# Unit tangent bundle

In Riemannian geometry, the **unit tangent bundle** of a Riemannian manifold (*M*, *g*), denoted by T^{1}*M*, UT(*M*) or simply UT*M*, 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 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

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 **S**^{n−1}, where *n* is the dimension of *M*. The unit tangent bundle is therefore a sphere bundle over *M* with fiber **S**^{n−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* : T*M* → **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.

## Structures

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

where is the pushforward along π of the vector *v* ∈ T_{u}UT*M*.

Geometrically, this contact structure can be regarded as the distribution of (2*n*−2)-planes which, at the unit vector *u*, is the pullback of the orthogonal complement of *u* in the tangent space of *M*. This is a contact structure, for the fiber of UT*M* is obviously an integral manifold (the vertical bundle is everywhere in the kernel of θ), and the remaining tangent directions are filled out by moving up the fiber of UT*M*. Thus the maximal integral manifold of θ is (an open set of) *M* itself.

On a Finsler manifold, the contact form is defined by the analogous formula

where *g*_{u} is the fundamental tensor (the hessian of the Finsler metric). Geometrically, the associated distribution of hyperplanes at the point *u* ∈ UT_{x}*M* is the inverse image under π_{*} of the tangent hyperplane to the unit sphere in T_{x}*M* at *u*.

The volume form θ∧*d*θ^{n−1} defines a measure on *M*, known as the **kinematic measure**, or **Liouville measure**, that is invariant under the geodesic flow of *M*. As a Radon measure, the kinematic measure μ is defined on compactly supported continuous functions *ƒ* on UT*M* by

where d*V* is the volume element on *M*, and μ_{p} is the standard rotationally-invariant Borel measure on the Euclidean sphere UT_{p}*M*.

The Levi-Civita connection of *M* gives rise to a splitting of the tangent bundle

into a vertical space *V* = kerπ_{*} and horizontal space *H* on which π_{*} is a linear isomorphism at each point of UT*M*. This splitting induces a metric on UT*M* by declaring that this splitting be an orthogonal direct sum, and defining the metric on *H* by the pullback:

and defining the metric on *V* as the induced metric from the embedding of the fiber UT_{x}*M* into the Euclidean space T_{x}*M*. Equipped with this metric and contact form, UT*M* becomes a Sasakian manifold.