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.

A Kähler Einstein structure on the tangent bundle of a space form

Abstract: We obtain a Kahler Einstein structure on the tangent bundle of a Riemannian manifold of constant negative curvature. Moreover, the holomorphic sectional curvature of this Kahler Einstein structure is constant. Similar results are obtained for a tube around zero section in the tangent bundle, in the case of the Riemannian manifolds of constant positive curvature.

Spherical means on compact Riemannian manifolds of negative curvature

Abstract: We show that the spherical mean of functions on the unit tangent bundle of a compact manifold of negative curvature converges to a measure containing a vast amount of information about the asymptotic geometry of those manifolds. This measure is related to the unique invariant measure for the strong unstable foliation, as well as the Patterson-Sullivan measure at infinity. It turns out to be invariant under the geodesic flow if and only if the mean curvature of the horospheres is constant. We use this measure in the study of rigidity problems.

Limiting distributions of curves under geodesic flow on hyperbolic manifolds

Abstract: We consider the evolution of a compact segment of an analytic curve on the unit tangent bundle of a finite volume hyperbolic n-manifold under the geodesic flow. Suppose that the curve is not contained in a stable leaf of the flow. It is shown that under the geodesic flow, the normalized parameter measure on the curve gets asymptotically equidistributed with respect to the normalized natural Riemannian measure on the unit tangent bundle of a closed totally geodesically immersed submanifold. Moreover, if this immersed submanifold is a proper subset, then a lift of the curve to the universal covering space T 1 (H n ) is mapped into a proper subsphere of the ideal boundary sphere @H n under the visual map. This proper subsphere can be realized as the ideal boundary of an isometrically embedded hyperbolic subspace in H n covering the closed immersed submanifold. In particular, if the visual map does not send a lift of the curve into a proper subsphere of @H n , then under the geodesic flow the curve gets asymptotically equidistributed on the unit tangent bun- dle of the manifold with respect to the normalized natural Rie- mannian measure. The proof uses dynamical properties of unipotent flows on finite volume homogeneous spaces of SO(n,1).

An intrinsic measure for submanifolds in stratified groups

Abstract: For each submanifold of a stratified group, we find a number and a measure only depending on its tangent bundle, the grading and the fixed Riemannian metric. In two step stratified groups, we show that such number and measure coincide with the Hausdorff dimension and with the spherical Hausdorff measure of the submanifold with respect to the Carnot-Caratheodory distance, respectively. Our main technical tool is an intrinsic blow-up at points of maximum degree. We also show that the intrinsic tangent cone to the submanifold at these points is always a subgroup. Finally, by direct computations in the Engel group, we show how our results can be extended to higher step stratified groups, provided the submanifold is sufficiently regular.

Effective Circle Count for Apollonian Packings and Closed Horospheres

Abstract: The main result of this paper is an effective count for Apollonian circle packings that are either bounded or contain two parallel lines. We obtain this by proving an effective equidistribution of closed horospheres in the unit tangent bundle of a geometrically finite hyperbolic 3-manifold, whose fundamental group has critical exponent bigger than 1. We also discuss applications to affine sieves. Analogous results for surfaces are treated as well.