Smooth Vector Field as Sum of Smooth Horizontal and Vertical Vector Fields

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 $\tilde M$.

Then $W$ can be uniquely decomposed as a sum:

$W = W^H + W^V$

where $W^H$ and $W^V$ are smooth horizontal and vertical vector fields.

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