The definition of unit sphere bundle can easily accommodate Finsler manifolds as well.
The unit tangent bundle carries a variety of differential geometric structures.
This is given in terms of a tautological one-form, defined at a point u of UTM (a unit tangent vector of M) by where
Geometrically, this contact structure can be regarded as the distribution of (2n−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 UTM 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 UTM.
Equipped with this metric and contact form, UTM becomes a Sasakian manifold.