Smooth Vector Field has Unique Smooth Horizontal Lift
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Theorem
Let $\tilde M, M$ be smooth manifolds.
Let $\pi : \tilde M \to M$ be a smooth submersion.
Let $\tilde g$ be a Riemannian metric on $\tilde M$.
Let $W$ be a smooth vector field on $M$.
Then $W$ has the unique smooth horizontal lift to $\tilde M$.
Proof
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Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Methods for Constructing Riemannian Metrics