Unit Tangent Bundle is Connected iff Manifold is Connected
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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