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 $G$.
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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