Unit sphere bundle
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:
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
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:
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.
The unit tangent bundle carries a variety of differential geometric structures. The metric on M induces a contact structure on UTM. This is given in terms of a tautological one-form, defined at a point u of UTM (a unit tangent vector of M) by
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