 Unit tangent bundle

The unit tangent bundle of a Riemannian manifold $M$ is the sphere bundle of its tangent bundle. Its points are pairs $(x,v)$ with $x\in M$, $v\in T_{x}M$, and $\Vert v\Vert =1$.

The unit tangent bundle is a fundamental concept in differential geometry and is used in the study of manifolds, particularly in the context of differential geometry and geodesic flows. Given a smooth manifold \( M \), the unit tangent bundle, denoted as \( U(TM) \), consists of all unit tangent vectors at every point in \( M \).