Unit Tangent Bundle is Connected iff Manifold is Connected

Theorem

Let $\struct {M,g}$ be a Riemannian manifold of dimension $n > 1$.

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

Then $UTM$ is connected if and only if $M$ is connected.

Proof

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Sources

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