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Unit tangent bundle
About: Unit tangent bundle is a research topic. Over the lifetime, 1056 publications have been published within this topic receiving 15845 citations.
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01 Nov 2002
TL;DR: In this paper, it was shown that there is a hypercomplex structure in the neighbourhood of the zero section of the tangent bundle TX of any complex manifold X with a real-analytic torsion-free connection compatible with the complex structure whose curvature is of type (1, 1).
Abstract: Using twistor techniques we shall show that there is a hypercomplex structure in the neighbourhood of the zero section of the tangent bundle TX of any complex manifold X with a real-analytic torsion-free connection compatible with the complex structure whose curvature is of type (1, 1). The zero section is totally geodesic and the Obata connection restricts to the given connection on the zero section.We also prove an analogous result for vector bundles: any vector bundle with real-analytic connection whose curvature is of type (1, 1) over X can be extended to a hyperholomorphic bundle over a neighbourhood of the zero section of TX.
22 citations
Journal Article•
TL;DR: In this paper, the authors considered the special case of the Riemannian metric induced from the original metric of a geodesic geodesics and characterized Γ in terms of these y and C.
Abstract: By a space form we mean in this paper any one of the Euclidean «-space £\", the unit «-sphere S\"[l] in E and the hyperbolic «-space /T[— 1] of sectional curvature — 1 . For brevity, we denote these manifolds by E\", S\" and H\" respectively. If we denote the set of unit tangent vectors of a space form ΑΓ by T1(M}, then Tl(M} with natural topology is the total space of the tangent sphere b ndle π: 71 (M\") — > M. The set ?1 (ΛΓ) keeps a natural Riemannian metric induced from the original metric of M\" [3], Each geodesic Γ of 71 (Af) is then interpreted s a certain vector field y along a curve C = nF in Af. The purpose of this paper is to characterize Γ in terms of these y and C. The special case of
22 citations
01 Jan 1999
22 citations
22 citations
01 Jan 1988
TL;DR: In this article, the authors 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.
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.
22 citations