Unit tangent bundle

In mathematics, the unit tangent bundle of a Finsler manifold ("M", || . ||), denoted by UT("M") or simply UT"M", is a fiber bundle over "M" given by the disjoint union

:$mathrm\{UT\}\; (M)\; :=\; coprod\_\{x\; in\; M\}\; left\{\; v\; in\; mathrm\{T\}\_\{x\}\; (M)\; left|\; |\; v\; |\_\{x\}\; =\; 1\; ight.\; ight\},$

where T_"x"("M") denotes the tangent space to "M" at "x". Thus, elements of UT("M") can be viewed as 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

:$pi\; :\; mathrm\{UT\}\; (M)\; o\; M,$:$pi\; :\; (x,\; v)\; mapsto\; x,$

which takes each point of the bundle to its base point.

If "M" is a finite-dimensional manifold of dimension "n", then the fiber "π"⁻¹("x") over a point "x" ∈ "M" is an ("n"−1)-sphere S^"n"−1, so the unit tangent bundle is a sphere bundle over "M" with fiber S^"n"−1. More precisely, the unit tangent bundle UT("M") is the unit sphere bundle for the tangent bundle T("M").

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 fibre "π"⁻¹("x") over "x" is then an infinite-dimensional sphere, and is certainly no longer a finite-dimensional sphere of dimension one less than that of "M".

Since a Riemannian manifold ("M", "g") is also a Finsler manifold with respect to the usual induced norm

:$|\; v\; |\_\{x\}\; :=\; sqrt\{g(v,\; v)\_\{x\; mbox\{\; for\; \}\; v\; in\; mathrm\{T\}\_\{x\}\; M,$

the unit tangent bundle UT("M") is also defined for Riemannian manifolds.

The unit tangent bundle is useful in the study of the geodesic flow.