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.