Definition:Vertical Group Action

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

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

Let $G$ be a group acting on $\tilde M$.

Suppose each element of $G$ takes each fiber to itself:


 * $\forall \phi \in G : \forall p \in \tilde M : \map \pi {\phi \cdot p} = \map \pi p$

Then the action is called to be vertical.