Definition:Left-Invariant Riemannian Metric
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Definition
Let $G$ be a Lie group.
Let $\struct {G, g}$ be a Riemannian manifold.
Let $L_\phi : G \to G$ be a left translation.
Suppose $g$ is invariant under all left translations:
- $\forall \phi \in G : {L_\phi}^* g = g$
where $*$ denotes the pullback of $g$ by $L_\phi$.
Then $g$ is said to be left-invariant.
Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 3$: Model Riemannian Manifolds. Invariant Metrics on Lie Groups