '" = (M, L) there exist the nonlinear connections determined only by the fundamental function L. One of them is given by the dual coefficients (M ) , M \) from the formulae (3.12), (4.13). (1)
(2)
The nonlinear connection mentioned in the previous theorem will be called canonical for the space Z/2)>".
12.5 Legendre mapping determined by a space If a Lagrange space of order two, L(Tin^= (M, L(x,y ( 1 ) ,y ( 2 ) )) is given, then it determines a local diffeomorphism tp : T2M —> T*2M, which preserves the fibres. The mapping tp transforms the canonical semispray S of L (2)n in the dual semispray 5{, where £ =
d2L
Ch.12. Hamilton spaces of order 2
297
Proposition 12.5.1. If L is the fundamental function of a Lagrange space of order two^lP)n, then the following mapping > : {y(0), yw, y{2)) € T2M —>• (x,y,p) € 2 T* M, given by
(5.1) Pi
~
_
2
is a local diffeomorphism which preserve the fibres. Proof. The mapping i s differentiableand its Jacobian has the determinant equal to det ||a^;||, which do not vanish on T2M. Of course, we have v2(y{0\yM,yW) = v*2 o
ft
=
^
Clearly,
U = (**2)-l(U),
UCM.
We have the following identities
(5.4)
MJ,«W>
and dxi
Jl%
^
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The differential
d
d
d
d
(
of the diffeomorphism > is expressed d
dipm
d ' Oyi dpm' d
Theorem 12.5.1. The mapping
~
dx'
in the dual semispray S^ on T*2M:
which has the following coefficients (5.7)"
/, = -al
Proof. If X e XiTHd) have the local expression y - ym
d
dyW
+ X(l)i
d
dyW
+ X(2)i
then
Consequently <^«5 = 5^ holds. Corollary 12.5.1. The dual semispray Sf (5.7)´, determines on T*2M a nonlinear connection with the coefficients: (5.8)
N\ = fffj, Ntj = djj.
Ch.12. Hamilton spaces of order 2
299
We have to ask ourselves if by means of the mapping ip we can transform a regular Lagrangian L(y (0) ,y (1) ,y (2) ) in a regular Hamiltonian. Notice that y(2)l = C{x,y,p) is not a vector field. Therefore the product i s not a scalar field as in the classical case of the Hamilton spaces # ( 1 ) n = (M,H(x,p)). o o . Let us fix a nonlinear connection N with the coefficients N ){x,y) on T M C C T2M. Then on THI we get the d-vector field
This d-vector field is transformed by ip in the following d-vector field on T*2M : 1 0 .
Let us consider the following Hamiltonian: H(x,y,p) = 2JHZW -
(5.10)
L(x,y,t(x,y,p)).
Then we have: Theorem 12.5.2. The Hamiltonian function H, (5.10), is the fundamental function of a Hamilton space H^n and its fundamental tensor field a*s(x,y,£(x,y,p)) is the contravariant of the fundamental tensor field a^ of the space Z/ 2 '" = {M,L). Proof. From the formula (5.10) we deduce (5.10)' Therefore, we get 9i3(x,y,p) = \&&H = ^ = ^ = a>i(x,y,t;(x,y,p))i op op Consequently, the pair H^n = (M,H), (5.10), is an Hamilton space or order two. (5.10)"
q.e.d.
The space H&n = (M, H), (5.10), is called the dual of the space L(2>" = (M, L). o Of course, this dual depends on the choice of the nonlinear connection of iV .
12.6
Legendre mapping determined by H^n
Now let us pay attention to the inverse problem: Being given a Hamilton space of order two, i/ ( 2 ' n = (M,H(x,y,p)), let us determine its dual, i.e., a Lagrange space of order two.
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300
& Lagrange Spaces
In this case, we will start from H^n and try to determine a local diffeomorphism of form (5.3) by means of the fundamental function H(x,y,p) of ff ( 2 K But 3/(2)l is not a vector field. Therefore we cannot define it only by -zd'H, which is a d-vector o
field. As in the previous section we assume that the nonlinear connection N, with o
coefficients N lj(x,y), Consequently, (6.1)
which does not depend on the momenta pi, is apriori given.
| #
is a d-vector field on ^ The mapping & : T*2M —> T2M defined by j/W* = x\ y{1)i = y\ yW = Ci(x,y,p),
(6.2) where
l & ' t f
(6.2)'
o
is the Legendre transformation determined by the pair (H(x, y,p), )
.
Theorem 12.6.1. The mapping given by (6.2), (6.2)´, is a local diffeomorphism, which preserves the fibres of T*2M and T2M. Proof.
The determinant of the Jacobian of £x is equal to det \\g%i{x, y,p)\\ and
The formula (6.1), (6.2), (6.2) ´ imply: (6.3)
H{=^{x,y,p),
(6.4)
zi(x,y,t\) = ±diH(x,y,p).
Let us consider the inverse mapping ipi of the Legendre transformation ^ : (6.5)
x< = y(°'' y* = y { l ) i ,
Pi =
It follows (6-6)
In a regular Lagrangian it is interesting to remark that the Hamiltonian H(x, y,p) is transformed by ^i exactly as in the classical case: (6.7)
L{x,yW,yW) = 2PiJ-H{x,yip),
Pi
=
Ch.12. Hamilton spaces of order 2
301
Theorem 12.6.2. The Lagrangian Lfrom (6.7) is a regular one. Its fundamental tensor field is given by 9ij(x,y^\ipi). Proof. Because
, . = 6), it results - - 5 - ^ = <£>»(£, 2/ (1) ,j/ (2) ) and from (6.6) we J Oy\')J I oyW1
The space L ( 2 ) n = (M, L) with L given in (6.7) is called the dual of the space
12.7
Canonical nonlinear connection of the space
H{2)n
The Lagrange space of order two, l/ 2 ) " = (M, L(x,y (1) ,y' 2) ), with the fundamental function (7.1)
L(x,y1,y2) = 2Pizi-H(x,y,p),
Pi
=
is the dual of the Hamilton space of ordertwo, H^n = (M, H(x, y,p)). Its canonical semispray
is transformed by the Legendre transformation £j in the canonical dual semispray 5{I:
The relation between the coefficients Gl and /j is as follows (7.2)
/i = y ™ ^ + 2 f » ! p ; - SG^^y.eiJSmi, 2/(2)i - ^(x.l/.p).
And since ^1 and ip\ are the inverse mappings, respectively, we have dxm
9is 9is
Substituting in (7.2) we get
(7.2)'
- ^ 7 . = y~<*
dxmm'
dy dymm
dy
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The expression (4.13) of G^z,?/ 1 ',?/ 2 ') give
The last formula and (7.2) ´ leads to fm = \
(7-3)
Because £J is in (6.2)´, we have: Theorem 12.7.1. The coefficients fi of the canonical semispray S^ of the Hamilton spaces H^n are given by: fJ(x,y,p) =
(7.4)
Finally, applying Theorem 9.5.2., we can formulate: Theorem 12.7.2. The coefficients of the canonical nonlinear connection N of the Hamilton space # ( 2 ) n are as follows: (7.5) 3
2 dyi I
Remarks. dH
o
o
1° If — = 0, N y(x, y) =N 'jk(x)yk,
o
then N coincides with N .
2° The torsion r^ = Nij — Nji vanishes. The Theorem 12.7.2 is important in applications.
12.8
Canonical metrical N connection of space
For a Hamilton space of order two, H^n = (M,H(x,y,p)) let us consider the canonical nonlinear connection N determined in the previous section. We are going
Ch.12. Hamilton spaces of order 2
303
to investigate the N-linear connections which are metrical with respect to fundamental tensor field glJ of H^n, i.e.: 9is\h = 0, 9ij\h = 0, 9v\h = 0
(8-1)
Considering the space ff(2)n as a generalized space G # ( 2 ) " with the fundamental tensor g'j we can apply Theorems 11.2.1 and 11.2.2, one obtain: Theorem 12.8.1. For a Hamilton space of order two, H^n following properties hold:
= {M,H(x,y,p))
1° There exists a unique N-connection CF(N) = {Hljk,Cl3k,Cjk) (8.1), as well as the conditions (8.2) 2° The coefficients of
(8-3)
Hl3k = H%, &jk = &kj,
d*
the
which satisfies
= Ck>
are given by the generalized Christoffel symbols:
C V = \g"(dj9.k
+ dkgjs -
where the operators 6j = dj — Nkjdk+Njkdk nonlinear connection N.
ds9]k),
are constructed using the canonical
The connection CV(N) is called the metrical N-connection of Now, applying the theory from Ch.9, we can write the structure equations of the metrical N-connection CF(N), Ricci identities and Bianchi identities. The parallelism theory as well as, the theory of the special curves, horizontal paths etc. can be obtained. We can conclude that the geometry of the second order Hamilton spaces, can be constructed from the canonical connections N and CT(N). The geometrical model of the space H^n is determined the N-lift G, ((3.1), Ch.11) of the fundamental tensor g'j and by the (p)-almost contact structure IF, ((3.7), Ch.11). We obtain, without difficulties: Theorem 12.8.2. The pair (G, F ) is a Riemannian almost contact structure determined only by canonical nonlinear connection N and by fundamental function H of the space H^n. If N is torsion free (i.e. Nij = ) , then its associated 2-form 6 is canonical presymplectic structure 8 = dpi A dx\ The geometrical space %Zn = (T*2M, G, F ) is called the geometrical model of the space H^n = (M, H(x, y,p)). It can be used to study the main geometrical features of the space ^
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12.9 The Hamilton spaces H^n of electrodynamics Let us consider the Hamilton spaces of order two, # ( 2 ) " = (M,H(x,y,p)), fundamental function 1 (9.1) H{x,y,p) = —glJ(x,y)pipj me
with the
2e e2 PiA'(x,y) + —^A'{x,y)A:>(x,y)gij(x,y), m me*
where gij(x,y) is the fundamental tensor of a Finsler space Fn = (M,F(x,y)), A*(x, y) is a vector field, on TlM and m, c, e are the physical constants. The functions gij(x,y) can be considered as gravitational potential and A*(x,y) the electromagnetic potentials. In the classical theory of the electrodynamics H (x, p) is obtained from the known Lagrangian of electrodynamics via Legendre transformation, [97], [105]. The fundamental tensor field of the space H^n is given by: (9.2)
lvj{x,y,p) = —gli{x,y)
It is homothetic to the fundamental tensor g'j(x,y) of Finsler space Fn. This remark leads to the fact that i = (M,H), (9.1), s the Hamilton space of order two of the Electrodynamics. The covariant tensor i s given by (9.2)'
j{j(x, y,p) = megijix, y).
The tensor field Cijk, (1.3), vanishes. It follows: Cijk = 0, dk
(9.3)
- 0.
The tensor Cljk from (1.4) reads: (9-4)
C{jk{x, y) = l-gis{dj9sk + dk9sj - dsgjk).
Then, Theorem 12.1.3 leads to: Theorem 12.9.1. 1° The wi-paths of space H^n in the point x0 6 M are characterized by the system of differential equations
Ch.12. Hamilton spaces of order 2
305
2° The W2-path of the space W• >" in the point XQ € M are characterized by the system of differential equations xl = xl0, yl = 0, -^
= 0.
o
Let N y(x, y) be the Cartan nonlinear connection of the Finsler space Fn and zl the d-vector field on T2M: Remember that the Christoffel symbols 7*^(2;, j/) of Fn and YjkyjVk = 7oo gi v e s
Let us consider the functions
(9-6)
C^l
We obtain the Legendre transformation (6.2) determined by the Hamilton space #< 2 K Then, it follows:
(9.7)
z'(x,y,0 = ld'H =
^Pj~^-A'.
The dual space L (2)n = (M, L(x,yw,yW)) of the space i/ ( 2 ) n has the property (9.8) L{x, y,£l)= p&H -H = YJpiPj -A'A'-y^, c
and the canonical dual semispray S^ has the coefficients (7.4) given in our case by i
f)
Taking into account Theorem 12.7.2 we obtain the coefficients Nlj and iVy of the canonical nonlinear connection N of the space // (2) ":
Therefore the coefficients Nlj can be written: (9.10)'
Ni]=N)-A), A) = -\d^H.
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Clearly Alj is a d-tensor field of type (1,1). The adapted basis I -=—r.-^—»-^— 1 to the distributions N , l
8
\6x' dy dpi/
has the first
vector fields -=—: of the form Sx%
In other words: {
'
5x 5xkk
ee [dx [dxkk
kk
dy dymm
k
dym\ "
Therefore, the coefficients Hljk of the canonical connection CT(N) of the space i/ ( 2 ) " are given in the following Theorem 12.9.2. The canonical metrical connection CT(N) of the Hamilton space #(2)n of electrodynamics has the following coefficients (9.11)
H{jk = F V + #„,
&»,
Cijk = 0
where {F%jk>C%jk) is Cartan metrical connection of the Finsler space Fn and A'jk is a d-tensor field expressed by
Indeed, the last theorem follows from a straightforward calculus, using the formula (*) and the expression of H'jk from (8.3). We remark that the geometry of the Hamilton spaces of electrodynamics (9.1) can be developed by means of the canonical nonlinear connection N, (9.10) and on the canonical metrical connection CT(iV), (9.11).
Chapter 13 Cartan spaces of order 2 The Hamilton spaces H^n = (M,H(x,y,p)) for which the fundamental function H(x, y, p) is 2-homogeneous with respect to momenta pt form an interesting class of Hamilton spaces of order two, called Cartan spaces or order two. For these spaces it is important to determine the fundamental geometrical object fields, as canonical nonlinear connections and canonical metrical N-connections.
13.
–
spaces
Definition 13.1.1. A Cartan space of order two is a pair C (2) " = for which the following axioms hold:
(M,K(x,y,p)),
1° K is a real function on T*2M, differentiableon T*2M and continuous on zero section of the projection 7r*2. 2° K > 0 on Tt2M. 3° K is positively 1-homogeneous with respect to momenta p*. 4° The Hessian of K2, with elements: (1.1)
gi>(x,y,p)= is positively defined on T*2M.
It follows that 5 y from (1.1) is contravariant of order two, symmetric and nondegenerate d-tensor field. It is called fundamental (or metric) tensor of space C(2)n. K(x,y,p) is called fundamental function of C'2'". Let us start noticing: 307
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Theorem 13.1.1. If the base manifold M is paracompact, then there exist on T*2M functions K such that the pair (M, K) is a Cartan space of order two. Proof. The manifold M being paracompact, it follows that T*2M is paracompact, too, and therefore there exists a real function F on TM, which is fundamental for the Finsler space Fn = (M, F(x, y)). Let a^(x, y) be the fundamental tensor of Fn and a>:j(x,y) be its contravariant tensor. Obviously a y is positively defined. If we consider the function K(x,y,p) =
(1.2)
{a'i(x,y)piPj}1'2,
then we obtain the fundamental function of a Cartan space of order two.
q.e.d.
The Cartan spaces C' 2 ' n with fundamental function (1.2) are special. They can be characterized by the vanishing of the d-tensor field Cijk =
(1.3)
~didjdhK2
Proposition 13.1.2. For any Cartan spaces of order 2, we obtain: 1° The components g'j(x,y,p) respect to p*.
3° gij(x,y,p)piPj
=
of the fundamental tensor are 0-homogeneous with
K2(x,y,p).
4° piCijk = 0. Let 9ij(x,y,p) be the covariant tensor of gli(x,y,p). A similar theorem with that given in Ch.12, can be formulated: Theorem 13.1.2. For any Cartan space C^n = (M,K) the following d-tensors
y
(1-4)
\ Clik = --gis
+ dk9]S - d°gjk),
have the properties: 1° They are the (1.5)
— and
-coefficients of a canonical metrical connection, i.e. g% = 0, gi]\k = 0.
Ch.13. Cartan spaces of order 2
309
2° The following identities hold: (1.6)
(1.7) (1.8)
CJ* = ft|*
k
9lsC^
= ** = tf S'jk = Cl}k - C % = 0; 5,J* = Cijk - C,-*' = 0.
The proof is similar with that given in §, Ch.5. Let us consider a -vertical curve 7 : / —> T*2M in the point x0 e M (Ch.10). Applying Theorem 10.7.5., we have: Theorem 13.1.3. 1) The – vertical paths in the point XQ £ M of a Cartan space of order two, (j(2)n _ a ^ ^ r e characterized by the system of differential equations
2) /« f/ie space C^2'" = (M, K) the w2-paths in the point x0 € M are characterized by the system:
13.2
Canonical presymplectic structure of space
C{2)n
The natural geometrical object fields on the manifolds T*2M, in the case of Cartan space of order two, together with the fundamental function K(x, y, p) of space C^n give rise to some important properties, especially in the case of canonical equations ( ch.12). We have u> = ptdx1 and (2.1)
0 = doj = dplA dxi
is the canonical presymplectic structure. The canonical Poisson structure {, }0 and {, }1 from (2.4), ch.12, can be also considered. Remarking that (2.2)
H(x,y,p) = K2(x,y,p)
is a regular Hamiltonian, by means of Proposition 12.2.1, we get:
The Geometry of Hamilton & Lagrange Spaces
310 Proposition 13.2.1. hold:
In a Cartan space C (2) " — (M,K) the following equations
{x\K2}0 = ?f, {y\K% = 0, {«,K2}0 = -
- o i * ^ i - — f K2\
-—
Notice that the triple (T*2M, K2(x,y,p),6) is a Hamiltonian system. Let us consider the canonical section of TT*, given by a0 : (x,p) € T*M —> (x,0, p ) and E o = Imcro- Then EO is an (immersed) submanifold of the manifold T*2M and let us denote the restriction of 6 t o by 6Q. We remark that E o has the equation y% = 0. Theorem 12.2.1. affirmes that the pair is a symplectic manifold. Therefore the mapping SgD : <#(E0) —> A"(E 0 ) defined by (2.4)
Sgo(X)=ix60,VXeX(i:o)
is an isomorphism. We denote Ko ~ K\s0. Then we have Proposition 13.2.2. The pair (M, ) is a classical Cartan space. Indeed, in this case, the definition from Ch.6 is satisfied. Its fundamental tensor field is (2-5)
gij{xAp) = \
We obtain, also: Theorem 13.2.1. There exists a unique vector field XKi 6 (2.6)
SQ(XKi)
with the property
= ixKl9Q = -dK2.
In the local basis, the vector field XK2 has the expression y
' '
' "Kl
dPi dx{
dx> dPi
The integral curves of the vector field XK2 determines the of the space
-canonical equations
Ch.13. Cartan spaces of order 2 Theorem 13.2.2. The (2.8)
311
-canonical equations of the space C' 2 ' n are as follows: — =
-, — =
j - , y{ = 0.
Corollary 13.2.1. The equations (2.8) can be written in the form
Remark. It is clear that the Jacobi method, described in Section 2, ch.4, for integration of the equations (2.8), can be used in this case. Now, let Ei be the fibre, in the point e M, of the bundle (T*2M, n*,M). Then Si is an immersed submanifold in T*2M. Let us consider (cf. §2, ch.12) the following differential forms on Ei: u)l=pidyi,
(2.10)
By. = dpi/\dyi,
and the Poisson bracket: { L g h
( 2 n )
=
The relations between these canonical structures on the manifold Ei can be studied applying the same method as in the case of structures (E o ,^o,^o)If we denote Kx = K^, we get the following -canonical equations. Theorem 13.2.3. The -canonical equations of the Cartan space C^n following (212)
x»
'-x ~ ° ' dt ~ dPi'
= (M,K(x,y,p))
are the
d t ~ ~ dy>
Taking into account the formula (2.3) we obtain Corollary 13.2.2. The equations (2.12) are equivalent to the system of equations
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For the integration of the -canonical equations we can use the Jacobi method. Let us try to determine a solution curve a(i) in the fibre Ei at point XQ € M of the form ±
(2-13) where 5j e Substituting in (2.12) we get: x-x0, {
'
dPi _ 82S1 dK2 _ dK2 dt dyidyi dpi dyi
'
It follows from (2.14)
and therefore K2 [xQ,y,—^-\
(2.15)
= const.
By integration of (2.15), we can determine 5 : and from (2.14) we obtain the curve
a(t).
13.3
Canonical nonlinear connection of
We can associate the Hamilton space of order two, # ( 2 ) " = (M,K2(x,y,p)) to a Cartan space of order two, C (2)n = (M,K(x,y,p)). Then the canonical nonlinear connection N of the space H^n will be called the canonical nonlinear connection of the Cartan space C' 2 '". Therefore, we can apply the theory from the section 7 of the previous chapter. °
° •
Let TV be a fixed connection with the coefficients N 'j(x,y). have the vector field (3.1)
Thus, on T M, we
z^y^+N'jixM, o
and we obtain the Legendre £i mapping determined by C^n and by ./V: (3.2)
y{0)i = x\ yW = yi, 2/(2)i = £(*,!/,p),
Ch.13. Cartan spaces of order 2
313
where (3.2)'
= \{d*K2(x,y,p)-
tf(*,y,p)
N^ytf}.
It follows that (3.2), (3.2)' is a local diffeomorphism which preserves the fibres of T'2M and T2M. From (3.1), (3.2) we obtain: frt{=gv(x,y,p),
(3-3)
zi(x,y,p) = \diK2(x,y,p).
(3.4)
Let ip\ be the inverse mapping of £i. Then it is of the form z< = y<°>\ yi = y< 1 >\ P< = <My<0\y<1\y<2>).
(3.5) From (3.5) we get
l ^
(3-6)
The Lagrangian (6.7), determined from if2 by means of Legendre transformation £i, is given by (3.7)
i(ar,y {1) ,y (2) ) = 2Piz* - K2(x,y,p),
Pi
=
The formula (3.7) and the property of homogeneity of K2, with respect to momenta Pi, lead to the equation: (3-8)
L(x,y^,^(x,y,p))
= K2(x,y,p).
Indeed, from (3.4) and (3.7) it follows L(x,y,£1)=pidiK2-K2
= K2.
The canonical dual semispray of C (2) " is
(3-9)
stl=S±
+
2d± + MX,y,P)±.
Theorems 12.7.1 and 12.7.2 imply: Theorem 13.3.1. The canonical semispray S$ has the coefficients (3.10)
fj{x,V,p) = \
^-l
The Geometry of Hamilton & Lagrange Spaces
314
Theorem 13.3.2. The coefficients of the canonical nonlinear connection N of the Cartan space of order two, C^n are given by: (3.11)
^=\ h H'*2+hX^A N\
Remarks. 1° The coefficients of canonical nonlinear connection N depend on an a priori o
o
given nonlinear connection TV (TV lj(x,y)). The previous theory is more simple o
if TV is the Cartan nonlinear connection of a Finsler space Fn =
(M,F(x,y)).
2° The torsion r^ of the canonical nonlinear connection vanishes. In this case the canonical presymplectic structure 8, (2.1), can be written in the form: e = 6Pi/\dxl,
(3.12) where 5pt = dpt - NjidxK
3° Since 5y{ = dy{ + N^dxi, Prop. 6.3, ch.9, we have: Theorem 13.3.3. A horizontal curve on C' 2 '" is characterized by the system of differential equations: (3.13)
x^x^t),
6y* = 0, ~ = 0,
where xl = xl(t), t € /, are a priori given.
13.4
Canonical metrical connection of space
Let N be the canonical nonlinear connection N of the Cartan space of order two, 6x' dy' dpi and its dual basis is (cte\ 6y\6pi), where (4 1)
s - - - - - TW- — + TV —
Ch.13. Cartan spaces of order 2
315
and Syl = dy* + Nljdx3,
(4.1)'
5Pi = dPl -
N^dx*.
Now, using an already very known method, we can prove: Theorem 13.4.1. 1) In a Cartan space of order two, C^n connection CY{N) = {H%jk,Cljk,C^k)
= (M,K) there exists a unique N-linear which verifies the following axioms:
1° N is the canonical nonlinear connection with coefficients (3.11). 2° CT(JV) is metrical with respect to fundamental tensor gli of space i.e. (4.2)
S y |* = 0, 0*|fc = O, ffy|* = 0
3° The d-tensor of torsion T'jk,
S'jk, Sijh vanish.
2) This connection has the coefficients: ^ (4-3)
-
S,gjk),
C V = \gis{dj9sk + dk9js -
dsgjk),
The N-connection CT(N), (4.3), is called the canonical metrical connection of the space C ( 2 ) n . Let us consider the covariant curvature tensors (6.5)', Ch.10 of CV(N). Then, applying the Ricci identities (see 9.7) to the covariant of fundamental tensor field, gtj and taking into account the equations (4.2)'
Sui* = o, glj\k = o, gij\k = o
we obtain: Theorem 13.4.2. The tensors of curvature Rijkh, Pijkh, Pijkh and Sijkh, Sijkh, Sijkh are skew-symmetric in the first two indices. The (y)-deflection tensors of CT(N) are given by (5.5)', ch.10, and (p)-deflection of the same connection are (4.4)
Ay = Nl3 -
PmH™,
Stj = -PrnC^,
% = &\.
The Geometry of Hamilton & Lagrange Spaces
316
Now, let us solve the problem of determination of a metrical N-linear connection for which the p-deflection tensor Ay vanishes.
Theorem 13.4.3. 1) In a Cartan space C1'2'" = (M, K) there exists a unique N-linear connection DV(N) = for which the following axioms hold: o
1° Ay = 0 and the coefficients N lj{x,y)
of the nonlinear connection
o
N(N lj, Nij) are given a priori. 2° DT(N) is metrical with respect to gij, i.e. (4.2) holds. 2) 1°. The coefficients (Hijk,Cijk,Ci>'t) of DT{N) are given by (4.3). 2°. The coefficients Ntj of the nonlinear connection N are expressed by:
IV•• — V
- -Bmn
-I- -F) K2-
I 'v0
N
s
1 4-
(4-5)
where "fljk(x, y,p) are the Christoffel symbols of gij(x, y,p) and the index "0" means the contraction by Pi or by p' = g'jPj. o
Proof. If the nonlinear connection N N 1j(x,y),Nij(x,y,p)) is known, then Theorem 13.4.1 can be applied and it follows the existence and uniqueness of the coefficients (//'j^C'jfcjCV*) from (4.3) which satisfy the axioms 2° and 3°. Let us determine the coefficients N^ of N in the condition of axiom 1°, i.e. Ay = 0. Taking into account (4.4), Atj = 0 is equivalent to Ntj = Hmijpm. Since the s
.
.
•
S
d
ZT
i
d
*r
d
.
operators -r- have the expressions 7— = -^—r- N ,—— + Nij-^—, the equations dxi dx3 ox] oyl opi l gij]h = 0, T jh = 0 leads to Hhij = 7 % + \ \{N ?gsj+ N fgis)dmghs+ (4.6)
2L
-^[{Nimgsj + N]mgis)dmghs
Nsrngshdmgij}.
+
Thus Ni:) = ph.Hhi:j allows to write:
(4.7)
Ni} = 7 ° v + l-[(N ?gsj+ N fgis)dmPs+
N ™gshdm
N
Ch.13. Cartan spaces of order 2
317
A new contraction by p> = gjsps leads to Noi = 7°o + \dmK2 N ?. Substituting in the formula (4.7) we have the solution (4.5) of the equation Ay = 0, and the theorem is proved. q.e.d. For the canonical metrical connection CT(N), (4.3) the Ricci identities are given in Theorem 10.6.1, taking into account the axiom 3°, i.e.: (4.8)
Tjk = 0, Sljk = 0, # * = 0.
The torsions are given by (6.3), (6.3)'' and the curvature tensors are expressed by (6.4) and (6.4)'. Theorem 10.6.3 gives some important identities for CF(N), in which we take into account (4.6) and (4.4).
13.5
Parallelism of vector fields. Structure equations of CT(N)
Letus consider the canonical metrical connection CT(N), (4.3), and let 7 : / —> T*2M be a smooth parameterized curve as in section 7, Ch.10. For a vector field X e X{T*2M) given in the adapted basis (5i,di,di) by (5.1)
X =X {Si + X{di + Xid1
the covariant differential DX is expressed by: (5.2)
DX = (rfi; 1 + X'u\)&i + (dX{ + XsLo\)di + (dX, -
Xsusi)d\
where the 1-forms of connection ula of the Cartan space of order two are: (5.3)
w'j = Hijkdxk + CljkSyk + C/*«p t .
Therefore, Theorem 10.7.1 gives the necessary and sufficient condition for the parallelism of vector field X, (5.1) with respect to CT(N) along the parametrized curve 7Theorem 10.7.2 states that the Cartan space C (2)n is with the absolute parallelism of vectors if, and only if, all d-tensors of curvature vanish: Rhijk = 0, phijk
= 0, / y / = 0,
Sh'jk = 0, S/,y = 0, sh^k = 0
The Geometry of Hamilton & Lagrange Spaces
318
We recall Theorem 10.7.3, also: Theorem 13.5.1. A smooth parametrized curve 7 : / —> T*2M is autoparallel with respect to the canonical metrical connection CY(N) of Cartan space C^n if and only if the system of differential equations (7.13), Ch.7, is verified. Of course, a theorem of existence and uniqueness for the autoparallel curve one can formulate without difficulties. Taking into account Theorem 10.3.3, and (7.15), Ch.10, we get: Theorem 13.5.2. The horizontal paths of the canonical metrical connection of the space C' 2 ' n are characterized by the system of differential equations: dV
. dxi dxk
Sy{
CY(N)
6Pi
Finally, Theorem 10.7.5 has as consequence:
Theorem 13.5.3.
1° The -vertical paths of CT(iV) in the point XQ € M is characterized by the system of differential equations
2° The
-vertical paths of
in the point x0 G M is characterized by
Remark. We assume that the restrictions of the coefficients C1^ to the zero section of?r' :T*2M —> M exist. The structure equations of the canonical metrical connection CT(N) of the Cartan space of order two, C ( 2 ) n = (M,K(x,y,p)) are given in the section 7 of Ch.7, taking into account the particular properties of this connection. So, we obtain: Theorem 13.5.4. The canonical metrical connection CT(N) of the Cartan space of order two C^n — , has the following structure equations: d{dxl) -dxm (5 4)
Aujlm =
(°). -Q',
Ch.13. Cartan spaces of order 2
319
and dJj - ujmj A m =
(5.5)
-Wj
(°). ( i ) .
where the 2-forms of torsion Q\ fi ! ,fij and 2-forms of curvature f2!j are given by the formulae (8.6), ch.7, and (4.8). The Bianchi identities of the connection CT(N) can be obtained taking the exterior differential of the system of equations (5.4), (5.5) modulo the same system. Remark. Using the structure equations, we can study, without major difficulties, the theory of Cartan subspaces of order two in a Cartan space C' 2 '".
13.6
Riemannian almost contact structure of a C(2)n space
Consider a Cartan space of order two, C^>n = (M, K(x,y,p)) and its canonical nonlinear connection N, with coefficients (N*j, iV,j) from (3.11). The adapted basis (5i,di,dl) and its dual (dxl, Syl, 5pi) are determined by N.
The associated GH^n = (M, gij(x, y,p)) space of C^n is uniquely determined. Therefore, using the theory from section 3, ch.11, we can investigate the notion of Riemannian almost contact structure of the space C (2)n . We introduce the N-lift to T*2M of the fundamental tensor Glj(x,y,p) by: (6.1)
G = gijdx* O dxj + g ^
which is defined in every point u* £ T*2M. Theorem 13.6.1. 1° G is a tensor field on the manifold T*2M, of type (0, 2), nonsingular depending only on the fundamental function K of C^n and by the nonlinear connection 2° The pair (T*2M, G) is a Riemannian space. 3° The distributions N
,
are respectively orthogonal with respect to G.
Proof. Since g^ and
The Geometry of Hamilton
320
& Lagrange Spaces
Theorem 13.6.2. The tensor G is covariant constant with respect to canonical metrical connection CV(N) of the space C^n, i.e. DG = 0.
In other words, the canonical metrical connection CT(iV) is an N-linear metrical connection with respect to the Riemannian structure G. Let us consider the
-linear mapping
F : X{EfM) —
~
defined by (6.2)
F(Si) = -gijdi, F(9i) = 0,
Theorem 13.6.3. The mapping F has the following properties: 1° IF is globally defined on T**M. 2° IF is a tensor field of type (1,1) on T*2M, i.e. (6.3)
F = -g^
® dx> + gijSj ® Sp{.
3° kerF = V^i, I m F = N®W2. 4° rank|F| = In.
The proof is exactly like the case of Theorem 9.8.2.
o
Hence, F is called the (p)-almost contact structure determined by gtj and by ./V . The Nijenhuis tens or .A/jp and the condition of normality of IF can be explicitely written in adapted basis (see ch.11). Now if we remark the tensor (6.3) and the fact that gtj and gtj are covariant constant with respect to the canonical metrical connection, it follows: Theorem 13.6.4. The tensor field F is covariant constant with respect to the canonical metrical connection CT(N), i.e.
Finally, let us notice that the pair (G, F ) has some remarkable properties.
Ch.13. Cartan spaces of order 2
321
Theorem 13.6.5. 1° The pair (G, F ) is a Riemannian almost contact structure on the manifold T*2M determined only by the space C^n and by the nonlinear connection 2° The tensor fields G and F are covariant constant with respect to canonical metrical connection CT{N) of the space C^>n. 3° The associated 2-form of the structure ( G , F ) is the canonical presymplectic structure on T*2M : 6 = dpi A dx%. Indeed, we have G(WX,Y) - -G(FY,X), VX,Y e X(T'2M) and 9{X,Y) = Q(WX, Y). In adapted basis we get 6 = 8pi A dx'. But the torsion TV, = N,j - Nji vanishes, hence as consequence 6 = dpt A dx\ and therefore Theorems 13.6.2 and 13.6.4 implies 2°, etc. _ The Riemannian almost contact space (T* 2 M, G , F ) is called the Riemannian almost contact model of the Cartan space of order 2, C ( 2 ) n . It is extremely useful in applications.
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Index Absolute Energy Adapted basis Almost complex structure contact structure Hermitian structure Kähalerian model product strucutre symplectic structure Autoparallel curves
Cartan metrical connection 146 nonlinear connection 144 spaces 139 special spaces 152 Christoffel symbols 39 Coefficients of a non-linear connection 127 of an N-linear connection 129 dual of a non-linear connection292
274 15 59,72
320 74 156 71 67 19
nonlinear 10,127 N-metrical 75,129 Levi-Civita 59 Berwald 50 Hashiguchi 49 Chern-Rund 49 remarkable 49 Coordinates 2 of a point transformation of 3 Covariant 22 h- and v- derivatives h— and 257 257 h- and 257 257 differential 113 Curvature of an N-linear connection 24,111
Basis loral adapted 15 Berwald spaces 50 Bianchi identities 26,48 Brackets 242 Lie brackets Poisson brackets 89 Bundle differentiable 2 k-osculator 219 of higher order accelerations 219 dual 220 Canonical spray dual semispray semispray k-semispray metrical connections nonlinear connections N-linear connections
40 231 67 391
d-vectors Liouville d-vectors d-tensors Deflection tensors Direct decomposition
130,148 10,127 20,75
336
28,251
298 107,251
22 99,274
Index
337
Distributions vertical horizontal Dual coefficients semispray of 231 Energy of superior order Euler-Lagrange equations
3
292
297 231 66 294 294
85 84
39
Hamilton spaces 119 Hamilton-Jacobi equations 125 Hamiltonian systems 125 Hamiltonian systems of higher• order 288
Hamiltonian space of electrodynamics 134 130 Hamilton vector field Hamilton 1-form 130 Higher order Lagrange spaces 290 182 locally Minkowski spaces Integral of action
294
92
11
f-related connections 210 Finsler metrics 32 spaces 32 152 special spaces Cartan duality 173 Fundamental 32 function 32 tensor of a Finsler space tensor of a higher order Lagrange 290 spaces General Relativity Generalized Lagrange spaces Geodesics
Jacobi-Ostrogradski momenta Jacobi method
65
Lagrangian regular of higher order Lagrange spaces Lagrange-Hamilton duality Landsberg spaces 94duality 94dual nonlinear connections dual N-linear connections Legendre mapping Lie derivative Lift homogeneous horizontal Sasaki-Matsumoto N—lift Liouville vector fields 1-form Locally Minkowski space Lorentz equations
64 290
63 159
51 159 163
168 296 5
57 14
56 123 5 93 182 85
Main invariants
294
N-linear connection Nonlinear connection
129 127
Poisson structure Presymplectic structure Projector horizontal vertical
92,288
Relativistic Optics Ricci identities
85 25,130
Sections in 227 Sections in 227 Structure equations
289
14 14
227 227
29,119
338 Spaces Berwald Douglas Finsler Hamilton Kropina Landsberg Randers Symplectic structures transformations Symplectomorphism Synge equations Torsions
Variational problem Whitney sum
The Geometry of Hamilton & Lagrange Spaces
50 50 32 124 181 51 177 89
189 208 293 23,110 293 11
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P.P.J.M. Schram: Kinetic Theory of Gases and Plasmas. 1991 ISBN 0-7923-1392-5 A. Micali, R. Boudet and J. Helmstetter (eds.): Clifford Algebras and their Applications in Mathematical Physics. 1992 ISBN 0-7923-1623-1 E. Prugovefiki: Quantum Geometry. A Framework for Quantum General Relativity. 1992 ISBN 0-7923-1640-1 M.H. Mac Gregor: The Enigmatic Electron. 1992 ISBN 0-7923-1982-6 C.R. Smith, G.J. Erickson and P.O. Neudorfer (eds.): Maximum Entropy and Bayesian Methods. Proceedings of the 11th International Workshop (Seattle, 1991). 1993 ISBN 0-7923-2031-X D.J. Hoekzema: The Quantum Labyrinth. 1993 ISBN 0-7923-2066-2 Z. Oziewicz, B. Jancewicz and A. Borowiec (eds.): Spinors, Twistors, Clifford Algebras and Quantum Deformations. Proceedings of the Second Max Born Symposium (Wroclaw, Poland, 1992). 1993 ISBN 0-7923-2251-7 A. Mohammad-Djafari and G. Demoment (eds.): Maximum Entropy and Bayesian Methods. Proceedings of the 12th International Workshop (Paris, France, 1992). 1993 ISBN 0-7923-2280-0 M. Riesz: CliffordNumbers and Spinors with Riesz' Private Lectures to E. Folke Bolinder and a Historical Review by Pertti Lounesto. E.F. Bolinder and P. Lounesto (eds.). 1993 ISBN 0-7923-2299-1 F. Brackx, R. Delanghe and H. Serras (eds.): Clifford Algebras and their Applications in Mathematical Physics. Proceedings of the Third Conference (Deinze, 1993) 1993 ISBN 0-7923-2347-5 J.R. Fanchi: ParametrizedRelativistic Quantum Theory. 1993 ISBN 0-7923-2376-9 A. Peres: Quantum Theory: Concepts and Methods. 1993 ISBN 0-7923-2549-4 P.L. Antonelli, R.S. Ingarden and M. Matsumoto: The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology. 1993 ISBN 0-7923-2577-X R. Miron and M. Anastasiei: The Geometry ofLagrange Spaces: Theory and Applications. 1994 ISBN 0-7923-2591-5 G. Adomian: Solving Frontier Problems of Physics: The Decomposition Method. 1994 ISBN 0-7923-2644-X B.S. Kerner and V.V. Osipov: Autosolitons. A New Approach to Problems of Self-Organization and Turbulence. 1994 ISBN 0-7923-2816-7 G.R. Heidbreder (ed.): Maximum Entropy and Bayesian Methods. Proceedings of the 13th International Workshop (Santa Barbara, USA, 1993) 1996 ISBN 0-7923-2851-5 J. Peina, Z. Hradil and B. JurCo: Quantum Optics and Fundamentals of Physics. 1994 ISBN 0-7923-3000-5 M. Evans and J.-P. Vigier: The Enigmatic Photon. Volume 1: The Field B ( 3 ) . 1994 ISBN 0-7923-3049-8 C.K. Raju: Time: Towards a Constistent Theory. 1994 ISBN 0-7923-3103-6 A.K.T. Assis: Weber's Electrodynamics. 1994 ISBN 0-7923-3137-0 Yu. L. Klimontovich: Statistical Theory of Open Systems. Volume 1: A Unified Approach to Kinetic Description of Processes in Active Systems. 1995 ISBN 0-7923-3199-0; Pb: ISBN 0-7923-3242-3 M. Evans and J.-P. Vigier: The Enigmatic Photon. Volume 2: Non-Abelian Electrodynamics. 1995 ISBN 0-7923-3288-1 G. Esposito: Complex General Relativity. 1995 ISBN 0-7923-3340-3
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J. Skilling and S. Sibisi (eds.): Maximum Entropy and Bayesian Methods. Proceedings of the Fourteenth International Workshop on Maximum Entropy and Bayesian Methods. 1996 ISBN 0-7923-3452-3 C. Garola and A. Rossi (eds.): The Foundations of Quantum Mechanics Historical Analysis and Open Questions. 1995 ISBN 0-7923-3480-9 A. Peres: Quantum Theory: Concepts and Methods. 1995 (see for hardback edition, Vol. 57) ISBN Pb 0-7923-3632-1 M. Ferrero and A. van derMerwe (eds.): Fundamental Problems in Quantum Physics. 1995 ISBN 0-7923-3670-4 F.E. Schroeck, Jr.: Quantum Mechanics on Phase Space. 1996 ISBN 0-7923-3794-8 L. de l a and A.M. Cetto: The Quantum Dice. An Introduction to Stochastic Electrodynamics. 1996 ISBN 0-7923-3818-9 P.L. Antonelli and R. Miron (eds.): Lagrange and Finsler Geometry. Applications to Physics and Biology. 1996 ISBN 0-7923-3873-1 M.W. Evans, J.-P. Vigier, S. Roy and S. Jeffers: The Enigmatic Photon. Volume 3: Theory and Practice of the fl(3) Field. 1996 ISBN 0-7923-4044-2 W.G.V. Rosser: Interpretation of Classical Electromagnetism. 1996 ISBN 0-7923-4187-2 K.M. Hanson and R.N. Silver (eds.): Maximum Entropy and Bayesian Methods. 1996 ISBN 0-7923-4311-5 S. Jeffers, S. Roy, J.-P. Vigier and G. Hunter (eds.): The Present Status of the Quantum Theory of Light. Proceedings of a Symposium in Honour of Jean-Pierre Vigier. 1997 ISBN 0-7923-4337-9 M. Ferrero and A. van der Merwe (eds.): New Developments on Fundamental Problems in Quantum Physics. 1997 ISBN 0-7923-4374-3 R. Miron: The Geometry of Higher-Order Lagrange Spaces. Applications to Mechanics and Physics. 1997 ISBN 0-7923-4393-X T. Hakioglu and A.S. Shumovsky (eds.): Quantum Optics and the Spectroscopy of Solids. Concepts and Advances. 1997 ISBN 0-7923-4414-6 A. Sitenko and V. Tartakovskii: Theory of Nucleus. Nuclear Structure and Nuclear Interaction. 1997 ISBN 0-7923-4423-5 G. Esposito, A.Yu. Kamenshchik and G. Pollifrone: Euclidean Quantum Gravity on Manifolds with Boundary. 1997 ISBN 0-7923-4472-3 R.S. Ingarden, A. Kossakowski and M. Ohya: Information Dynamics and Open Systems. Classical and Quantum Approach. 1997 ISBN 0-7923-4473-1 K. Nakamura: Quantum versus Chaos. Questions Emerging from Mesoscopic Cosmos. 1997 ISBN 0-7923-4557-6 B.R. Iyer and C.V. Vishveshwara (eds.): Geometry, Fields and Cosmology. Techniques and Applications. 1997 ISBN 0-7923-4725-0 G.A. Martynov: Classical Statistical Mechanics. 1997 ISBN 0-7923-4774-9 M.W. Evans, J.-P. Vigier, S. Roy and G. Hunter (eds.): The Enigmatic Photon. Volume 4: New Directions. 1998 ISBN 0-7923-4826-5 M. Rédei: Quantum Logic in Algebraic Approach. 1998 ISBN 0-7923-4903-2 S. Roy: Statistical Geometry and Applications to Microphysics and Cosmology. 1998 ISBN 0-7923-4907-5 B.C. Eu: Nonequilibrium Statistical Mechanics. Ensembled Method. 1998 ISBN 0-7923-4980-6
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V. Dietrich, K. Habetha and G. Jank (eds.): Clifford Algebras and Their Application in Mathematical Physics. Aachen 1996. 1998 ISBN 0-7923-5037-5 J.P. Blaizot, X. Campi and M. Ploszajczak (eds.): Nuclear Matter in Different Phases and Transitions. 1999 ISBN 0-7923-5660-8 V.P. Frolov and I.D. Novikov: Black Hole Physics. Basic Concepts and New Developments. 1998 ISBN 0-7923-5145-2; Pb 0-7923-5146 G. Hunter, S. Jeffers and J-P. Vigier (eds.): Causality and Locality in Modern Physics. 1998 ISBN 0-7923-5227-0 G.J. Erickson, J.T. Rychert and C.R. Smith (eds.): Maximum Entropy and Bayesian Methods. 1998 ISBN 0-7923-5047-2 D. Hestenes: New Foundations for Classical Mechanics (Second Edition). 1999 ISBN 0-7923-5302-1; Pb ISBN 0-7923-5514-8 B.R. Iyer andB. Bhawal (eds.): BlackHoles, Gravitational Radiation and the Universe. Essays in Honor of C. V. Vishveshwara. 1999 ISBN 0-7923-5308-0 P.L. Antonelli and T.J. Zastawniak: Fundamentals ofFinslerian Diffusion with Applications. 1998 ISBN 0-7923-5511-3 H. Atmanspacher, A. Amann and U. Müller-Herold: On Quanta, Mind and Matter Hans Primas in Context. 1999 ISBN 0-7923-5696-9 M.A. Trump and W.C. Schieve: Classical Relativistic Many-Body Dynamics. 1999 ISBN 0-7923-5737-X A.I. Maimistov and A.M. Basharov: Nonlinear Optical Waves. 1999 ISBN 0-7923-5752-3 W. von der Linden, V. Dose, R. Fischer and R. Preuss (eds.): Maximum Entropy and Bayesian Methods Garching, Germany 1998. 1999 ISBN 0-7923-5766-3 M.W. Evans: The Enigmatic Photon Volume 5: O(3) Electrodynamics. 1999 ISBN 0-7923-5792-2 G . N . Afanasiev: Topological Effects in Quantum Mecvhanics. 1999 ISBN 0-7923-5800-7 V. Devanathan: Angular Momentum Techniques in Quantum Mechanics. 1999 ISBN 0-7923-5866-X P.L. Antonelli (ed.): Finslerian Geometries A Meeting of Minds. 1999 ISBN 0-7923-6115-6 M.B. Mensky: Quantum Measurements and Decoherence Models and Phenomenology. 2000 ISBN 0-7923-6227-6 B. Coecke, D. Moore and A. Wilce (eds.): Current Research in Operation Quantum Logic. Algebras, Categories, Languages. 2000 ISBN 0-7923-6258-6 G. Jumarie: Maximum Entropy, Information Without Probability and Complex Fractals. Classical and Quantum Approach. 2000 ISBN 0-7923-6330-2 B. Fain: Irreversibilities in Quantum Mechanics. 2000 ISBN 0-7923-6581-X T. Borne, G. Lochak and H. Stumpf: Nonperturbative Quantum Field Theory and the Structure of Matter. 2001 ISBN 0-7923-6803-7 J. Keller: Theory of the Electron. A Theory of Matter from START. 2001 ISBN 0-7923-6819-3 M. Rivas: Kinematical Theory of Spinning Particles. Classical and Quantum Mechanical Formalism of Elementary Particles. 2001 ISBN 0-7923-6824-X A.A. Ungar: Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession. The Theory of Gyrogroups and Gyrovector Spaces. 2001 ISBN 0-7923-6909-2 R. Miron, D. Hrimiuc, H. Shimada and S.V. Sabau: The Geometry of Hamilton and Lagrange Spaces. 2001 ISBN 0-7923-6926-2
Fundamental Theories of Physics 119. M. PavSifi: The Landscape of Theoretical Physics: A Global View. From Point Particles to the Brane World and Beyond in Search of a Unifying Principle. 2001 ISBN 0-7923-7006-6
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