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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: For every finite collection of curves on a surface, an associated (semi-)norm on the first homology group of the surface is defined in this article, where the authors give an interpretation of these points in terms of certain coorien-tations of the original collection of geodesics.
Abstract: For every finite collection of curves on a surface, we define an associated (semi-)norm on the first homology group of the surface. The unit ball of the dual norm is the convex hull of finitely many integer points. We give an interpretation of these points in terms of certain coorien-tations of the original collection of curves. Our main result is a classification statement: when the surface has constant curvature and the curves are geodesics, integer points in the interior of the unit ball of the dual norm classify isotopy classes of Birkhoff sections for the geodesic flow (on the unit tangent bundle to the surface) whose boundary is the symmetric lift of the collection of geodesics. These Birkhoff sections also yield numerous open-book decompositions of the unit tangent bundle.
1 citations
1 citations
Journal Article•
TL;DR: In this article, it was shown that a 3-dimensional Ricci-parallel Riemannian manifold is locally symmetric if and only if it is of dimension 2 and constant curvature 1.
Abstract: We investigate the tangent sphere bundle of a 2-dimensional Riemannian manifold M with the natural Riemannian structure g in the two classes, given by A.Gray([3]), including Ricci-parallel Riemannian manifolds. Also, we prove that is conformally flat if and only if is locally symmetric. The motivation of this paper are a fact that a 3-dimensional Ricci-parallel Riemannian manifold is locally symmetric and a result([2]) that the natural Riemannian structure of the tangent sphere bundle of a Riemannian manifold is locally symmetric if and only if either the baes manifold is flat or is of dimension 2 and constant curvature 1.
1 citations
TL;DR: In this paper, the index and co-index of the twisted tangent bundle of projective spaces were determined, and the stabilty of these indices was discussed, as well as the set of integers that can be realized as the stable coindex of a vector bundle over the projective space.
Abstract: We determine the index and co-index of the twisted tangent bundle of projective spaces. We also discuss the stabilty of them, and determine the set of integers that can be realized as the stable co-index of a vector bundle over the projective space.
1 citations
Posted Content•
TL;DR: In this article, the authors studied Tonelli Lagrangian systems on the 2-torus in energy levels above the strict critical value and analyzed the structure of global minimizers in the spirit of Morse, Hedlund and Bangert.
Abstract: We study Tonelli Lagrangian systems on the 2-torus in energy levels above Ma\~n\'e's strict critical value and analyize the structure of global minimizers in the spirit of Morse, Hedlund and Bangert. In the case where the topological entropy of the Euler-Lagrange flow on the fixed energy level vanishes, we show that there are invariant tori for all rotation vectors indicating integrable-like behavior on a large scale. On the other hand, using a construction of Katok, we give examples of reversible Finsler geodesic flows with vanishing topological entropy, but having ergodic components of positive measure in the unit tangent bundle.
1 citations