Definition:Horizontal Lift

Definition
Let $\tilde M$, $M$ be smooth manifolds.

Let $\tilde X$, $X$ be vector fields on $\tilde M$ and $M$ respectively.

Let $\pi : \tilde M \to M$ be a smooth submersion.

Let $\tilde X$ be a horizontal vector field.

Suppose $\tilde X$ is $\pi$-related to $X$:


 * $\forall x \in \tilde M : \map {\d \pi_x} {\tilde X_x} = X_{\map \pi x}$

Then $\tilde X$ is called a horizontal lift of $X$.