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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 article, the geodesic flow of a compact surface without conjugate points and genus greater than one and continuous Green bundles is studied and its quotient space is shown to carry the structure of a 3-dimensional compact manifold.
Abstract: We study the geodesic flow of a compact surface without conjugate points and genus greater than one and continuous Green bundles. Identifying each strip of bi-asymptotic geodesics induces an equivalence relation on the unit tangent bundle. Its quotient space is shown to carry the structure of a 3-dimensional compact manifold. This manifold carries a canonically defined continuous flow which is expansive, time-preserving semi-conjugate to the geodesic flow, and has a local product structure. An essential step towards the proof of these properties is to study regularity properties of the horospherical foliations and to show that they are indeed tangent to the Green subbundles. As an application it is shown that the geodesic flow has a unique measure of maximal entropy.
4 citations
TL;DR: In this article, it was shown that the Riemannian geometry of a tangent sphere bundle of a RiemANNian manifold (M,g) of constant radius r reduces essentially to the one of unit tangent spheres bundle (T1(m,g)) equipped with the respective induced Sasaki metrics.
Abstract: We show that the Riemannian geometry of a tangent sphere bundle of a Riemannian manifold (M,g) of constant radius r reduces essentially to the one of unit tangent sphere bundle of a Riemannian manifold equipped with the respective induced Sasaki metrics. Further, we provide some applications of this theorem on the ·-Einstein tangent sphere bundles and certain related topics to the tangent sphere bundles. g is compatible with the almost complex structure defined by taking account of the Levi-Civita connection with respect to the metric g and further, (J, ˜) gives rise to an almost Kahler structure on TM. Besides the Sasaki metric ˜, there is another well-known Riemannian metric on TM (denoted by ˆ) defined by Cheeger and Gromoll (5). In the sequel, we shall call it the Cheeger-Gromoll metric on TM. The explicit expression of the metricwas given by Musso and Tricerri (10). The tangent sphere bundle Tr(M,g) of (M,g) of constant radius r is regarded as a hypersurface of (TM, ˜) and in particular, T1(M,g) is called the unit tangent sphere bundle of (M,g). It is interesting and useful to study the relation between the geometric properties of (M,g) and Tr(M,g). We denote the induced Sasaki metric on Tr(M,g) by g 0 , T1(M,g) by g 0 and the rescaling metric (4r 2 ) i1 gr 0 by ¯ gr. By making use of the almost Kahler structure (J,˜), we can define the so-called standard contact metric structure ( ¯ gr,`,»,·). In our previous paper (4), we discussed the problem, "when is T1(M,g) equipped with the standard contact metric structure ·-Einstein?", and also raised the
4 citations
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TL;DR: In this paper, the authors give an exposition on the Holmes-Thompson theory developed by Alvarez and show that it can also be obtained from the symplectic structure on the tangent bundle of the Riemannian manifold, the manifold of the Minkowski unit sphere.
Abstract: In this article, we give an exposition on the Holmes-Thompson theory developed by Alvarez. The space of geodesics in Minkowski space has a symplectic structure which is induced by the projection from the sphere- bundle. we show that it can be also obtained from the symplectic structure on the tangent bundle of the Riemannian manifold, the tangent bundle of the Minkowski unit sphere. We give detailed descriptions and expositions on Holmes-Thompson volumes in Minkowski space by the symplectic structure and the Crofton measures for them. For the Minkowski plane, a normed two dimensional space, we express the area explicitly in an integral geometry way, by putting a measure on the plane, which gives an extension of Alvarez's result for higher dimensional cases. 1. Introductions
4 citations
TL;DR: In this article, the existence of homogeneous vectors on the fiber space of the tangent bundle τ G/K of a homogeneous differentiable manifold has been studied, for both cases of G semisimple and weakly semismple.
Abstract: Let M = G/K be a homogeneous differentiable manifold. We consider the homogeneous bundle \( \Im \) = (G, π, G/K, K) and the tangent bundle τ G/K of M = G/K, and give some results about the existence of homogeneous vectors on the fiber space of τ G/K, for both cases of G semisimple and weakly semisimple.
4 citations
TL;DR: In this paper, a study of prolongations of F-structure to the tangent bundle of order 2 has been presented, where the authors propose an extension of the T-bundle of order 3.
Abstract: A study of prolongations of F-structure to the tangent bundle of order 2 has been presented.
4 citations