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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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18 citations
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TL;DR: In this article, Oh and Shah showed that the skinning measures, when finite, of the equidistant hypersurfaces to C equidistribute to the Bowen-Margulis measure on the unit tangent bundle of M, under additional assumptions on the rate of mixing.
Abstract: Let C be a locally convex subset of a negatively curved Riemannian manifold M. We define the skinning measure on the outer unit normal bundle to C in M by pulling back the Patterson-Sullivan measures at infinity, and give a finiteness result of skinning measures, generalising the work of Oh and Shah, with different methods. We prove that the skinning measures, when finite, of the equidistant hypersurfaces to C equidistribute to the Bowen-Margulis measure on the unit tangent bundle of M, assuming only that the Bowen-Margulis measure is finite and mixing for the geodesic flow. Under additional assumptions on the rate of mixing, we give a control on the rate of equidistribution.
18 citations
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TL;DR: In this paper, it was shown that the flat product metric on a Riemannian manifold with boundary is scattering rigid, where the scattering data (loosely speaking) of the manifold is map $S:U+\partial M to U^-partial M$ from unit vectors $V$ at the boundary that point inward to unit vectors at the boundaries that point outwards.
Abstract: We prove that the flat product metric on $D^n\times S^1$ is scattering rigid where $D^n$ is the unit ball in $\R^n$ and $n\geq 2$. The scattering data (loosely speaking) of a Riemannian manifold with boundary is map $S:U^+\partial M\to U^-\partial M$ from unit vectors $V$ at the boundary that point inward to unit vectors at the boundary that point outwards. The map (where defined) takes $V$ to $\gamma'_V(T_0)$ where $\gamma_V$ is the unit speed geodesic determined by $V$ and $T_0$ is the first positive value of $t$ (when it exists) such that $\gamma_V(t)$ again lies in the boundary. We show that any other Riemannian manifold $(M,\partial M,g)$ with boundary $\partial M$ isometric to $\partial(D^n\times S^1)$ and with the same scattering data must be isometric to $D^n\times S^1$. This is the first scattering rigidity result for a manifold that has a trapped geodesic. The main issue is to show that the unit vectors tangent to trapped geodesics in $(M,\partial M,g)$ have measure 0 in the unit tangent bundle.
17 citations
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TL;DR: In this paper, it was shown that one of these two Gauss-Bonnet formulas can be generalized to an index formula for the bundle homomorphism under the assumption that the bundle admits only certain kinds of generic singularities.
Abstract: In a previous work, the authors introduced the notion of ‘coherent tangent bundle’, which is useful for giving a treatment of singularities of smooth maps without ambient spaces. Two different types of Gauss–Bonnet formulas on coherent tangent bundles on $2$-dimensional manifolds were proven, and several applications to surface theory were given. Let $M^n$ ($n\ge 2$) be an oriented compact $n$-manifold without boundary and $TM^n$ its tangent bundle. Let $\mathcal{E}$ be a vector bundle of rank $n$ over $M^n$, and $\phi:TM^n\to \mathcal{E}$ an oriented vector bundle homomorphism. In this paper, we show that one of these two Gauss–Bonnet formulas can be generalized to an index formula for the bundle homomorphism $\phi$ under the assumption that $\phi$ admits only certain kinds of generic singularities. We shall give several applications to hypersurface theory. Moreover, as an application for intrinsic geometry, we also give a characterization of the class of positive semi-definite metrics (called Kossowski metrics) which can be realized as the induced metrics of the coherent tangent bundles.
17 citations
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TL;DR: In this article, the Lagrange and Markov dynamical spectra associated to a geodesic flow on a surface of negative curvature have been studied and shown to have non-empty interior for a large set of real functions on the unit tangent bundle and for typical metrics with negative curvatures and finite volume.
Abstract: We consider the Lagrange and the Markov dynamical spectra associated to a geodesic flow on a surface of negative curvature. We show that for a large set of real functions on the unit tangent bundle and for typical metrics with negative curvature and finite volume, both the Lagrange and the Markov dynamical spectra have non-empty interior.
17 citations