# Definition:Metric Invariant under Group Action

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## Definition

Let $\struct {M, g}$ be a Riemannian manifold.

Let $G$ be a group.

Suppose $\forall \phi \in G$ the mapping $x \mapsto \phi \cdot x$ is an isometry.

Then the metric $g$ is said to be **invariant under $G$**.

## Sources

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