On Randers spaces of scalar curvature

Randers spaces of scalar curvature or of constant curvature have not yet been studied completely because of the complicated forms of the connection coefficients and the curvature tensors. A necessary and sufficient condition for a Randers space to be of scalar curvature, found under some assumptions by this time, is given in this paper. Also a condition for the space to be of constant curvature, found previously in a complicated form, is given in simple form with a geometrical meaning. It is shown that a Randers space is locally Minkowskian if and only if it is a space of vanishing constant curvature.     

On classification of conformally flat spaces

Classification of conformally flat n-dimensional pseudo-Riemannian spaces via Plebanski's method is discussed. It is based on embedding these spaces into a flat (n + 2)-dimensional space and on finding their minimal isometry groups which are subgroups of the conformal group. In particular, the case n = 4 is given in full detail and compared with incomplete results known in the literature. The found conformally flat spacetimes are identified with the associated solutions of the Einstein equations and with the spacetimes used in various cosmological considerations.     

On Randers spaces of constant flag curvature

This paper concerns a ubiquitous class of Finsler metrics on smooth manifolds of dimension n. These are the Randers metrics. They figure prominently in both the theory and the applications of Finsler geometry. For n ≥ 3, we consider only those with constant flag curvature. For n = 2, we focus on those whose flag curvature is a (possibly constant) function of position only. We characterize such metrics by three efficient conditions. With the help of examples in 2 and 3 dimensions, we deduce that the Yasuda-Shimada classification of Randers space forms actually addresses only a special case. The corrected classification for that special case is sharp, holds for n ≥ 2, and follows readily from our three necessary and sufficient conditions.     

Nearly everywhere flat spaces

We discuss some spacetimes, which are flat everywhere except for a thin shell of matter or a string of matter, in the framework of the Israel formalism. First we study spherically symmetric universes with a single sheet of matter. Then we show that the construction of a cosmic string as a limit of various thin shell distributions of matter leads to identical results.     

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 the projectivity of conformally flat spaces created by a hydrodynamical energy-momentum tensor

It is shown that under the condition _ju _k = _k u_j imposed on the mapping function the geodesics in conformai gravitational fields are the same. The following fact is also established: all conformally flat spaces satisfying this condition correspond to the gravitational fields of an ideal fluid.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 16–19, 1972.     

Conformally flat spaces and gravitation

This article considers the theory of gravity which is defined by R ² as the free Lagrangian. The resulting equations are conformally invariant, and their equivalence to Einstein's equation is demonstrated (provided the stress tensor is traceless). The possibility of adapting this theory to massive point particles on a conformally flat background is discussed.     

Point perturbations in constant curvature spaces

Point perturbations of the free Hamiltonian in two- and three-dimensional spaces of constant curvatures are considered. The study of the spectral properties of perturbed Hamiltonian and various asymptotics for its point levels are presented. It is shown that the binding energy in comparison with the case of zero curvature reduces in the case of Lobachevsky plane and rises in the case of 2D-sphere, when the scattering length is much less than the curvature radius.     

Some Berwald spaces of non-positive flag curvature

In this paper, by using left invariant Riemannian metrics on some three-dimensional Lie groups, we construct some complete non-Riemannian Berwald spaces of non-positive flag curvature and several families of geodesically complete locally Minkowskian spaces of zero constant flag curvature.     

Superintegrable systems on spaces of constant curvature

Construction and classification of two-dimensional (2D) superintegrable systems (i.e. systems admitting, in addition to two global integrals of motion guaranteeing the Liouville integrability, the third global and independent one) defined on 2D spaces of constant curvature and separable in the so-called geodesic polar coordinates are presented. The method proposed is applicable to any value of curvature including the case of Euclidean plane, sphere and hyperbolic plane. The main result is a generalization of Bertrand’s theorem on 2D spaces of constant curvature and covers most of the known separable and superintegrable models on such spaces (in particular, the so-called Tremblay–Turbiner–Winternitz (TTW) and Post–Winternitz (PW) models which have recently attracted some interest).     

Note on asymptotically flat empty spaces

The algebraic programming system FORMAC is used to extend asymptotic solutions of the Newman-Penrose equations. The expansions are then applied to metrics with geodesic rays and to the Newman-Penrose conserved terms. It is shown how the expansion may prove useful in finding new solutions to the field equations.This paper presents results of one phase of research carried out at the Jet Propulsion Laboratory, California Institute of Technology, under Contract No. NAS7-100, sponsored by the National Aeronautics and Space Administration.     

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.     

Topology of zero riemann curvature flat space in supergravity

The equation of motion of the gravitino is investigated in a flat space reduced from the Kerr geometry by taking the massless limit of the gravitational source. We adopt the ansatz ψ_μ(x)=δ_μoψ(x), i.e., the Coulomb potential analogue to the Rarita-Schwinger Majorana field of the gravitino. A non-trivial exact classical solution of ψ(x) is obtained and it is interpreted as the source of the intrinsic topology of the background flat space. This flat space is spanned by two Minkowski sheets interconnected through a disk of radius a (the angular momentum parameter in the Kerr geometry).     

Real structures in asymptotically flat ℋ spaces

A real structure is defined in asymptotically flat ? spaces and investigated in connection with the equations of motion in ? space.     

Metric characterization of Berwald spaces of non-positive flag curvature

We show the equivalence of the non-positivity of the flag curvature with the non-positive curvature properties of Busemann and Pedersen for (not necessarily reversible) Berwald manifolds. So an analytical property is characterized by synthetic concepts of non-positively curved metric spaces.     

A characterization of Robertson-Walker spaces by null sectional curvature

ForN a null vector andA a vector perpendicular toN, define the null sectional curvature, with respect to TV, of the planeN A ask _N(N A) = R(N,A)A,N<A,A.Then Robertson-Walker metrics can be locally characterized as those for whichk _n at each point is a constant for all the null plans at that point (in each null direction,N must be appropriately chosen). A global characterization of Robertson-Walker spaces is achieved by adding completeness and causality hypotheses.     

Magnetic field correlation tensor in spaces of constant curvature

The concept of statistical homogeneity and isotropy for vector fields in spatial sections of constant curvature was analyzed. Solenoidality conditions for a corresponding correlation tensor were obtained for positive and negative curvature. It was shown that these conditions differ from the corresponding condition for fields in Euclidean space.     

A note on asymptotically flat spaces. II

After brief reviews of the Geroch and spin-coefficient formalism approaches to null infinity, we present a dictionary which translates between the two formalisms.