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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, a template with two ribbons that describes the topology of all periodic orbits of the geodesic flow on the unit tangent bundle to any sphere with three cone points with hyperbolic metric was constructed.
Abstract: We construct a template with two ribbons that describes the topology of all periodic orbits of the geodesic flow on the unit tangent bundle to any sphere with three cone points with hyperbolic metric. The construction relies on the existence of a particular coding with two letters for the geodesics on these orbifolds.
3 citations
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TL;DR: In this paper, the authors studied the ergodicity of geodesic flows on surfaces with no focal points under the assumption that the set of points with negative curvature has at most finitely many connected components.
Abstract: In this article, we study the ergodicity of the geodesic flows on surfaces with no focal points. Let $M$ be a smooth connected and closed surface equipped with a $C^\infty$ Riemannian metric $g$, whose genus $\mathfrak{g} \geq 2$. Suppose that $(M,g)$ has no focal points. We prove that the geodesic flow on the unit tangent bundle of $M$ is ergodic with respect to the Liouville measure, under the assumption that the set of points on $M$ with negative curvature has at most finitely many connected components.
3 citations
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TL;DR: In this paper, it was shown that the absolute value at zero of the Ruelle zeta function defined by the geodesic flow coincides with the higher-dimensional Reidemeister torsion for the unit tangent bundle over a 2D hyperbolic orbifold and a non-unitary representation of the fundamental group.
Abstract: We show that the absolute value at zero of the Ruelle zeta function defined by the geodesic flow coincides with the higher-dimensional Reidemeister torsion for the unit tangent bundle over a 2-dimensional hyperbolic orbifold and a non-unitary representation of the fundamental group. Our proof is based on the integral expression of the Ruelle zeta function. This integral expression is derived from the functional equation of the Selberg zeta function for a discrete subgroup with elliptic elements in PSL(2;R). We also show that the asymptotic behavior of the higher-dimensional Reidemeister torsion is determined by the contribution of the identity element to the integral expression of the Ruelle zeta function.
3 citations
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TL;DR: For rank-one convex projective manifold with a compact convex core, this article showed that there exists a unique probability measure of maximal entropy on the set of unit tangent vectors whose geodesic is contained in the core, and that it is mixing.
Abstract: For any rank-one convex projective manifold with a compact convex core, we prove that there exists a unique probability measure of maximal entropy on the set of unit tangent vectors whose geodesic is contained in the convex core, and that it is mixing. We use this to establish asymptotics for the number of closed geodesics. In order to construct the measure of maximal entropy, we develop a theory of Patterson--Sullivan densities for general rank-one convex projective manifolds. In particular, we establish a Hopf--Tsuji--Sullivan--Roblin dichotomy, and prove that, when it is finite, the measure on the unit tangent bundle induced by a Patterson--Sullivan density is mixing under the action of the geodesic flow.
3 citations
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TL;DR: In this article, the horizontal splitting of the vertical tangent bundle of a composite fibred manifold is studied, and the splitting defines the modified covariant differential and implies the special fashion of Lagrangian densities of field models on composite manifolds.
Abstract: In classical field theory, the composite fibred manifolds Y -> Z -> X provides the adequate mathematical formulation of gauge models with broken symmetries, e.g., the gauge gravitation theory. This work is devoted to connections on composite fibred manifolds. In particular, we get the horizontal splitting of the vertical tangent bundle of a composite fibred manifold, besides the familiar one of its tangent bundle. This splitting defines the modified covariant differential and implies the special fashion of Lagrangian densities of field models on composite manifolds. The spinor composite bundles are examined.
3 citations