Unit tangent bundle

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

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

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

$\pi : \mathrm{UT} (M) \to M,$

$\pi : (x, v) \mapsto x,$

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 Sⁿ⁻¹, so the unit tangent bundle is a sphere bundle over M with fiber Sⁿ⁻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

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