Showing papers on "Unit tangent bundle published in 1988"

Riemannian Metrics on Tangent Bundles

Abstract: Some «natural» metrics on the tangent and on the sphere tangent bundle of Riemannian manifold are constructed and studied via the moving frame method.

Riemannian geometry, fiber bundles, Kaluza-Klein theories and all that

Abstract: This book discusses the geometrical aspects of Kaluza-Klein theories. The ten chapters cover topics from the differential and Riemannian manifolds to the reduction of Einstein-Yang-Mills action. It would definitely prove interesting reading to physicists and mathematicians, theoretical and experimental. Contents: Generalities; Riemannian Geometry of Lie Groups; Riemannian Geometry of Homogeneous Spaces; Riemannian Geometry of a (Right) Principal Bundle; Riemannian Geometry of a Bundle with Fibers and a Given Action of a Lie Group G; Geometry of Matter Fields; Harmonic Analysis and Dimensional Reduction; Dimensional Reduction of the Orthogonal Bundle and of the Spin Bundles; G-Invariance of Einstein-Yang-Mills Systems; Action of a Bundle of Groups.

Ergodic theory of foliations and a theorem of Sacksteder

Abstract: We introduce the leafwise geodesic flow of a foliation, a flow on the unit tangent bundle to the leaves which preserves the natural foliation on this manifold. The transverse dynamics of this flow closely mirror the dynamics of the original foliation, and in this paper we outline a program for the study of foliation dynamics based on this observation. For example, the topological entropy of a foliation is defined to be the toplogical entropy of this flow relative to the invariant foliation. This yields a topological entropy close to that defined by Ghys-Langevin-Walczak. The metric entropies of a foliation are defined to be the corresponding relative metric entropies of the leafwise geodesic flow, with respect to invariant measures for the flow. The topological entropy then dominates the metric entropies, and the supremum of the metric entropies over the space of probability measures equals the topological entropy. This extends to foliations the relative variational principle of Ledrappier and Walters. Upper estimates of foliation metric entropies via transverse Lyapunov exponents are given, extending work of Strelcyn, from which we deduce a generalization of a theorem of Sacksteder concerning the existence of linearly contracting holonomy in exceptional minimal sets for codimension-one foliations of differentiability class Holder C1.

The restricted tangent bundle of a rational curve in P2

Abstract: (1988). The restricted tangent bundle of a rational curve in P2. Communications in Algebra: Vol. 16, No. 11, pp. 2193-2208.

Invariant derivation of the Euler-Lagrange equation

Abstract: The tangent bundle geometry is used to obtain a coordinate-free derivation of the Euler-Lagrange equation.

Distribution of closed geodesics with a preassigned homology class in a negatively curved manifold

Abstract: Let M be a compact Riemannian manifold whose geodesic flow φ i : UM→UM on the unit tangent bundle is of Anosov type. In this paper we count the number of φ i -closed orbits and study the distribution of prime closed geodesies in a given homology class in H 1 ( M, Z ). Here a prime closed geodesic means an (oriented) image of a φ i -closed orbit by the projection p : UM → M .

The visibility axiom on a Hadamard manifold whose geodesic flow is of Anosov type

Abstract: For a Hadamard manifold M, the set of points at infinity M(oo) is defined. If the geodesic flow on the unit tangent bundle of M is of Anosov type, then with a certain curvature condition M satisfies the Visibility Axiom. To prove this result, we use the Tits metric on M(oo).