Showing papers on "Distribution (differential geometry) published in 1988"

On the Value of a Distribution at a Point

Abstract: We show connections between the notion of the value of a distribution at a point in the Łojasiewicz sense and the integrability of its Fourier transform. We consider the one-dimensional case.

d f -cohomology of Lagrangian foliations

Abstract: The cohomology of various sheaves associated with a Lagrangian foliation is discussed. Obstructions for the foliation either to behave like in the case of a cotangent bundle or to have an affine transversal distribution are found in this cohomology. Furthermore, one proves a formula for all the symplectic structures of a cotangent bundle such that the fibers constitute a Lagrangian foliation.

On Extremal Properties of Compact Parabolic Surfaces in a Riemannian Space

Abstract: The structure of parabolic surfaces in a Riemannian space is studied. Conditions are determined for compact parabolic surfaces to be totally geodesic. The proof uses the normal bundle of the surface with the Sasaki metric. A differentiable horizontal distribution arises on this bundle. It is proved that the distribution is holonomic and totally geodesic.Bibliography: 23 titles.

Magnetized cosmological model in lyra manifold

Abstract: A spatially-homogeneous and anisotropic magnetized cosmological model in Lyra's manifold is obtained when the source of the gravitational field is a perfect fluid distribution. The magnetic field is due to an electric current produced along thex-axis. The physical behaviour of the model is discussed.

It is shown that the gravitational singularities described as space-time folia- tion singularities are caustics and topological reconstructions.

Abstract: The criterion of gravitational singularities continues to be a subject of discussion [i, 2]. It was proposed to describe gravitational singularities as space-time foliation singularities [4-6], as an alternative to the known Penrose-Hocking-Ellis b-criterion [3] and which would, in contrast to the b-criterion, permit establishment of the singularity structure. It is shown in this paper that the space-time foliation singularities are topological reconstructions anc caustics. Our approach is based on the following theorems. THEOREM i. For any gravitational field g there exist a 1-form ~ and a Riemannian metric gR in the manifold X ~ such that g=gR--2~ | ~I1~1 =' (i) where I~I 2 = gR(~, ~) = -g(~, ~) be a non-singular 1-form on X ~. For any Riemann metric gR on X ~ a pseudo-Riemann metric g exists such that the relationship (i) is satisfied. The form ~/I~l agrees with the tetrade form h ~ = h~ of the gravitational field g. THEOREM 2. A mutually one-to-one correspondence exists between the nonsingular forms on X ~ and smooth orientatable distributions F of 3-dimensional subspaces of the tangential spaces in X 4 which are determined by the equation ~(F) = 0. The distribution F is called space-time if it is generated by a tetrade of the form = h ~ of a certain gravitational field. COROLLARY. Any gravitational field determines the space-time distribution on the manifold X ~. Conversely, every orientable 3-dimensional distribution on X ~ is space-time relative to a certain gravitational field. The space-time distribution F determines the space-time structure on the manifold X 4 consistent with g, and a corresponding Riemann metrix gR is a locally Euclidean topology on X ~ consistent with g. Let us call a space-time structure causal if the space-time distribution F is integrable, i.e., ~ A d~ = 0, and its generating form is exact, i.e., ~ = df. In this case foliation of the manifold X ~ into hypersurfaces holds such that hyperplanes of the distribution F are tangents to the layers of this foliation. Such a causality condition is equivalent to stable Hocking causality [3]. Layers of a causal foliation are level surfaces of the generating functions f and some curve transversal to the foliation layers does not intersect any layer more than once.