Unit tangent bundle
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In mathematics, the unit tangent bundle of a Riemannian manifold (M, g), denoted by UT(M) or simply UTM, is a fiber bundle given by the disjoint union
where Tx(M) denotes the tangent space to M at x. Thus, elements of UT(M) can be viewed as pairs (x, v), where x is some point of the manifold and v is some tangent direction (of unit length) to the manifold at x. The unit tangent bundle is equipped with a natural projection
which takes each point of the bundle to its base point. The fiber over a point x in M is an (n−1)-sphere, where n is the dimension of M, so the unit tangent bundle is sphere bundle over M. More precisely, the unit tangent bundle UT(M) is the unit sphere bundle for the tangent bundle T(M).
The unit tangent bundle is useful in the study of the geodesic flow.
