Homogeneous geodesics in homogeneous Finsler spaces

In this paper, we study homogeneous geodesics in homogeneous Finsler spaces. We first give a simple criterion that characterizes geodesic vectors. We show that the geodesics on a Lie group, relative to a bi-invariant Finsler metric, are the cosets of the one-parameter subgroups. The existence of infinitely many homogeneous geodesics on the compact semi-simple Lie group is established. We introduce the notion of a naturally reductive homogeneous Finsler space. As a special case, we study homogeneous geodesics in homogeneous Randers spaces. Finally, we study some curvature properties of homogeneous geodesics. In particular, we prove that the S-curvature vanishes along the homogeneous geodesics.     

Finsler generalization of the hypothesis of Riemannian geodesics for the motion of test objects. II

In a continuation of a study begun in an earlier paper [1], the Lagrangian density is constructed for the gravitational field in the Finsler approach on the basis of the Einstein Lagrangian density. A restricted two-body problem is discussed in the Finsler approach.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 19–22, October, 1981.     

Left-flat spaces and null geodesics

Equations are derived giving the null geodesics in any left-flat space as the intersection of hypersurfaces. The connection with null geodesics given in terms of the good-cut functions forH spaces is established.     

On weakly symmetric Finsler spaces

In this paper, we study weakly symmetric Finsler spaces. We first study an existence theorem of weakly symmetric Finsler spaces. Then we study some geometric properties of these spaces and prove that any such space can be written as a coset space of a Lie group with an invariant Finsler metric. Finally we show that each weakly symmetric Finsler space is of Berwald type.     

Riemannian geometry in spaces with Grassman coordinates

The use of spaces containing Grassman (anticommuting) coordinates (in addition to the usual space-time coordinates) as a framework for unified gauge theories is described. The theory developed represents a local gauge-invariant extension of conventional (global) supersymmetry. Aside from containing the usual general coordinate invariance group of gravitational theory, the gauge supersymmetry group is seen to also encompass other symmetries of particle physics, e.g., electromagnetic (or Yang-Mills) invariance. The role of spontaneous symmetry breaking and the field equations unifying the Einstein, Maxwell, and Dirac interactions are discussed.Research supported in part by the National Science Foundation.Invited talk at the conference, The Riddle of Gravitation, on the Occasion of the 60th Birthday of Peter G. Bergmann, Syracuse, New York, March 1975.     

Closed geodesics on Riemannian manifolds via the heat flow

The main topic discussed in this paper is the following question: Given a Riemannian manifold M and a closed C¹ curve f: S¹ → M does there exist a (unique) solution of the heat equation ?_tf_t = τ(f_t) defined for all t ≧ 0 which is continuous at t = 0 along with its first S¹-derivative and which coincides with f at t = 0.     

Homogeneous Finsler spaces of negative curvature

We prove that a homogeneous Finsler space with non-positive flag curvature and strictly negative Ricci scalar is a simply connected manifold.     

On C3-like Finsler spaces

It is well-known that Cartan's torsion tensor C_ijk of any two-dimensional Finsler space is of a simple form and an n(?3)-dimensional Finsler space with the tensor of such a simple form is Riemannian owing to Brickell's theorem. A. Moór showed that the tensor of any three- dimensional Finsler space is of a special form. The purpose of the present paper is to study n(?4)-dimensional Finsler spaces with the tensor of such a special form.     

Path integrals in Riemannian spaces

The covariant path integral for a free particle in curved space will be evaluated by means of a spectral analysis of smooth paths. No discretization rule will be required to put the action on a lattice. The connection between the resulting quantum hamiltonian and the Onsager-Machlup lagrangian for diffusion processes willbe discussed. The present treatment corrects an earlier version.     

Marginally trapped surfaces in spaces of oriented geodesics

We investigate the geometric properties of marginally trapped surfaces (surfaces which have null mean curvature vector) in the spaces of oriented geodesics of Euclidean 3-space and hyperbolic 3-space, endowed with their canonical neutral Kaehler structures. We prove that every rank one surface in these four manifolds is marginally trapped. In the Euclidean case we show that Lagrangian rotationally symmetric sections are marginally trapped and construct an explicit family of marginally trapped Lagrangian tori. In the hyperbolic case we explore the relationship between marginally trapped and Weingarten surfaces, and construct examples of marginally trapped surfaces with various properties.     

Separable coordinates for four-dimensional Riemannian spaces

We present a complete list of all separable coordinate systems for the equations and with special emphasis on nonorthogonal coordinates. Applications to general relativity theory are indicated.     

Canonical connections of Finsler metrics and Finslerian connections on Riemannian metrics

A concept of canonical connection of a Finsler metric is developed. Connections that are compatible with Finsler metrics are compared with the canonical connection itself. They are also compared with the corresponding Cartan connection. A necessary and sufficient condition on metric Finsler connections is given for the metric to be Riemannian. This study unearths different ways in which Finsler geometry could be used to generalize the theory of general relativity.     

Tensor of conformal correspondence of Riemannian spaces

From the difference of the Christoffel symbols of two Riemannian spaces one can construct a third-rank tensor whose vanishing is a necessary and sufficient condition for conformal correspondence of the spaces. The connection between this new tensor and the symbols of Thomas and Weyl's conformal curvature tensor is pointed out.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 115–120, April, 1977.I am grateful to N. M. Bogatov and N. V. Smirnovaya, who were of great assistance in the preparation of the present paper.     

Canonical distributions on Riemannian homogeneous k-symmetric spaces

It is known that distributions generated by almost product structures are applicable, in particular, to some problems in the theory of Monge–Ampère equations. In this paper, we characterize canonical distributions defined by canonical almost product structures on Riemannian homogeneous $k$-symmetric spaces in the sense of types AF (anti-foliation), F (foliation), TGF (totally geodesic foliation). Algebraic criteria for all these types on $k$-symmetric spaces of orders $k=4,5,6$ were obtained. Note that canonical distributions on homogeneous $k$-symmetric spaces are closely related to special canonical almost complex structures and $f$-structures, which were recently applied by I. Khemar to studying elliptic integrable systems.     

Principal mappings of 3-dimensional Riemannian spaces into spaces of constant curvature

As is well-known, the Gauss theorem, according to which any 2-dimensional Riemannian metric can be mapped locally conformally into an euclidean space, does not hold in three dimensions. We define in this paper transformations of a new type, that we call principal. They map 3-dimensional spaces into spaces of constant curvature. We give a few explicit examples of principal transformations and we prove, at the linear approximation, that any metric deviating not too much from the euclidean metric can be mapped by a principal transformation into the euclidean metric.     

Path integrals for diffusion processes in Riemannian spaces

The Onsager-Machlup lagrangian for general continuous Markov processes in curved spaces will be derived invoking (i) continuous and differentiable trajectories, (ii) a Fourier series analysis of stochastic paths and (iii) the principle of general covariance. No discretization rule will be required in order to put the continuous action on a lattice.     

Lie derivatives and deviation equations in Riemannian spaces

The connection between Lie derivatives and the deviation equations has been investigated in Riemannian spacesV _n. On this basis the deviation equations of the geodesies have been obtained, in spaces with symmetries, as well as deviation equations of nongeodesic trajectories, through imposing certain conditions on the Lie derivatives with respect to the tangential vector of the basic trajectory.     

Integration of Dirac equation in Riemannian spaces with five-dimensional group of motions

In the present article a classification of Riemannian spaces with five-dimensional group of motion is described from the point of view of a solution of the Dirac equation. A class of spaces is identified in which the Dirac equation does not admit a complete separation of variables, and exact solutions of the Dirac equation are obtained in these spaces by means of the method of noncommutative integration. Omsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 24–28, August, 1997.