Unit tangent bundle

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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:

${\displaystyle \mathrm {UT} (M):=\coprod _{x\in M}\left\{v\in \mathrm {T} _{x}(M)\left|g_{x}(v,v)=1\right.\right\},}$

where T_x(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

${\displaystyle \pi :\mathrm {UT} (M)\to M,}$

${\displaystyle \pi :(x,v)\mapsto x,}$

which takes each point of the bundle to its base point. The fiber π⁻¹(x) over each point x ∈ M is an (n−1)-sphere Sⁿ⁻¹, where n is the dimension of M. The unit tangent bundle is therefore a sphere bundle over M with fiber Sⁿ⁻¹.

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:

${\displaystyle \mathrm {UT} _{x}(M)=\left\{v\in \mathrm {T} _{x}(M)\left|F(v)=1\right.\right\}.}$

If M is an infinite-dimensional manifold (for example, a Banach, Fréchet or Hilbert manifold), then UT(M) can still be thought of as the unit sphere bundle for the tangent bundle T(M), but the fiber π⁻¹(x) over x is then the infinite-dimensional unit sphere in the tangent space.