Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://shamolin2.imec.msu.ru/art-166-1.pdf Äàòà èçìåíåíèÿ: Tue Feb 3 22:31:20 2015 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 00:46:54 2016 Êîäèðîâêà:

DOI 10.1007/s10958-015-2213-z Journal of Mathematical Sciences, Vol. 204, No. 6, February, 2015

FINSLER SPACES, BINGLES, POLYINGLES, AND THEIR SYMMETRY GROUPS R. R. Aidagulov and M. V. Shamolin UDC 517.925

In [22], the notions of bingles and tringles in the space H3 were introduced. Their definitions are based on an imp ortant additivity principle. However, the additivity principle itself is applicable only under a certain "coplanarity" condition. Therefore, based on this principle, one cannot compare bingles (and tringles, etc.) b etween incomparable angles, i.e., it is p ossible to take different values of missing coefficients of prop ortionality for different directions preserving the additivity principle. The definition of bingles accepted in [22] p ossesses the following "strange" p eculiarity: for any two vectors a and b from the same octant, there exists a vector c from this octant such that the bingles b etween a and c and b etween c and b are equal to zero. It is desirable that the definition of a bingle p ossesses the following prop erty: if the bingle b etween a and b is equal to zero, then these vectors are prop ortional; in this case, the "strange" prop erty cannot hold for any nonprop ortional vectors. Keeping the additivity principle, we can define the notion of a bingle so that the following, stronger metric prop erty holds: (1) (a, b) 0; (2) (a, b) = 0 a b; (3) (a, b) (a, c)+ (c, b). First, we consider the additivity principle in the most general case. Let X b e a smooth manifold. Categories of smooth manifolds are studied within the framework of differential top ology. Geometric considerations app ear when one introduces a way of calculating lengths (areas, volumes, etc.) as additive functionals on one-dimensional (two-dimensional, three-dimensional, etc.) submanifolds of the manifold X . Consider the one-dimensional case (length). Let x( ), 0 1,

b e a smooth curve on a manifold X . The additivity of the length functional L means that L x( ) = L x1 ( ) + L x2 ( ) , where the curve x( ) is the sum of the curves x1 ( ) and x2 ( ): x( ) = x1 ( )+ x2 ( ), i.e., for some numb er 0 < < 1 we have x x( ) = x (1)

1 2

, - 1-

0 , , 1.

Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 88, Geometry and Mechanics, 2013.

732

1072­3374/15/2046­0732 c 2015 Springer Science+Business Media New York


Naturally, the length of a curve must b e indep endent of its parametrization. These two requirements imply that
1

L x( ) =
0

ds,

ds = g (x, dx),

g (x, a dx) = ag (x, dx), a > 0.

(2)

Finsler geometry also requires the p ossibility of expressing velocities (unit vectors of the tangent space) through momenta (tangent hyp erplanes to the unit sphere) that are elements of the cotangent bundle, and vice versa. Functions of velocities b ecome functions of momenta, while velocities and momenta themselves are related by the Legendre transforms that are tropical analogs of the Fourier transforms. As is well known, the Legendre transform (see [3]) establishes a corresp ondence b etween any function of the variable x and a function of another variable p by the following rule: f (p) = max(px - f (x)).
x

(3)

Since the maximum in the last expression is attained exactly at the p oint x where the tangent hyp erplane to the graph of the function f (x) is orthogonal to the vector p, the Legendre transform can b e interpreted as a transition to tangent variables. The inverse transform is defined by the same formula by interchanging the variables x and p. Surely, the variables x and p can b e multi-dimensional; then the variable p b elongs to the dual space and px =
i

p i xi .

We can also consider the case where p and x are elements of functional spaces; then functionals play the role of functions. As is well known, in classical mechanics, the Lagrange transform establishes a corresp ondence b etween Lagrange functions dep ending on velocities and Hamilton functions dep ending on momenta. However, in classical mechanics the kinetic energy is a quadratic function of velocities and in this case the transition from the Lagrange function to the Hamilton function is relatively simple and can b e p erformed without using the Legendre transform. In pure mathematics, the Legendre transform are used for obtaining various inequalities. For example, if f (x) and f (p) are related by Legendre transforms, then for any x and p, the following inequality holds: pi xi f (x)+ f (p).
i

In particular, if a (Finsler) metric is defined by the formula x f (x) =
i

, aR a (here we omit the index i for brevity), then the corresp onding metric in the dual space is defined as follows: pb 11 + = 1, b R, f (p) = i i , b ab and for any vector x and any covector p, the corresp onding inequality holds. Such transforms in b oth directions are p ossible if the function is convex or concave. The dimensions of the domains of the corresp onding variables for smooth functions coincide if and only if they strictly convex or concave. Tropical mathematics is based on the replacement of multiplication by addition and replacement of addition by the binary op eration min(a, b) or max(a, b); in this case, distributivity is not violated. Although this extraordinary mathematics gave nothing to conventional science, it deserves attention 733

a i


due to new p oints of view on usual things. For example, the sp ectral Fourier transform in tropical mathematics is reduced to the Legendre transform. It may well turn that in Finsler geometry tropical ideas can lead to new results. The first case (of convexity of a metric) is realized if and only if the triangle inequality is valid: g(x, a + b) g(x, a)+ g(x, b), (4)

and equality occurs only in the case where the vectors a and b are parallel. Now it b ecomes clear that the length functional determines a metric if and only if the function g(x, a) is p ositive for all nonzero vectors a. Then Finsler geometry b ecomes an appropriate generalization of Euclidean geometry. The second case is realized if and only if the inverse triangle inequality holds for any two measurable vectors from the same connected comp onent: g(x, a + b) g(x, a)+ g(x, b), (5)

and equality holds only for parallel vectors of nonzero length. The second case is a Finsler generalization of Minkowski space and is of great interest in the physical context. Note that inequality (5) cannot hold for all vectors. Therefore, for each p oint x, one can consider the set of all admissible vectors that are said to b e measurable. The set of measurable vectors is constrained by the following conditions: (1) if vectors a and b are measurable, then for any p ositive numb ers x and y , the vector xa + yb is also measurable; (2) the set of measurable vectors has maximal dimension, i.e., there exists a basis consisting only of measurable vectors. The first condition, in particular, implies that the set of measurable vectors is a convex cone. In inequality (5), we must additionally stipulate that strong equality is imp ossible in the case of nonparallel measurable vectors of nonzero length (resp ectively, from the inner domain of measurable vectors) or assume that b oundary vectors of zero length are either nonmeasurable or parallel to all measurable vectors (thus, we extend the notion of parallelism). We also note that Finsler generalizations do not generate other pseudo-Euclidean metrics with other signatures. In this case, we can assume that measurability is defined only for "p ositive" vectors, i.e., we are restricted to one connected comp onent where measurability is defined; the corresp onding vectors are said to b e time-like. Thus, the notion of length is defined only for time-like world lines. The validity of conditions (4) or (5) is also necessary for the existence and uniqueness of a geodesic emanating from a given p oint in a given measurable direction. In the first case, geodesics yield a minimum whereas in the second case a maximum. As an example, consider metrics associated with hyp er-complex numb ers Hn , i.e., metrics that are invariant with resp ect to automorphisms of the algebra Hn , which coincides with the symmetric group Sn . This class includes metrics defined by symmetric k -order p olynomials of n variables. It turns out that all nondegenerate metrics of this typ e b elong to the first class, where the triangle inequality holds, or to the second case, where prop erty (5) holds. In the general case, the proof of this assertion is difficult since one must apply gradual rotations of the bingle b etween a and b. For example, we verify the last assertion for the Berwald­Moor metric. In this case, measurable vectors are n-tuples of p ositive numb ers and inequality (5) is equivalent to the well-known Minkowski inequality [4]:
1/n 1/n 1/n

(ai + bi )
i


i

a

i

+
i

bi

.

734


For the symmetric metric
1/

|a| =
i

|ai |



,

the following inequality is known: (1 - ) (a + b) -|a|-|b| 0, where equality (for p ositive vectors) holds only in the case where the vectors are prop ortional (parallel). Thus, the transition from Euclidean Finsler geometry ( > 1) to Minkowski-typ e geometry ( < 1) implies the replacement of the triangle inequality by the inverse inequality, which must hold for vectors with p ositive comp onents, more precisely, when ai bi 0 i. If we divide the metric by the constant n1/ and let tend to zero, then we obtain the Berwals­Moor metric as a particular case of such a metric for = 0. Unfortunately, in the monograph [6] devoted to Finsler geometry, the two sp ecial typ es of Finsler spaces describ ed ab ove and the corresp onding triangle inequalities (4) and (5) are not emphasized. In the first case, the signature of the metric (which can also b e defined for non-quadratic metrics) is + + ··· + and in the second case is + - - ··· -.
n times n - 1 times

To prove that other signatures are imp ossible, consider the simplest pseudo-Euclidean metric with a different signature: ds2 = dx2 + dx2 - dx2 . 1 2 3 The indicatrix of this metric is a hyp erb oloid of one sheet, which is neither convex nor concave surface. We calculate the coordinates of the dual space (see [6]): dx1 dx2 dx3 , p2 = , p3 = - , ds = dx2 + dx2 - dx2 . 1 2 3 ds ds ds It is easy to see that there is a unique functional dep endence b etween the coordinates: p1 = p2 + p2 - p2 = 1, 1 2 3 and that this space completely corresp onds to the definition of a Finsler space by [6]. However, in this space geodesics are not defined: one can move from p oint (0, 0, 0) to p oint (x1 ,x2 ,x3 ) in many ways lying in the vertical plane containing the initial p oint and the endp oint, and the length of the curve can equal zero. Thus, this space is not a Finsler space of Euclidean typ e. Also, we can easily verify that in this space, there exist arbitrarily long paths, i.e., the space is not a Finsler space of Minkowski typ e. In fact, this refers to all pseudo-Euclidean geometries with signatures different from the signatures of Euclidean space and Minkowski space. It is interesting that in courses of geometry and physics this elementary fact that geometries of other signatures are unsuitable for the definition of geodesics from the variational principle is not mentioned. Thus, the variational method of determining distances based on the additive principle (the definition of a minimum or maximum of some integral taken along a curve connecting two p oints) leads to the convexity or concavity of the indicatrix. The uniqueness condition for geodesics requires strict convexity or concavity, which can b e given by the triangle inequalities (4) and (5) with equality conditions only for parallel vectors. In addition, for (twice) smooth indicatrices, these conditions also provide the requirement of coincidence of the dimension of the tangential variables on a sphere with dimension n - 1. Pavlov and his disciples study commutative p olynumb er Finsler geometries. In this case, all commutative sets of p olynumb ers are direct sums of some copies of the sets of real and complex numb ers. 735 (6)


We show that if the set of complex numb ers is contained in a set of p olynumb ers as a gebra, then for this set of p olynumb ers, neither Finsler geometry of Euclidean typ e nor Minkowski typ e can b e defined. In the sum R + C, we consider two vectors. Let a nonzero vector a = (1,a2 + ia3 ) b Then the vector b = (b1 ,a2 + ia3 ) is also a measurable nonzero vector if b1 is close to for any p ositive t the vector a + tb = (t +1)a +(b1 - 1, 0+ i0) is also measurable and has the length l(t) = Since |a| = we have l(t) = t|b| + for large t and
3 3

prop er subala geometry of e measurable. 1. Therefore,

(1 + tb1 )(t +1)2 a2 + a2 . 2 3 |b| = c|a|, c= 1 t
3

a2 + a2 , 2 3

b1 ,

1+2c3 |a| + O 3c3

,

2+ c3 |b| + O(t) 3c for small t. This shows that if c < 1, then for large t the inequality triangle holds while for t the opp osite inequality holds. For c > 1, the opp osite situation holds. It is easy to show that the last prop erty holds for any algebra of commutative p olynumb ers containing complex numb ers as a prop er subalgebra. Thus, Finsler geometry of Minkowski typ e is realized only on a direct sum of some (clearly, more than one) copies of the algebra of real numb ers, while Finsler geometry of Euclidean typ e is realized on irreducible algebras without divizors of zero consisting of real numb ers, complex numb ers, and-- in the noncommutative case--quaternions. Note that in this case, nonassociative algebras are not considered. Summarizing the discussion of Finsler spaces, we sp ecify the new terminology. A metric Finsler space is a smooth manifold equipp ed with the p ositive length functional defined by (2). In this case, the metric function g (x, a) vanishes only for a = 0 and satisfies the triangle inequality (4), which b ecomes an equality only for parallel vectors, since homogeneity implies the equality l(t) = |a| + g (x, a + b) = g(x, a)+ g(x, b)

for b = ra, r > 0. A Finsler­Minkowski space is a smooth manifold on which the length functional is defined by formula (2). In this case, the metric function satisfies the "triangle anti-inequality" (5), which b ecomes an equality only for vectors of p ositive length a and b that differ by a p ositive factor. Precisely these spaces are imp ortant in physics. Now we consider a general Finsler space. We distinguish one of the coordinates a0 of a vector and denote the ratio of all other coordinates to a0 by ai vi = , i = 1,... ,n - 1. a0 Then from (2) we obtain the decomp osition g(x, a) = c(x)a0 + c1 (x)v1 + ··· + cn Passing to prop er time dt = c(x)dx0 +
i -1

(x)vn

-1

+ f (x, v ).

(7)

ci (x)dxi ,

736


we can eliminate the comp onents of the velocity in a first-order decomp osition. Further, assuming that the Hessian is nondegenerate, we uniquely distinguish the spatial coordinates such that the metric has the form v2 + O v3 (8) g(x, dx) = dt 1+ 2 in the first case and the form v2 + O v3 g(x, dx) = dt 1 - (9) 2 in the second case. Other signatures are imp ossible due to conditions (4) or (5). All other coordinates that reduce the metric to the same form are related to rotations in the first case and to Lorentz b oosts in the second case. Up to third order, isometries always exist. More exact isometries, in general, b ecome anisotropic and/or nonlinear and may b e absent altogether except for the identical isometry. Relations (8) and (9) can b e locally obtained near any direction defined by a unit vector, but, unlike the case of quadratic metrics, the coordinates of the hyp erplane pi dai = 0
i

do not corresp ond to the coordinates of the vector a. For example, for a metric of rank k, the coordinates of a hyp erplane are defined as p olynomials of degree k - 1 of the coordinates of the vector pi = gj
1

...j

k -1

i

aj1 ... aj

k -1

(as usual, summation over rep eated upp er and lower indices is meant). The notion of locality significantly differs from the traditional, where considerations near a chosen p oint are assumed. In our case, all vectors emanate from the same p oint, and locality relates to the set of vectors under consideration with directions "close" to a given direction a. Therefore, Finsler geometry has infinitely many degrees of freedom at each p oint and hence can serve as a bridge to the unification of the general theory of relativity with quantum mechanics. The normalization of the Euclidean metric on the tangent space of the sphere in conditions (8) and (9) is defined by the normalization inherited from the Finsler space, with resp ect to which the length of the vector a is equal to 1. This allows one to define the length of a curve on the sphere. In the metric case length is defined by the formula dr 2 = g2 (x, dx) - dt2 , |dx| dt,

and geodesics coincide with that defined by the induced metric. In Finsler spaces of Minkowski typ e we obtain the following: 1, dr 2 = dt2 - g2 (x, dx), dt = 1, |dx| and geodesics on the sphere are defined by the minimum of this new functional on the sphere, which b ecomes a Riemannian space. Here the first argument is a constant (a p oint does not vary) and the sphere is defined by the endp oints of direction vectors dx. Now we consider the notion of k -dimensional volumes for k-dimensional submanifolds of X . We can assume that the parameters determining the surface run over the simplex 1 + 2 + ··· + k = 1, i 0.

We denote the vertices of the simplex by A0 ,A1 ,... ,Ak . Let O b e a p oint of this simplex; then the additivity condition can b e written in the form
k

Vk (A0 ,A1 ,... ,Ak ) =
i=0

Vk ... ,Ai

-1

,O ,Ai

+1

,... .

(10)

737


The sum on the left-hand side consists of k -dimensional volumes of partitions of the simplex when the ith vertex is replaced by the p oint O. Using also the natural condition of indep endence of kdimensional volumes of the parametrization, we find that k-dimensional volumes can b e calculated as the integrals Vk = dvk , dvk = g(x, Jk )d1 d2 ··· dk , (11)

where Jk is a set of k -dimensional minors xi / j . Note that gk is a homogeneous function of degree one (as in the case k = 1) with resp ect to these minors. In Riemannian geometry, the definition of length for curves automatically implies the definitions of measures for areas, volumes, etc. In the present pap er, we restrict ourselves to the study of the two-dimensional case (area); measures in higher dimensions (e.g., volume) are defined similarly. As is well known, the area is the sum of absolute values of some bilinear skew-symmetric form on elementary pairs of vectors. All skew-symmetric forms of n vectors in an n-dimensional space are defined up to a constant factor. In the Riemannian case, this factor is chosen so that for the orthonormed basis the area is equal to 1. However, in Minkowski space this normalization leads to problems. A vector orthogonal to another vector with endp oints on the sphere can b e nonmeasurable; moreover, all vectors orthogonal to the time vector (1, 0, 0, 0) are nonmeasurable. In measuring vectors that are tangent to the indicatrix, using an isometry (see [22]) we reduce the problem to the measurement of lengths of such nonmeasurable vectors. In this sense, in calculating bingles, we consider, rather than the area of the sector of the unit sphere b etween the vectors a and b, the length of the geodesic arc on the sphere connecting them. Note that in calculating angles (bingles, tringles, etc.) b etween vectors at the same p oint we can assume that the space is a flat Finsler space if the metric is invariant with resp ect to translations. In this case, the radius of a sphere equal to 1 has only conditional character since we do not calculate induced distances on the sphere from the Finsler space considered, and only calculate rotation angles b etween vectors at a given p oint. First, we consider bingles. As was noted ab ove, we cam assume that the Finsler space is flat. The sphere is defined by the condition g 2 (x, dx) = f (y ) = 1, y = dx.

Therefore, the direction dx b ecomes a p osition of a p oint y on the sphere y , and on the sphere a Riemannian metric is defined. To describ e this metric near the direction of a vector a of length 1, we introduce the covector defining the tangent hyp ersurface at the p oint a: p = grad f = f (y ) y i .
y =a

Tangent vectors z to the sphere at a p oint a are annihilated by this covector: pi z i = 0. We pass to a coordinate system in which the vector y has the form y = at + z i ei , where ei are tangent vectors to the sphere, i = 1,... ,n - 1. Expanding the function f (y ) in a series up to second order, we obtain 1 = f (y ) = f (a)+ 738 1 2f f dt + dz i dz j . t 2 z i z j


This defines a Riemannian metric on the sphere in local coordinates: 1 2f . 2 z i z j In the last expression, the sign "+" corresp onds to the metric Finsler case and the sign "-" to the case of a Finsler­Minkowski space. Thus, the metric on the sphere needed for defining angles (bingles, tringles, etc.) is always a p ositive-definite Riemannian metric. Using bingles, one can introduce new coordinates on the sphere. Performing this procedure on a sphere or a pseudosphere, we obtain local isomorphisms b etween them: dr 2 = gij dz i dz j , gij (a) = ± A : S S , where S is the sphere of directions of the Finsler space and S is an (n - 1)-dimensional sphere (for the metric Finsler spaces) or a pseudosphere (for Finsler­Minkowski spaces). Any isometry of a Finsler space (we can sp eak only of isometry of directions at a given p oint) isometrically transforms the sphere. Therefore, under such transforms, all angles (bingles, tringles, etc.) are preserved. By definition, they are also invariant under scaling. Denote by Gk the group of transforms preserving k-ingles. Then for transforms F generated by small translations we obtain a local transform AF A-1 of the sphere or the pseudosphere preserving (k - 1)-dimensional areas. Therefore, in the case k < n, this group coincides with the motion group of the sphere in the metric case or the motion group of the pseudosphere (i.e., the Lorentz group) in the case of a Finsler­Minkowski space. For n-ingles, when (n - 1)-dimensional volumes in an (n - 1)-dimensional sphere (pseudosphere) are measured, this group is substantially wider since any divergence-free field on this sphere generates a transform preserving volumes. Therefore, it is infinite-dimensional. In addition, one can also use scaling transforms that turn a vector a into c(a)a (multiplication by a p ositive factor dep ending on the vector itself ). Nevertheless, it may occur that there is no linear transforms preserving bingles, tringles, etc. (except for the identity transform) among them. There is another way of defining tringles and higher ingles in arbitrary Finsler spaces. For this purp ose, we expand the function g k (x, dx) up to the kth degree and obtain a form of degree k of translations in the tangent space and calculate k-dimensional volumes of sectors by using such a metric of rank k . Since expansion terms of all degrees coincide under an isometry, such k-ingles are also conformally invariant. They also p ossess the additivity prop erty. In this case, 1-ingles are defined by linear expansions and coincide with the lengths of vectors. As ab ove, bingles are prop ortional to quadratic forms on vectors tangent to the sphere. However, the isometry group for higher k-ingles can turn out to b e narrower than for the previous definition based on the Riemannian metric on the sphere. Now we consider a sp ecific metric of the Berwald­Moor space, which is a Finsler­Minkowski space. In this case, multiplication by hyp er-complex numb ers with norm 1 is an isometry. Therefore, the sphere is a commutative Lie group of dimension n - 1. By construction, the metric is invariant under the action of this group. For the metric defined by a metric tensor of rank k we have f (y ) = 1+ k (a,a,... ,dy )+ k (k - 1) (a,a,... ,dy ,dy ) 2
2/k

+ O dy

3 3

= 1 + 2(a,a,... ,dy )+(k - 1)(a,a,... ,dy ,dy )+(2 - k)(a,a,... ,dy )2 + O dy This implies that the Riemannian metric for this Finsler­Minkowski space has the form dr 2 = gij dz i dz j = -(k - 1)(a,a,... ,ei ,ej ), where ei are basis vectors in the tangent space of the sphere at the p oint a.

.

(12)

739


Note that if the initial metric is quadratic (k = 2), then it coincides with the corresp onding induced metric on the tangent space up to sign and is indep endent of the p oint a. Assuming that k = n, where n is the dimension of the space, and using the Berwald­Moor metric, we obtain the invariance of the metric considered. We find the coordinates of an orthonormal system of tangent vectors at the p oint 1 = (1, 1,... , 1). Up to a constant factor, we can take ei = c (1, 1,... , 1) + d(1, 0,... , 0) + r (0,... , 0, 1, 0,... , 0) . From the orthogonality of vectors we obtain d = ± n - 1, r = -n - d, n . n + n - 2+2nd
2

c=

Then, to find the bingle b etween two p ositive vectors a and b reduced to the same norm, it suffices to represent them in the form b = a exp(c), c= ln b1 b2 bn , ln , ... , ln a1 a2 an the sum of squares of coordinates in the logarithmic coordinates and p olyingles as symmetry group of bingles etc., without n-ingles the symmetry group is infinite-

and calculate the Euclidean length c as the square root of orthonormal basis ei . Similarly, one can calculate tringles as areas of triangles in the corresp onding volumes in this space of logarithms. The scaling, coincides with the Lorentz group. As ab ove, for dimensional (see [1, 2, 5, 7­21, 23­27]).

Acknowledgment. This work was partially supp orted by the Russian Foundation for Basic Research (Pro ject No. 12-01-00020-a). REFERENCES 1. M. Anastasiei and R. Miron, "Generalized Finsler metrics," in: Finsler Geometry. Joint Summer Research Conf., July 16­20, 1995, Seattle, WA, USA (D. Bao et al., ed.), Contemp. Math., 196, Am. Math. Soc., Providence, Rhode Island (1996), pp. 157­159. 2. A. K. Aringazin and G. S. Asanov, "Problems of finslerian theory of gauge fields and gravitation," Repts. Math. Phys., 25, 183­241 (1988). 3. V. I. Arnol'd, Mathematical Methods of Classical Mechanics [in Russian], Nauka, Moscow (1989). 4. E. F. Beckenbach and R. Bellman, Inequalities, Springer-Verlag, Berlin­G¨ ottingen­Heidelb erg (1961). 5. T. Q. Binh, "Cartan-typ e connections and connection sequences," Publ. Math., 35, Nos. 3­4, 221­229 (1985). 6. G. I. Garas'ko, Foundations of Finsler Geometry for Physicists [in Russian], Moscow (2009). 7. A. Grey and L. M. Hervella, "The sixteen classes of almost hermitian manifolds and their linear invariants," Ann. Math. Pure Appl., 123, No. 4, 35­58 (1980). 8. M. Hashiguchi, "On generalized Finsler spaces," An. Sti. Univ. Iasi. Sec. 1a, 30, No. 1, 69­73 (1984). 9. M. Hashiguchi, "Some topics on Finsler geometry," Conf. Semin. Mat. Univ. Bari., 210 (1986). 10. S. Ikeda, "On the Finslerian metrical structures of the gravitational field," An. Sti. Univ. Iasi. Sec. 1a, 30, No. 4, 35­38 (1984). 11. S. Ikeda, "Theory of fields in Finsler spaces," Semin. Mec. Univ. Timi¸ ara, No. 8, 1­43 (1988). so 12. S. Ikeda, "On the theory of gravitational field in Finsler spaces," Tensor, 50, No. 3, 256­262 (1991). 740


13. S. Kikuchi, "On metrical Finsler connections of generalized Finsler spaces," in: Proc. 5 Natl. Semin. on Finsler and Lagrange Spaces in Honour of 60th birthday of Prof. Radu Miron, Brasov, February 10­15, 1988, Brasov (1989), pp. 197­206. 14. M. S. Kneb elman, "Collineations and motions in generalized space," Am. J. Math., 51, 527­564 (1929). 15. M. Matsumoto, "On Einstein's gravitational field equation in a tangent Riemannian space of a Finsler space," Repts. Math. Phys., 8, No. 1, 103­108 (1975). 16. M. Matsumoto, Foundation of Finsler Geometry and Special Finsler Spaces, Kais. Press. Otsu. Ipan (1986). 17. M. Matsumoto, "The Tavakol van den Bergh conditions in the theories of gravity and pro jective changes of Finsler metrics," Publ. Math. Debrecen, 42, Nos. 1­2, 155­168 (1993). 18. R. Miron, "Metrical Finsler structures and metrical Finsler connections," J. Math. Kyoto Univ., 23, 219­224 (1983). 19. R. Miron, "On the Finslerian theory of relativity," Tensor, 44, No. 1, 63­81 (1987). 20. R. Miron, R. K. Tavakol, V. Balan, and I. Roxburgh, "Geometry of space time and generalized Lagrange gauge theory," Publ. Math. Debrecen, 42, Nos. 3­4, 215­224 (1993). 21. A. Moor, "Entwicklung einer Geometrie der allgemeiner metrischen Linienelement R¨ me," Acta au Sci. Math., 17, Nos. 1­2, 85-120 (1956). 22. D. G. Pavlov and S. S. Kokarev, "Polyangles and their symmetries," in: Hyper-Complex Numbers in Geometry and Physics [in Russian], No. 11 (2009). 23. S. Sasaki, "On the differential geometry of tangent bundles of Riemannian manifolds, I," T^ u ohok Math. J., 10, No. 3, 338­354 (1958). 24. I. Tashiro, "A theory of transformation groups on generalized spaces and applications to Finsler and Cartan spaces," J. Math. Soc. Jpn., 11, No. 11, 42­71 (1959). 25. H. S. Wang, "On Finsler spaces with completely integrable equations of Killing," J. London Math. Soc., 22, 5­9 (1947). 26. S. Watanab e, "Generalized Finsler spaces conformal to a Riemannian space and the Cartan like connections," An. Sti. Univ. Iasi. Sec. 1a, 30, No. 4, 95­98 (1984). 27. K. Yano and E. T. Davies, "On the tangent bundles of Finsler and Riemannian manifolds," Rend. Circ. Math., 12, No. 2, 211­228 (1963). R. R. Aidagulov M. V. Lomonosov Moscow State University, Moscow, Russia E-mail: a rust@bk.ru M. V. Shamolin Institute of Mechanics of the M. V. Lomonosov Moscow State University, Moscow, Russia E-mail: shamolin@imec.msu.ru

741