Characterization of Unit Tangent Bundle

Theorem

Let $\struct {M, g}$ be a Riemannian manifold with or without boundary.

Let $TM$ be the tangent bundle of $M$.

Let $UTM$ be the unit tangent bundle of $M$.

Let $\pi : UTM \to M$ be the canonical projection.

Then $UTM$ is a smooth, properly embedded submanifold of the codimension-$1$ with boundary $\map \partial {UTM} = \map {\pi^{-1}} {\partial M}$ in $TM$.