# Definition:Transitive Group Action on Fibers

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## 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**.

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## Sources

- 2018: John M. Lee:
*Introduction to Riemannian Manifolds*(2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Methods for Constructing Riemannian Metrics