Definition:Unit Tangent Bundle

Definition

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:

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

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Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$. Riemannian Metrics. Definitions