The Geometry of Hamilton and Lagrange Spaces

<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:

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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.

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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 {x) the following 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:

The Geometry of Hamilton & Lagrange Spaces

54

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- /.

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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
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
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° {/,
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
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).

{/,
(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+

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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.

The Geometry of Hamilton & Lagrange Spaces

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 Sp,.

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)


-\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.

The Geometry of Hamilton & Lagrange Spaces

144

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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146

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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148

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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150

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

The Geometry of Hamilton & Lagrange Spaces

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)

U

(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 , (/
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162

(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*.

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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 6y*> then K* = {KiftYg^Si ® dc ® d ^ ® <5*pd. Remark. We have A""/), = {g^gbdK'fc)* = g*acgtbd{Kl / : c )*. Therefore the components of the -dual of K in (x,p) are obtained from the components Kxjab of K in (x , p°a = daL(x, y), unchanging the horizontal part and raising and lowering of indices for vertical part by using gab.

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\

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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 dxi - da ® Sya on U is P* = 6* ® dxl - 9" ® 8pa. Let V be a linear connection on U and V* its -dual on U*, given by (1.18). Definition 7.3.1. A linearconnection V (V*) on TM (T*M) is called d-connection if V P = 0 (V*P* = 0). Proposition 7.3.1. V w a d-connection if and only if

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)

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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.

The Geometry of Hamilton & Lagrange Spaces

180

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).

Ch.7. The duality between Lagrange & Hamilton spaces

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.

Ch. 7. The duality between Lagrange & Hamilton spaces

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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186

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{-x\ (j>{x) = fax1 + ... + <j>nxn (fa = const.), we get the ecological metric of Antonelli ([11], [13], [152]):

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 (/ - ) is supplementary to VT*M. HT*M = Ker (/-<£) will be called the horizontal bundle and WT'M = 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 -). Conversely, if a connection F is given on T*M, we can get a connection-pair on T*M by taking a complementary subbundle of Ker (/ - F). Let (j> be a connection-pair on T*M and F the associated connection. We will denote by h and v the projections induced by F :

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 Sxj,

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 on T*M is called symmetric if Fy = Fj, and ji

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 *6 = - 6 .

(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 on T*M induces two almost symplectic forms, globally defined on T*M : (1.24)

= 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 6ph

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 -connection if (2.3)

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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198

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 -connection such that the following properties are verified: (i) 9n\k - 0 , (iii) T)k

=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


Sp,.

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) = f*f*1 is a connection-pair (ii) f is F - regular. Proof. (j> is clearly an almost product structure on T*M and it is a connection-pair iff Ker (7 — ) is transversal to VT*M, or equivalently, nt : Ker (I - 4>) -> TA/ is an isomorphism. But this last condition is equivalent to (ii) because / . ( K e r ( J - 0 ) ) = Ker ( / - ? ) . Definition 8.4.2. The connection-pair ^ given by (i) above is called the pushforward of by f. The connection F associated to will also be called the pushforward of byf. Also we will say that