Definition:Metric Invariant under Group Action

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 $G$.

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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