 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 formula_6 is the pushforward along π of the vector "v" ∈ T"u"UT"M".