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.     

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.     

Finsler spaces with Riemannian geodesics

In Finsler spaces the intervalds=F(x ⁱ ,dx ⁱ ) is an arbitrary function of the coordinatesx ⁱ and coordinate incrementsdx ⁱ withF homogeneous of degree one in thedx ⁱ . It is shown that for Riemannian spacesds _R ²=g _ij dx ⁱ dx ⁱ which admit a non trivial covariantly constant tensorH _i .(x ^k ) there is an associated Finsler space with the same geodesic structure. The subset of such Finsler spaces withH _i .(x ^k ) a vector or second rank decomposable tensor is determined.     

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.     

Separable dynamical systems: Characterization of separability structures on Riemannian manifolds

This paper is a direct continuation of the short note [1] on separability structures on Riemannian manifolds. A separability structure on a V_n is characterized by the existence of r Killing vectors and n–r Killing 2-tensors whose properties are briefly collected in a theorem. A general discussion on the form of the metric tensor and the Killing tensors components is given.     

Algebraic classification of four-dimensional locally Euclidean spaces

The algebraic classification of locally Euclidean four-dimensional spaces (having positive-definite metric) is considered in terms of dual complex variables, which, like ordinary complex numbers, are a type of hypercomplex numbers. It is shown that only four space subtypes exist: 1, 1a, D, 0 and there are no wave types. Transition between space subtypes is described by a CUSP catastrophe analogous to a second-order phase transition in a solid. Introducing an imaginary coordinate, we obtain the usual four-dimensional space—time, and our classification reduces to the well-known Petrov—Penrose algebraic classification of gravitational fields.The work was carried out under the auspices of the Interindustry Science and Technology Astronomy Program.Krasnoyarsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, 90–95, November, 1994.     

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.     

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.     

Submanifolds in five-dimensional pseudo-Euclidean spaces and four-dimensional FRW universes

Equations for submanifolds, which correspond to embeddings of the four-dimensional FRW universes in five-dimensional pseudo-Euclidean spaces, are presented in convenient form in general case. Several specific examples are considered.     

Transformation to pseudo-Cartesian coordinates in locally flat pseudo-Riemannian spaces

A tractable method is presented for obtaining transformations to pseudo-Cartesian coordinates in locally flat pseudo-Riemannian spaces. The procedure is based on the properties of parallel covector fields. As an illustration, the method is applied to obtain certain transformations that arise in the Hamilton–Jacobi theory of separation of variables.     

Flat twistor spaces,conformally flat manifolds and four-dimensional field theory

A definition is proposed of four-dimensional conformal field theory in which the Riemann surfaces of two-dimensional CFT are replaced by (Riemannian) conformally flat four-manifolds and the holomorphic functions are replaced by solutions of the Dirac equation. The definition is investigated from the point of view of twistor theory, allowing methods from complex analysis to be employed. The paper fills in the main mathematical details omitted from the preliminary announcement [15].     

Superintegrable potentials on 3D Riemannian and Lorentzian spaces with nonconstant curvature

A quantum sl(2,ℝ) coalgebra (with deformation parameter z) is shown to underly the construction of a large class of superintegrable potentials on 3D curved spaces, that include the nonconstant curvature analog of the spherical, hyperbolic, and (anti-)de Sitter spaces. The connection and curvature tensors for these “deformed“ spaces are fully studied by working on two different phase spaces. The former directly comes from a 3D symplectic realization of the deformed coalgebra, while the latter is obtained through a map leading to a spherical-type phase space. In this framework, the nondeformed limit z → 0 is identified with the flat contraction leading to the Euclidean and Minkowskian spaces/potentials. The resulting Hamiltonians always admit, at least, three functionally independent constants of motion coming from the coalgebra structure. Furthermore, the intrinsic oscillator and Kepler potentials on such Riemannian and Lorentzian spaces of nonconstant curvature are identified, and several examples of them are explicitly presented.     

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.     

Separation of variables in Einstein spaces. II. No ignorable coordinates

We prove that the only Einstein spaces which admit a coordinate system with no ignorable coordinates which separates the Hamilton-Jacobi equation are certain symmetric spaces of Petrov typeD due to Kasner and the constant-curvature de Sitter spaces. We also show that a space admitting a coordinate system with no ignorable coordinates which separates the Helmholtz (Schrödinger) equation must be of Petrov type