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.

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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