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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: The index of symmetry as mentioned in this paper measures how far a Riemannian manifold from being a symmetric space is from being one of the group types, and it is defined as a measure of how close a manifold is to being a group type.
Abstract: We introduce a geometric invariant that we call the index of symmetry, which measures how far is a Riemannian manifold from being a symmetric space. We compute, in a geometric way, the index of symmetry of compact naturally reductive spaces. In this case, the so-called leaf of symmetry turns out to be of the group type. We also study several examples where the leaf of symmetry is not of the group type. Interesting examples arise from the unit tangent bundle of the sphere of curvature 2, and two metrics in an Aloff-Wallach 7-manifold and the Wallach 24-manifold.
2 citations
TL;DR: In this paper, effective equidistribution of non-closed horocycles in the unit tangent bundle of infinite-volume geometrically finite hyperbolic surfaces was proved.
Abstract: We prove effective equidistribution of non-closed horocycles in the unit tangent bundle of infinite-volume geometrically finite hyperbolic surfaces.
2 citations
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TL;DR: In this paper, it was shown that the geodesic flow on the unit tangent bundle of a Riemannian manifold with genus greater than one and no focal points is Bernoulli.
Abstract: If $(M,g)$ is a smooth compact rank $1$ Riemannian manifold without focal points, it is shown that the measure $\mu_{\max}$ of maximal entropy for the geodesic flow is unique. In this article, we study the statistic properties and prove that this unique measure $\mu_{\max}$ is mixing. Stronger conclusion that the geodesic flow on the unit tangent bundle $SM$ with respect to $\mu_{\max}$ is Bernoulli is acquired provided $M$ is a compact surface with genus greater than one and no focal points.
2 citations
TL;DR: In this article, the tangent bundle over a Finslerian manifold M of n-dimension endowed with the Cartan connection ∇ is made into a 2n dimensional affinely connected manifold by assigning a connection ∆c to T(M).
Abstract: Let T(M) be the tangent bundle over a Finslerian manifold M of n-dimension endowed with the Cartan connection ∇. One makes T(M) into a 2n dimensional affinely connected manifold by assigning a connection ∇c to T(M). The cross-section\(\mathfrak{B}\) of a vector field V defined in M reveals in T(M) an n-dimensional submanifold and its geometry is developed by means of the affine subspace theory and of the affine collineations in the base Finsler manifold.
2 citations
01 Jan 1976
TL;DR: In this article, a family of algebraic manifolds which are hyperbolic in the sense of Kobayashi, but whose tangent bundles are not negative, is constructed.
Abstract: We construct a family of algebraic manifolds which are hyperbolic in the sense of Kobayashi, but whose tangent bundles are not negative.
2 citations