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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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TL;DR: In this paper, a connection between extreme values of stable random fields arising in probability and groups G acting geometrically on CAT(-1) spaces is established, mediated by the action of the group on its limit set equipped with the Patterson-Sullivan measure.
Abstract: We establish a connection between extreme values of stable random fields arising in probability and groups G acting geometrically on CAT(-1) spaces X The connection is mediated by the action of the group on its limit set equipped with the Patterson-Sullivan measure Based on motivation from extreme value theory, we introduce an invariant of the action called extremal cocycle growth and show that its non-vanishing is equivalent to finiteness of the Bowen-Margulis measure for the associated unit tangent bundle U(X/G) provided X is not a tree whose edges are (up to scale) integers As a consequence, we establish a dichotomy for the growth-rate of a partial maxima sequence of stationary symmetric $\alpha$-stable ($0 < \alpha < 2$) random fields indexed by groups acting on CAT(-1) spaces We also establish analogous results for normal subgroups of free groups This enables investigation of extremes for tree-indexed stable random fields
3 citations
01 Jan 1998
TL;DR: Two natural means to lift a distribution from a manifold to its tangent bundle, and they are shown to agree if and only if the original distribution is integrable, are provided in this article.
Abstract: Two natural means are provided to lift a distribution from a manifold to its tangent bundle, and they are shown to agree if and only if the original distribution is integrable. Two special cases, the case when the manifold is the total space of a vector bundle, and the case when the manifold is the total space of a bration over R, are dealt with in particular. For the latter case, the two constructions interact with the ane structure of the corresponding jet bundles in the \same" way.
3 citations
TL;DR: In this article, it was shown that the projection map of the unit normal bundle of a minimal surface in a Riemannian manifold is a harmonic morphism with totally geodesic fibres.
Abstract: Let \(\) be an isometric immersion between Riemannian manifolds and \(\) be the unit normal bundle of f. We discuss two natural Riemannian metrics on the total space \(\) and necessary and sufficient conditions on f for the projection map \(\) to be a harmonic morphism. We show that the projection map of the unit normal bundle of a minimal surface in a Riemannian manifold is a harmonic morphism with totally geodesic fibres.
3 citations
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TL;DR: In this paper, the authors consider the geodesic flow of reversible Finsler metrics on the 2-sphere and the 2torus, and show that ergodicity is restricted to strict subsets of tubes between flow-invariant tori in the unit tangent bundle.
Abstract: We consider the geodesic flow of reversible Finsler metrics on the 2-sphere and the 2-torus, whose geodesic flow has vanishing topological entropy. Following a construction of A. Katok, we discuss examples of Finsler metrics on both surfaces, which have large ergodic components for the geodesic flow in the unit tangent bundle. On the other hand, using results of J. Franks and M. Handel, we prove that ergodicity and dense orbits cannot occur in the full unit tangent bundle of the 2-sphere, if the Finsler metric has positive flag curvatures and at least two closed geodesics. In the case of the 2-torus, we show that ergodicity is restricted to strict subsets of tubes between flow-invariant tori in the unit tangent bundle of the 2-torus.
3 citations
TL;DR: In this article, the authors consider a product preserving functor F of order r and a connection Γ on a manifold M and introduce horizontal lifts of tensor fields and linear connections from M to F(M) with respect to Γ.
Abstract: In this paper we consider a product preserving functor F of order r and a connection Γ of order r on a manifold M. We introduce horizontal lifts of tensor fields and linear connections from M to F(M) with respect to Γ. Our definitions and results generalize the particular cases of the tangent bundle and the tangent bundle of higher order.
3 citations