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.

Characterization of Geodesic Flows on T 2 with and without Positive Topological Entropy

Abstract: In the present work we consider the behavior of the geodesic flow on the unit tangent bundle of the 2-torus T2 for an arbitrary Riemannian metric. A natural non-negative quantity which measures the complexity of the geodesic flow is the topological entropy. In particular, positive topological entropy implies chaotic behavior on an invariant set in the phase space of positive Hausdorff-dimension (horseshoe). We show that in the case of zero topological entropy the flow has properties similar to integrable systems. In particular, there exists a non-trivial continuous constant of motion which measures the direction of geodesics lifted onto the universal covering \({\mathbb{R}^{2}}\) . Furthermore, those geodesics travel in strips bounded by Euclidean lines. Moreover, we derive necessary and sufficient conditions for vanishing topological entropy involving intersection properties of single geodesics on T2.

Entropy estimates for geodesic flows

Abstract: Let M be a compact Riemannian manifold of (variable) negative curvature. Let h be the topological entropy and h μ the measure entropy for the geodesic flow on the unit tangent bundle to M . Estimates for h and h μ in terms of the ‘geometry’ of M are derived. Connections with and applications to other geometric questions are discussed.

The Boy–Gauss–Bonnet Theorems for C∞-Singular Surfaces with Limiting Tangent Bundle

Abstract: Using C∞-singularity theory, we describe natural conditions on nonimmersivemappings of a compact orientable 2-manifold into Euclidean 3-space (e.g., theexistence of a global limiting tangent bundle). These conditions are weaker thanthose of caustic theory and imply that the image is a CW-complex. In general, the domainmanifold is not homeomorphic to the image CW-complex. We then adapt classical methods to this setting. (This adaptation is simplified by freely moving betweenthe Gauss, Cartan, and Kozul formulations for surface theory.) As an application,we show that part of the extrinsic topology of the image is determined by the firstfundamental form, i.e., a global `theorem egregium'.

Chern classes of the tangent bundle on the Hilbert scheme of points on the affine plane

Abstract: SAMUEL BOISSI`EREAbstract. The cohomology of the Hilbert schemes of points on smooth pro-jective surfaces can be approached both with vertex algebra tools and equi-variant tools. Using the first tool, we study the existence and the structure ofuniversal formulas for the Chern classes of the tangent bundle over the Hilbertscheme of points on a projective surface. The second tool leads then to nicegenerating formulas in the particular case of the Hilbert scheme of points onthe affine plane.

Lagrangian submanifolds and the Euler-Lagrange equations in higher-order mechanics

Abstract: A direct construction of the Euler-Lagrange equations in higher-order mechanics as a submanifold of a higher-order tangent bundle is given, starting from the Lagrangian submanifold defined by the Lagrangian function. This construction uses higher-order tangent bundle geometry, derives the Euler-Lagrange equations as the constraint equations of a submanifold, and makes no assumptions about the regularity of the Lagrangian.