Definition:Unit Tangent Bundle

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Let $\struct {M, g}$ be a Riemannian manifold.

Let $T M$ be the tangent bundle of $\struct {M, g}$.

Let $p \in M$ be a point in $M$.

Let $\tuple {p, v}$ be a geometric tangent vector.

The unit tangent bundle $UTM \subseteq TM$ is the subset of unit vectors:

$UTM = \set {\tuple {p, v} \in TM : \norm {v}_g = 1}$