Definition:Transitive Group Action on Fibers

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

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

Let $p,q \in \tilde M$ be base points such that $\map \pi p = \map \pi q$.

Let $G$ be a group.

Suppose:


 * $\exists \phi \in G : \phi \cdot p = q$

Then the group action is said to be transitive on fibers.