Unit tangent bundle

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 :\theta_u(v) = g(u,\pi_* v)\, where \pi_* is the pushforward along π of the vector v ∈ TuUTM. 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. 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 :\theta_u(v) = g_u(u,\pi_*v)\, where gu is the fundamental tensor (the hessian of the Finsler metric). Geometrically, the associated distribution of hyperplanes at the point u ∈ UTxM is the inverse image under π* of the tangent hyperplane to the unit sphere in TxM at u. The volume form θ∧dθn−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 UTM by :\int_{UTM} f\,d\mu = \int_M dV(p) \int_{UT_pM} \left.f\right|_{UT_pM}\,d\mu_p where dV is the volume element on M, and μp is the standard rotationally-invariant Borel measure on the Euclidean sphere UTpM. The Levi-Civita connection of M gives rise to a splitting of the tangent bundle :T(UTM) = H\oplus V into a vertical space V = kerπ* and horizontal space H on which π* is a linear isomorphism at each point of UTM. This splitting induces a metric on UTM by declaring that this splitting be an orthogonal direct sum, and defining the metric on H by the pullback: :g_H(v,w) = g(v,w),\quad v,w\in H and defining the metric on V as the induced metric from the embedding of the fiber UTxM into the Euclidean space TxM. Equipped with this metric and contact form, UTM becomes a Sasakian manifold. ==Bibliography==