The geometry of Ingarden spaces

Vol. 54 (2004)

REPORTS ON MATHEMATICAL PHYSICS

No. 2

THE GEOMETRY OF INGARDEN SPACES RADU MIRON Department of Mathematics, "A1. I. Cuza" University, 6600 Iasi, Romania (e-mail: [email protected]) (Received February 5, 2003)

The Randers spaces R F n were introduced by R. S. Ingarden [8]. They are considered as Finsler spaces F n (M, ~ q - t ) equipped with the Caftan nonlinear connection. In the present paper we define and study what we call the Ingarden spaces, 1F n, as Finsler spaces F n = (M, c~ + t ) equipped with the Lorentz nonlinear connection. The spaces RF n and I F n are completely different. For I F n we discuss: the variational problem, Lorentz nonlinear connection, canonical N-metrical connection and its structure equations, the Cartan 1-form o9, the electromagnetic 2-form Dr and the almost symplectic 2-form 0. The formula dw = - ~ + 0 is established. It has as a consequence the generalized Maxwell equations. Finally, the almost Hermitian model of I F n is constructed. :

Keywords: Randers space, Ingarden space, Lagrange geometry, Finsler geometry.

Introduction An important class of Finsler spaces suggested by the theories of gravitation and electromagnetism is formed by Randers spaces [5, 10]. They have the fundamental function F ( x , y ) = or(x, y ) + f l ( x , y), with ~ = ~f a i j ( x ) y i y j and fl = b i ( x ) y i, a i j ( x ) being a Riemannian metric and b i ( x ) a covector field on the manifold M. These spaces have been discovered by R. S. Ingarden in his Ph.D. thesis. He wrote [8, p. 26]: "The special kind of Finsler's space with the metric d s = v / a i j ( x ) d x i d x j + ai ( x ) d x i

we shall call a Randers space, since Randers (1941) seems to be the first to consider this kind of spaces, but as affinelly c o n n e c t e d R i e m a n n s p a c e s which is rather confusing notion". Actually, the Randers spaces are considered as Finsler spaces F n = (M, o~ + t ) equipped with the Cartan nonlinear connection ]V [2-7, 10]. They are denoted by c

R E n = ( M , ot + fl, N ) .

The importance of the space R F n in geometry and in theoretical physics is very well underlined in the books [2, 4, 10] and in the papers [8, 11, 19]. [131]

132

R. MIRON

The variational problem of the functional I ( c ) = f ( a + ~ ) d t , in the natural parametrization of the curve c with respect to the Riemannian structure oe, leads to the Lorentz nonlinear connection N, which is determined only by the fundamental function F -- o~ +/3 [13]. The triple I F ~ = (M, a + f i , N) we shall call the Ingarden space. The name comes from Ph.D. thesis of Ingarden where the electromagnetic and other phenomena have been studied. And, of course, because of his great merit in the introduction of the notion of Randers spaces. The geometries of spaces R F ~ and I F n are completely different. First of all, some fundamental geometrical fields of the space I F ~ do not have a Finslerian meaning, since they are not homogeneous with respect to the variables yi. This is the case of the Lorentz nonlinear connection N. Also, the space R F n has a natural symplectic structure, while I F n has a nonintegrable symplectic structure. These differences show that Ingarden spaces do not form a new class of Finsler spaces. But it is an important class of Lagrange spaces [1, 14, 16]. For the Ingarden spaces we study: the variational problem which allows to introduce the Lorentz nonlinear connection N, canonical N-metrical connection and its structure equations, curvatures and torsions, the Cartan 1-form co, the electromagnetic 2-forms 5c and the almost symplectic structure 0; the formula do) = - 5 c + 0 is established, which shows that 0 is nonintegrable and that it allows to determine the generalized Maxwell equations. The spaces I F ~ can be lifted to the tangent manifold T M in an almost Hermitian space H 2". The notion of Ingarden space was presented by the author during the "Workshop on Finsler Geometry and its Applications", Debrecen, 11-15 August, 2003. In a forthcoming paper we will study Ingarden spaces endowed with the homogeneous Lorentz nonlinear connection.

1.

The f u n d a m e n t a l tensor of Finsler space F '~ = (M, a + t )

We define the Randers and Ingarden spaces starting with the Finsler fundamental function F ( x , y) = or(x, y ) + fi(x, y). If we equip the Finsler space F n = ( M , o: q- t ) c

with canonical Cartan nonlinear connection N, we obtain the space R F n = ( M , ol + c

t , N), called the Randers space. The space F n endowed with the Lorentz nonlinear connection N will be called Ingarden space and will be denoted by I F n = ( M , ot +

/~, N). In this respect we consider the fundamental tensor field gij of F n. It was thoroughly studied by many authors. Especially, in [2-5, 7, 10] one finds a good analysis of this notion. Let M be a real, n-dimensional C~-manifold, ( T M , Jr, M ) be its tangent bundle, (x i) the local coordinates of the points x ~ M and (x i, yi) the local coordinates of the points u = (x, y ) c T M . The indices i, j . . . . run over the set {1, 2 . . . . . n} and the Einstein convention of summation is adopted.

133

THE GEOMETRY OF INGARDEN SPACES

On M we consider a Riemannian metric ~" = a i j ( x ) d x i @ d x j and 1-form b = b i ( x ) d x i. These two geometric fields determine a function F " T M > I~:

F(x, y) = or(x, y) + fi(x, y), where

(1.1)

i

a(x, y) = v / a i j ( x ) y i y j,

fl(X, y) = b i ( x ) y i.

(1.2)

The function F is differentiable on the manifold T M = T M \ {0} and is continuous on the null section of the projection re • T M ,> M. One proves [5] that if

b = v/aij(x)bi(x)bJ(x)

< 1,

b i =- aiJbj,

(1.3)

then F is a fundamental Finsler function. So, F n = (M, a + t ) is a Finsler space. The fundamental tensor field gij of the space F n, 1 02F 2 gij -- 2 0 y i O y j

(1.4)

on T M is given by [5, 13] O

giy = p ( a i j where

F ot

P -

--

a +fi - - , ol

li

-

(1.5)

ilj) + lilj,

OF - Oy i

--

li

~-bi,

li----

Ool Oy i"

(1.6)

It follows that

det(gij) = pn+l det(aij).

(1.7)

The contravariant tensor gU of gij has the following form 1 i' 1 [fi+oeb 2 . . . . ] gij = --a y + lil j - ( l i bJ--[- l j b i) p -~ Fz 0

,,

(1.8)

0

where l ' = a 'J lj.

2.

Randers space

Let us consider a Finsler function F = o ~ + f l . If c : t e [ 0 , 1] > (xi(t))eUCM is a smooth parametrized curve with the property Im c C U, U being a domain of a local chart of the manifold M, then we can consider the integral of action of the Lagrangian L(x, y ) = F2(x, y) defined by 1

dx

134

R. MIRON

Thus, the Euler-Lagrange equations are

0[Of + fl]2

d 0[0~ "q- fl]2

Ox i

dt

3y i

--0,

yi _

dx i

(2.2)

dt

These equations can be written in the form dZx i

_ _

dx i

dx j dx k ---0,

i

dt z + Pjk(x' Y) dt

yi _

dt

(2.3)

dr'

where I'jk(X i , y) are the Christoffel symbols of the fundamental tensor gij.

REMARKS 1. F}k(x, y) are not the coefficients of a d-connection on T M . 2. In the canonical parametrization of the curve c, (2.3) are the equations of geodesics of the Finsler space F n. c

It is a well-known fact that "the Cartan nonlinear connection" N has the coefficients c i 1 3 (i.i h k) (2.4) N s--20yJ hkY Y • c

The coefficients of ~r are calculated in [5]. The expressions for Nj are rather complicated. DEFINITION 2.1. The Finsler space F n = (M, oe + fi) equipped with a Cartan c

nonlinear connection N is called a Randers space. It is denoted by R F n = (M, ot + c

fi, N). The existence o f the nonlinear connection N allows one to split the vector tangent space Tu(TM) in the form Tu(TM) = Nu ~3V,,

Vu e T M ,

(2.5)

where V is the vertical distribution on the tangent bundle ( T M , re, M). Therefore a local adapted basis of the splitting (2.5) is determined by 3

3 )

~X i ' Oy i

(i = 1, 1.1

n), ....

where ~X i

u= O-~?u-

c (x,y)a-~d ' Nj oy u

One proves [13, 14] the following result.

Jr(u)

= x.

THE GEOMETRYOF INGARDENSPACES c

THEOREM 2.1. C

135 c

1) There exists a unique N-metrical connection C F ( N )

=

c

(Fj~, Cjk) of the Randers space RF n which satisfies the following axioms: cH Vk gij : 0 ,

cV Vk gij : O,

c

C

T~jk : O,

si jk : O.

c

2) The connection CI'(N) has the coefficients expressed by the generalized

Christoffel symbols: ~1 is(~gjs ~g~k Fjk-- ~ g \ ~x k + ~X j

CSgjk~, -~-~ /]

(2.6)

cj - g kW;y + - OyJ -

O'-y; ]"

REMARKS c

1. The N-linear connection is the famous Cartan metrical connection of the space F n. c

2. Calculation of the coefficients Fjik is extremely complicated. But Berwald's formula for the spray curvature provides an efficient way to access the essence of this geometry. This is the opinion of the known geometer David Bao expressed during the "Workshop of Finsler Geometry and Applications", Debrecen, July 11-15, 2003. c c 3. Clearly, these two connections N and CF(N) are basic in the geometry of Randers spaces. 3.

I n g a r d e n spaces

For the Finsler spaces F n = (M, o t + ~ ) instead of the Cartan canonical nonlinear C

connection N, there are many other nonlinear connections determined only by the fundamental function F = ot ÷ j6. An important one was introduced by the author of this paper in [13] and was called the Lorentz nonlinear connection since it was determined by the Lorentz equation of the space F n. It is an important notion allowing to define the concept of Ingarden space. Consider an arc curve c • t E [0, 1] > (xi(t)) E U C M with the length

l(c) = f01 [oe(x, 2) +/~(x, 2)]dt.

(3.1)

136

R. MIRON

The variational problem for the functional l(c) leads to the Euler-Lagrange equations:

d O(Ot+ fl) -- O, dt Oyi

0(~ -b fl) Ox i

yi =

dx i = k i, dt

(3.2)

which can be written in the form

0¢~

d (Oot)

Ox i

dt

(Obi

~y i

Ohj \

~x J

.

yi

dxi

~x i ) y : '

or in the equivalent form:

'F 2ol L axi

"(o:/1

1 d°~-i O°t2 2 dt Oy i

dt k OYi .]J

( Obi

Objlj

yi

~,OxJ

-~xiJ y '

dxi = dt

(3.2a)

The function cr • t ~ [0, 1] . > a(t) 6 R defined by

a(t)=

fo'( "xl r

(3.3)

a X, d r I

is invertible on the interval [0, 1]. It allows us to take o- as a parameter along the curve c. We obtain o~ x,

= 1.

(3.4)

In this parametrization of the curve c, Eqs. (3.2a) become 0oe2

OXi

d (OOt2t = 2Fu(x)yJ

d a \ Oy' I

where

Fij --

Obi OxJ

Obj Ox i'

yi

'

dx i -- d a '

F i j ( x ) = ai~ F~j,

(3.5)

(3.6)

is the electromagnetic tensor field of the space F n. O

Denoting by yijk (X) the Christoffel symbols of the Riemannian structure ~ = aij(x)dxi @ dx j, the Euler-Lagrange equations (3.5) become

d2x i

o dx j dx k _ F i dxJ

-d-a- T + YJik d a d a

J-d~"

(3.7)

This is the Lorentz equation known in electrod_d~namics, see [13, 14]. Its solution are the integral curves of the semispray S on T M [14],

S = YiO-~Oxi

-

-

2Gi(x' Y)O-~

(3.8)

137

THE GEOMETRY OF INGARDEN SPACES

with the coefficients O

2Gi (x, y) = gi.ik (x)YJY k - F i j (x)Y j"

(3.8a)

Of course, S depends only on Finsler fundamental metric a + 19. Evidently, it is not homogeneous with respect to yi. We note in passing that Eqs. (3.4), (3,5) and (3.7) have been previously treated in [5], albeit in the context of projective geometry. This semispray determines a remarkable nonlinear connection N with the coefficients: O 1 0G i Ni __ yk_ Fi LF i J -- OyJ -- Yjk F j, j (x) = 2 j ( x ) . (3.9) N is called the Lorentz nonlinear connection of the Finsler space F n = (M, ot +/3). Some properties of the Lorentz nonlinear connection are as follows: 1. N is not homogeneous with respect to yi, since its coefficients N i j (3.9) are not homogeneous functions. 2. N gives rise to other splitting of the tangent bundle T ( T M ) : T , ( T M ) = Nu @ Vu,

(3.10)

Yu c T M .

0 ]\/ ( i = 1 . . . . . n) to (3.10)is given by An adapted basis /SkX[3_i' Oyi

- - -~x i Ox i

(3.11)

N j. - - -t- FJi z Oy j (~xi Oyj,

with 0

~X i

-

-

a OX i

Nj

o

a i Oy j '

0

(3.11a)

NJi = yJik y~.

Therefore: 3. The weak torsion t i j k - -

ONi J

ONi k

Oy k

OyJ

4. The curvature tensor R i jk

=

¢SNi k of N is ¢~xj

3Ni J ¢~xk

0

of N vanishes.

o

0

Ri jk = yS Rs ijk _ ( V # F i j

0

_ V#Fi

0

(3.12)

),

o

o

where R s ijk (x) is the curvature tensor of the Riemann structure fi" and V~ is the O.

covariant derivation with respect to the connection C 1~ (N) = (Y]k, 0). Consequently we have the following result.

138

R. MIRON

THEOREM 3.1. The Lorentz nonlinear connection N is integrable if and only if 0

0

the Riemann space 7U~= (M,~d) is locally fiat and V ~ F i j =

0.

Now we can give the following definition. DEFINITION 3.1. The Finsler space F ~ = (M, o~ + fi) equipped with a Lorentz nonlinear connection N is called an Ingarden space. It is denoted by I F ~ = (M, a + fi, N). An important result concerning I F ~ is expressed in the following theorem. THEOREM 3.2. For any Ingarden space I F n the Berwald connection BF(N) has 0

the coefficients (Yjk, 0). N i j , ONij Oy k , 0

Proof: Indeed, BF(N) is

( N i j Yjk (x), O).

[]

Of course, there is a big difference between the spaces R F ~ and I F " because: C

1. Nonlinear connections N and N are completely different. 2. Geodesics of Ingarden spaces are given by the Lorentz equations (3.7) while the geodesics of Randers spaces are given by

c: ( dx ~ dxa ds 2 tN* --j x , d s } ds = O. d2x i

3. As we will show the £r_ and N-metrical connections of these two spaces are different, too. 4. The geometrical model on T M of a Randers space is an almost KO_hlerian space. This_model of an Ingarden space is more general. It is an almost Hermitian space on TM. 5. On T M , R F ~ determines a symplectic structure, while I F n determines a nonintegrable almost symplectic structure.

4.

The N-metrical connection of the space I F n

Let us consider an Ingarden space I F n = (M, ot + fi, N). Its fundamental tensor gij is given by (1.5) and its contravariant gij has the form (1.8). We determine a canonical N-linear connection D P ( N ) = (Fj~, CjK), metric with respect to gij. If we denote by V n and V v the h- and v-covariant derivative with respect to DF(N): V H gij _ ~gij

6x k

V V gij _

Ogij

Oy ~

(4.1) CiSkgsj -- C}k s gsi ,

THE GEOMETRY OF INGARDEN SPACES

139

then the tensor gij is covariant constant with respect to D F ( N ) if and only if VHgij O, VVgij 0. In this case DY(N) is said to be a metrical N-linear connection of the space II "~. The h- and v-torsion tensors of D r ( N ) are =

=

Ti jk ~-- Fi jk -- Fikj ,

si jk = Ci jk -- Ci kj"

(4.2)

The following important results hold [14]. THEOREM 4.1. For any Ingarden space I F n there exists a unique d-connection I F ( N ) = (Fjik, Cjk) having the following properties: 1) I F ( N ) is a metrical nonlinear connection: VHgij = 0 ,

VVgij = 0 ,

(4.3)

si jk = 0.

(4.4)

2) I Y ( N ) is h- and v-torsion free:

Tijk = o,

THEOREM 4.2. The connection IF(N) with the properties (4.3) and (4.4) depends only on the fundamental function F = ot + ft. Its coefficients are given by the following generalized Christoffel symbols:

Fj k = ~ giS 1~,~X k q- ~

i

((~gjs

~gjk

~gjk~, ~ /I

Ogjk Cjk = ~g ~,~y~ + - -

agjk)

(4.5) 1 is (agJs

OyJ

Oy s

The connection I F ( N ) depending only on the fundamental function F = ot + fl will be called a canonical metrical connection of I F n. The expressions for the coefficients of the connection I F ( N ) are given in the following theorem. THEOREM 4.3. The canonical metrical connection II TM has the coefficients:

gjih =

Yj~ +~g

grh-- V~ gjh

gjr+ 0

0

(4.6)

+ gir( F~h Csjr+ FSj Csrh-- ~°'sr C~jh), Cjh = 1 ir o o o ~--dg [(Otbh -- fl lh)kjr + (olbj -- fl lj)krh -- (otbr - fl lr)kjh], where O O

Cijh -- 2] 0Ogij y h

'

kij = a i j - lilj

are the Cartan tensor and the angular tensor of the space F n.

(4.7)

140

R. MmON

Proof: We have .

.

.

Ogij

.

8gi 3X___j k _y= V ~ gij 4-ghj Y/; q-gih yjh 4- Fhk Oy h Substituting it into (4.5) we obtain Fj~ from (4.6). Remarking that Cjk = gisCsj k and the Cartan tensor Cijk is given by (4.7), by a straightforward calculation we obtain the second equality (4.6). [] Now, using the main geometrical objects: the fundamental function F = ot + fi, the Lorentz nonlinear connection and the canonical metrical N-linear connection I F ( N ) we can construct the geometry of Ingarden space I F n [14]. We will give some elements of the geometrical theory of these spaces, for instance, the structure equations of II'(N). The connection 1-forms of I F ( N ) are:

~°i i = FJi J x k + CJfi yk' where (dx i,

3yi),

i = 1 .....

n,

(4.8)

is the dual basis of the adapted basis

i' ~ i

•

So, we have

3y i = dx i + N i jdx j. The exterior differentials of the 1-forms 1

d(6y i) = - ~ R

i

(4.9)

3y i are

" jkdX J A dx k -

O__O_Ni

Oy k

jdx j A 8y k ,

(4.10)

where R i jk is given in (3.12) and, from (3.9), ~k N i j --

o

ONij

(4.10a)

_ ~/i jk"

Oy~

Using a known result [14] we obtain the following. THEOREM 4.4. The structure equations of the canonical connection I F ( N ) are given by d ( d x i) - dx k A (z)i k = _~-~i, (4.11) d ( 3 y i) - 6y k A o) i

_~i

k

~

do) i j - cokj A coi k -'=

_S2i

j,

(4.12)

where the 2-forms of torsion f2 i and ~2i are f2 i = CjkdXJ A 8yk,

f2 ~i = 1-Ri jkdX j A dx k + pijkdXJ A 3y k, 2

(4.13)

141

THE G E O M E T R Y OF I N G A R D E N SPACES

while the 2-forms of curvature are

~--~i j = ~1R j lkhdxk . . 1 i 3yk A ~yh. A dx h + Pj lkhdxk A 6y h q - ~ S j kh

(4.14)

Here the d-tensors of torsion and curvature have the following forms: o

p i j k = yijk - F j ki

(4.15)

and

Rh i

~ F~j ~X k

¢~Fhk s i ~xJ + FhjFsk --

_ OF~j Oy k

Vj Chk -[- CjsP~k ,

jk-

Ph i jk

H

Sh ijk = aC j

i

i

s

- OyJ -.-J-

Oy k

F s Fsji

i s + Chs Fjk,

s

i

(4.16) s

i

ChjCsk -- ChkCsj.

Taking into account the expressions for the coefficients Fjik and C)k from (4.16) the d-curvature tensors are extremely difficult to calculate. In the next section, in order to obtain the explicit expression for these tensors, we will use the method developed in [13]. The curvatures and torsions of the canonical connection I F ( N )

5.

The metrical N-linear connection I F ( N ) has the coefficients (4.6), where Fjik are the h-coefficients and Cjk the v-coefficients of I F ( N ) . But Fjk can be written in the form O

Fj/k =

yi

(5.1)

i

where Bj~ is the following d-tensor field,

Bj~

=

1 ir~_°H o o o o o ~g tv~ gjr+ V ff grk-- V ~ gjk) + gir(Fsk Csjr+ FSj Csrk- FSr Csjk).

(5.2)

Clearly, we deduce that

Bjk = Bikj

(5.2a)

and 1

0

o

o

0

i Y j Bjki = ~2 o6i r ,.r, J (~v Hk gJr -JV v f t grk-- V ff gjk) -I- yJ F s j Csk. Consider transformations of the linear connection on T M [13, 14] o

o

C 1~ (N, Y, C)

> C['(/V, F, C)

> I V ( N , F, C)

(5.2b)

142

R. MIRON

analytically expressed by o

o

_

-Nj = Uj - Fj,

.

o.

Elk

=

~

o

jk

o

o

i s 4- C},F;,

-i C]k

=

i C}k,

1

(5.3)

N] : with

o

i Oaij _ O.

Cijk: 2 Oyk

_Denoting by ~H, @v the h- and v-covariant derivation with respect to CF(N, F, C) and applying them to a tensor field T we obtain o

o

o

o

Therefore, the h- and v-covariant derivations with respect to I F ( N , F, C), V H and V V, are as follows: s (7~Tij + T s j B si k - T i sB~k, i _ Ti sWk" t'.s V V T i j = V V T i j 4- T sjCst:

V~Tij

:

According to these formulae we can determine the relations between the curvatures o

of d-connections C F, Cf' and IF. So, we have o

o

•

o

i

o

s

o

i

o

s

~,

o

h

o

i~'r it?s

[~j ~kh = Rj ikh 4- Pj ks F; -- Pj hsF; 4- ~'j ,-,*h*k, o

o

o

(5.4)

['j '~kh= Pj ~kh + Sj ik,F;" , o ~---

kh

and

(5.5)

e j ikh = Pj Zkh 4- ~j ikh,

where . . . i r ~)jzkh = (ThH Bijk --. VH Bijh. 4- BrjkBZrh -- BrjhB*rk + B~jrTrkh 4- CjrRkh,

•

e,r

~ j ~ = (~:B~j~ - V~C'j~ + BO~C% - C%8~r~ + B'j~ %

~j%, = 9V C~j~ - 9 [ cijh + CO, C% - crjhc%,

i -r +C~P ~,

(5.6)

143

THE GEOMETRY OF INGARDEN SPACES

and where T and /b denote O

O. o

0 0 ~ Fjs = O, -- Cks

~ijk = C}st~

o

~ j ~ = ej~ -

0fj

i

+ c~

=0.

O

Taking into account the fact that the tensor C}k vanishes, we deduce o.

o

i

Pj ;kh= O,

(5.7)

Sj kh =0"

Looking at Eqs. (5.4) we obtain: O

Rj'~h = Rj; kh '

[gj tkh = 0,

~;j ~

o.

Finally, we obtain ~j'kh, ~j;kh and o'j;kh i r ~)jZkh ~-" v H nijk -- v~I nijh + nrjknirh -- nrjhnirk "4- CjrRkh,

(5.8)

~jikh = VV Bijk -- v H fijh -'l- BrjkCirh -- CrjhBirk, ~jikh = v V f i j k -- v V fijh -~- Crjkfirh -- Crjhfirk ' where CF has the coefficients O

N- ij = N j ,

i

~ = ×;~, ej~=o

(5.9)

Substituting into (5.5) we get O

Rj tkh = Rj tkh +t)j ~kh,

(5.10) Sj Ikh = ~j tkh.

These are the formulae for the curvature tensors of the canonical connection I F ( N ) of Ingarden spaces. REMARKS

1. The covariant derivation ~ n

is given by means of (5.9) in the form

o f Bijk o

0

0

VkH Bljh -- ~Bijh . Fs . 8X k + BSjh )/sk --Blsh )"jk - B l j s hk •

_

_

/

"

Also, we have -V i OOJ h V~ B)h -- OY k .

S

'

144

R. MIRON

2. In the case when a = a i j ( x ) d x i @ dx j is semidefinite we can study, using the previous formulae, the Einstein equations [13] of the space 1F n. 3. Also, we can define the scalar curvature of I F ~ and it is possible that a Yasuda-Shimada theorem holds. The original version [19] of this theorem, for RF", is not correct. The corrected version is given in [6] and [12].

6.

The interior electromagnetic tensor of the Ingarden space c

In a Randers space, the h-deflection tensor D'j of the metrical canonical conc

nection C F ( N ) vanishes. Therefore, in this space does not exist the interior electromagnetic tensor field [17]. Contrary, in Ingarden space the h-deflection tensor D i of the canonical metrical J connection I F ( N ) does not vanish. It gives rise to an interior electromagnetic tensor o

which is not identical with to the exterior electromagnetic tensor Fij (x) provided by the 1-form fl = bi(x)dx i. The h- and v-deflection tensors D'j and d'j are given by

Dij = Vff y i,

dij = V V y i.

(6.1)

They have the forms o

d'j =

D*j = F'j + y* B'sj,

~'j.

(6.2)

But the covariant h- and v-deflection tensors

Dij = gisDS],

dij = gisdSj

(6.2a)

are o

o

o

1 ,. (Vr~ gij+ v H gir-- V/jgjr)_t_y ~ FSr Csij, Dij = gis FSj +-~Y o

(6.3)

dij -----gij. As in the Lagrange spaces [14, 17] we can give the following definition. DEFINITION 6.1. The h- and v-interior electromagnetic tensors f'ij and f/j of the space I F n are respectively: 1

1

~.j = -~(Dij - Dji),

J~j = ~(dij - dji).

(6.4)

According to the expressions (6.3) for Dij and d/j we get o

= (gis -% fij =0.

o

+

l

_ _o

yr(V

o

gir-

gjr),

(6.5)

THE GEOMETRY OF INGARDEN SPACES

145

PROPOSITION 6.1. The interior electromagnetic tensor field .~j depends only on the fundamental function F = ot + ~ on the space I F n. Let us denote Rijk = gisRSjk • Thus the main result o n .~/j is given by the next theorem. THEOREM 6.1. The interior electromagnetic tensor field ~ij of the Ingarden space I F n satisfies the following generalized Maxwell equations: VkH f'ij q- V f f ~jk -]- v H f'ki = --(Rijk -'[- Rjki q- Rkij),

v[ /j +

+ vyfk; =0.

(6.6)

Proof: Applying the Ricci identities to the Liouville vector field yi and taking into account the Bianchi identities, we obtain (6.6). [] There is other way to prove (6.6) by using the 2-form ov = f i j d x i A dx j ,

(6.7)

the Cartan I-form co co . . . .

1

OF 2

20y

d x i,

(6.8)

i

and the Poincar6 2-form 0 [14], 0 = gijSY i A dx j.

(6.9)

Evidently, the forms )c, co and ® have geometrical meaning. THEOREM 6.2. The forms ~ co and ® have the following important properties: (i) ~, co, and ® depend only on the fundamental function F = ot + ft. (ii) do) = - ~ + ®. (6.10) (iii) The 2-form - ~ + ® is closed, i.e. d® = dU.

(6.11)

Proof: (i) is evident because ~, co and ® do not depend on local transformations of coordinates on T M . (ii) The exterior differentials d of the 1-form co and F 2 = gijyiy j lead to (6.10). (iii) The exterior differential of do~ gives (6.10). [] Now (6.11) implies, by exterior differentiation, dgij A (~yi A d x j 'l- gijd(~y i) = d~ij A d x i A d x j .

146

R. MIRON

Remarking that dgij -- ~gij ~x k d x k + Ogij(~ OYk y k , the previous equality allows one to obtain the generalized Maxwell equations (6.6). In the case when the Lorentz nonlinear connection N is integrable, then Rijk = 0 and Eqs. (6.6) have the form =0,

E V~A-/j = 0, (ijk)

(ijk)

(6.12)

where Z is a cyclic sum. (ijk)

7.

The almost Hermitian structure on T M generated by Ingarden structure

The 2-form ® introduced in (6.9) plays an important role in applications of Ingarden spaces, since on the phase space T M there is an almost symplectic structure. THEOREM 7.1. (i) The Poincard 2-form ® determines on T M an almost symplectic structure. (ii) ® is a symplectic structure if and only if the electromagnetic 2-form f is closed. The proof is based on the form of ® gijc3yi A d x j and on the fact that d ® = d5c. The Lorentz nonlinear connection N allows us to define an almost complex structure on T M by: =

F

8

0 ay"

?-fi

-

. . . (i. =. .1

n).

(7.1)

Clearly F has a geometrical meaning on T M and F o F = - I . Since the weak torsion of N vanishes another result follows [14]. THEOREM 7.2. The almost complex structure F is integrable if and only if the Lorentz nonlinear connection N is integrable. Now, let us consider the tensor field G on T M , G = g i j d x i @ d x j + gij(3y i @ (3y j.

(7.2)

Then G is a tensor field of type (0,2), symmetric and nondegenerate. Moreover, G is a Riemannian metric if the fundamental tensor gij is positively defined. It is semi-Riemannian if gij has this property. Finally, is not difficult to prove the next theorem. THEOREM 7.3. (i) The structures G and ]Fe depend only on the fundamental function F = ot + ft.

THE GEOMETRY OF INGARDEN SPACES

147

(ii) The pair (G, IF) is an almost Hermitian structure on TM. (iii) The almost symplectic 2-form associated with the almost Hermitian structure (G, IF) is the Poincar~ 2-form ® = gij~Y i A d x j.

The almost Hermitian space H 2n = (T'M, G, ~r) is called the almost Hermitian model of the Ingarden space I F n.

H 2n is an almost Kiihlerian space only in some special cases. Remarking that in Randers spaces the corresponding space H 2n is always almost K~ihlerian, the great difference between the geometrical (and physical) nature of the spaces R F ~ and I F n is visible. REFERENCES [1] M. Anastasiei and H. Shimada: Deformations of Finsler metrics, in vol.: Finslerian Geometries, Kluwer Acad. Publ., FTPH, no. 109, (2000), 53. [2] P. L. Antonelli, R. S. Ingarden and M. Matsumoto: The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology, Kluwer Acad. Publ., FTPH no. 58, 1993. [3] P. L. Antonelli and R. Miron (eds.), Lagrange and Finsler Geometry. Applications to Physics and Biology, Kluwer Acad. Publ. FTPH no. 76, 1996. [4] G. S. Asanov, Finsler Geometry, Relativity and Gauge Theories, D. Reidel Publ. Co. and Kluwer Acad. Publ., Dordrecht, FTPH no. 12, 1985. [5] D. Bao, S. S. Chern and Z. Shen: An Introduction to Riemann-Finsler Geometry (Graduate Texts in Mathematics; 200), Springer, 2000. [6] D. Bao and C. Robles: On Randers spaces of constant flag curvature, Rep. Math. Phys. 51 (2003), 9. [7] A. Bejancu: Finsler Geometry and Applications, Ellis Harwood Ltd., 1990. [8] R. S. Ingarden: On the geometrically absolute optical representation in the electron microscope, Trav. Soc. Sci. Lettr. Wroclaw B45 (1957), 60. [9] R. S. Ingarden: Differential geometry and physics, Tensor N.S. 30 (1976), 201. [10] M. Matsumoto: Foundations of Finsler Geometry and Special Finsler Spaces, Kaisheisha Press, Otsu 1986. [11] M. Matsumoto: On Finsler spaces with Randers metric and special forms of important tensors, J. Math. Kyoto Univ. 14 (1974), 477. [12] M. Matsumoto and H. Shimada: The corrected fundamental theorem on Randers spaces of constant curvature, Tensor N.S. 63 (2002), 43. [13] R. Miron: General Randers Spaces, Lagrange and Finsler Geometry, Ed. by P. L. Antonelli and R. Miron, FTPH no. 59, Kluwer Acad. Publ. 1996. [14] R. Miron and M. Anastasiei: The Geometry of Lagrange Spaces: Theory and Applications, Kluwer Acad. Publ., FTPH no. 59, 1994. [15] R. Miron, D. Hrimiuc, H. Shimada and V. S. Sab~u: The Geometry of Hamilton and Lagrange Spaces, Kluwer Acad. Publ., FTPH no. 118, 2000. [16] R. Miron and T. Kawaguchi: Relativistic Geometrical Optics, Int. J. Theor. Phys. 30, no. 11 (1991), 1521. [17] R. Miron and M. Tatoiu-Radivoiovici: Extended Lagrangian Theory of Electromagnetism, Rep. Math. Phys. 21 (1988), 193. [18] I. Vaisman: Symplectic Geometry and Secondary Characteristic Classes, Birkah~iuser, Basel 1987. [19] H. Yasuda and H. Shimada: On Randers spaces of scalar curvature, Rep. Math. Phys. 11 (1977), 347.