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High Quality Content by WIKIPEDIA articles! In Riemannian geometry, a branch of mathematics, the unit tangent bundle of a Riemannian manifold (M, g), denoted by UT(M) or simply UTM, is the unit sphere bundle for the tangent bundle T(M). It is a fiber bundle over M whose fiber at each point is the unit sphere in the tangent bundle: mathrm{UT} (M) := coprod_{x in M} left{ v in mathrm{T}_{x} (M) left g_x(v,v) = 1 right. right}, where Tx(M) denotes the tangent space to M at x. Thus, elements of UT(M) are 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 pi : mathrm{UT} (M) to M, pi : (x, v) mapsto x, which takes each point of the bundle to its base point. The fiber 1(x) over each point x M is an (n 1)-sphere Sn 1, where n is the dimension of M.