 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".