\begin{definition}[Definition:Horizontal Lift]
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 $X$ in the following way:
:$\forall x \in \tilde M : \map {\d \pi_x} {\tilde X_x} = X_{\map \pi x}$
Then $\tilde X$ is called a '''horizontal lift of $X$'''.
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\end{definition}