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.

The Gauss-Bonnet theorem for vector bundles

Abstract: We give a short proof of the Gauss-Bonnet theorem for a real oriented Riemannian vector bundle E of even rank over a closed compact orientable manifold M This theorem reduces to the classical Gauss-Bonnet-Chern theorem in the special case when M is a Riemannian manifold and E is the tangent bundle of M endowed with the Levi-Civita connection The proof is based on an explicit geometric construction of the Thom class for 2-plane bundles

When are the tangent sphere bundles of a Riemannian manifold η -Einstein?

Abstract: We study the geometry of a tangent sphere bundle of a Riemannian manifold (M, g). Let M be an n-dimensional Riemannian manifold and TrM be the tangent bundle of M of constant radius r. The main theorem is that TrM equipped with the standard contact metric structure is η-Einstein if and only if M is a space of constant sectional curvature \({\frac{1}{r^2}}\) or \({\frac{n-2}{r^2}}\).

Generalised spin structures on 2-dimensional orbifolds

Abstract: Generalised spin structures, or $r$-spin structures, on a $2$-dimensional orbifold $\Sigma$ are $r$-fold fibrewise connected coverings (also called $r$\textsuperscript{th} roots) of its unit tangent bundle $ST\Sigma$. We investigate such structures on hyperbolic orbifolds. The conditions on $r$ for such structures to exist are given. The action of the diffeomorphism group of $\Sigma$ on the set of $r$-spin structures is described, and we determine the number of orbits under this action and their size. These results are then applied to describe the moduli space of taut contact circles on left-quotients of the $3$-dimensional geometry $\widetilde{\mathrm{SL}}_{2}$.

A totally geodesic property of Hopf vector fields

Abstract: We prove that the Hopf vector field is unique among geodesic unit vector fields on spheres such that the submanifold generated by the field is totally geodesic in the unit tangent bundle with Sasaki metric As an application, we give a new proof of stability (instability) of the Hopf vector field with respect to volume variation using standard approach from the theory of submanifolds and find exact boundaries for the sectional curvature of the Hopf vector field

Homogenisation for anisotropic kinetic random motions

Abstract: We introduce a class of kinetic and anisotropic random motions $(x_t^{\sigma},v_t^{\sigma})_{t \geq 0}$ on the unit tangent bundle $T^1 \mathcal M$ of a general Riemannian manifold $(\mathcal M,g)$, where $\sigma$ is a positive parameter quantifying the amount of noise affecting the dynamics. As the latter goes to infinity, we then show that the time rescaled process $(x_{\sigma^2 t}^{\sigma})_{t \geq 0}$ converges in law to an explicit anisotropic Brownian motion on $\mathcal M$. Our approach is essentially based on the strong mixing properties of the underlying velocity process and on rough paths techniques, allowing us to reduce the general case to its Euclidean analogue. Using these methods, we are able to recover a range of classical results.