Topic
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 Jan 2006
TL;DR: In this article, the complete lift conformal vector field on a Finsler manifold is homothetic, where the tangent bundle is a Riemannian metric on the manifold.
Abstract: Let (M,g) be a Finsler manifold, TM its tangent bundle and ˜ a Riemannian metric on TM derived from g. Then every complete lift conformal vector field on M is homothetic.
8 citations
TL;DR: In this article, the authors give an upper bound for the volume of a periodic geodesic on a surface with respect to the unit tangent bundle, which is linear in the geometric length.
Abstract: A periodic geodesic on a surface has a natural lift to the unit tangent bundle; when the complement of this lift is hyperbolic, its volume typically grows as the geodesic gets longer. We give an upper bound for this volume which is linear in the geometric length of the geodesic.
8 citations
TL;DR: The local limit theorem for ℘:limt→∞t3∕2eλ0t℘(t,x,y) is shown in this article, where λ 0 is the bottom of the spectrum of the geometric Laplacian and C(x, y) is a positive λ0-harmonic function which depends on x,y∈ M˜.
Abstract: Consider the heat kernel ℘(t,x,y) on the universal cover M˜ of a closed Riemannian manifold of negative sectional curvature. We show the local limit theorem for ℘:
limt→∞t3∕2eλ0t℘(t,x,y)=C(x,y),
where λ0 is the bottom of the spectrum of the geometric Laplacian and C(x,y) is a positive λ0-harmonic function which depends on x,y∈M˜. We also show that the λ0-Martin boundary of M˜ is equal to its topological boundary. The Martin decomposition of C(x,y) gives a family of measures {μxλ0} on ∂M˜. We show that {μ xλ0} is a family minimizing the energy or Mohsen’s Rayleigh quotient. We apply the uniform Harnack inequality on the boundary ∂M˜ and the uniform three-mixing of the geodesic flow on the unit tangent bundle SM for suitable Gibbs–Margulis measures.
8 citations
19 Jun 2015
TL;DR: In this article, the curvature of Legendre curves in the unit spherical bundle is defined by using a moving frame, and an existence and uniqueness condition of the evolute of a frontal is given.
Abstract: In order to consider singular curves in the unit sphere, we consider Legendre curves in the unit spherical bundle. By using a moving frame, we define the curvature of Legendre curves in the unit spherical bundle. As applications, we give a relationship among Legendre curves in the unit spherical bundle, Legendre curves in the unit tangent bundle and framed curves in the Euclidean space, respectively. Moreover, we define not only an evolute of a front, but also an evolute of a frontal in the unit sphere under certain conditions. Since the evolute of a front is also a front, we can take evolute again. On the other hand, the evolute of a frontal if exists, is also a frontal. We give an existence and uniqueness conditions of the evolute of a frontal.
8 citations
TL;DR: In this paper, the authors study embeddings of complex vector bundles, especially line bundles, in the complexification of the tangent bundle of a manifold and understand implications of properties of interest in partial differential equations.
Abstract: We study embeddings of complex vector bundles, especially line bundles, in the complexification of the tangent bundle of a manifold The aim is to understand implications of properties of interest in partial differential equations
8 citations