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In Riemannian geometry, the unit tangent bundle of a Riemannian manifold , denoted by TM, UT or simply UTM, is the unit sphere bundle for the tangent bundle T. It is a fiber bundle over M whose fiber at each point is the unit sphere in the tangent bundle:
where Tx denotes the tangent space to M at x. Thus, elements of UT are pairs , where x is some point of the manifold and v is some tangent direction 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 π over each point x ∈ M is an -sphere S, where n is the dimension of M. The unit tangent bundle is therefore a sphere bundle over M with fiber S.
The definition of unit sphere bundle can easily accommodate Finsler manifolds as well. Specifically, if M is a manifold equipped with a Finsler metric F : TM → R, then the unit sphere bundle is the subbundle of the tangent bundle whose fiber at x is the indicatrix of F: