Unit tangent bundle
In Riemannian geometry, the unit tangent bundle of a Riemannian manifold (M, g), denoted by T1M, UT(M), UTM, or SM 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 space: U T ( M ) := ∐ x ∈ M { v ∈ T x ( M ) | g x ( v , v ) = 1 } , {\displaystyle \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.