Definition:Right-Invariant Riemannian Metric

Definition

Let $G$ be a Lie group.

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

Suppose $g$ is invariant under all right translations $R_\phi : G \to G$:

$\forall \phi \in G : {R_\phi}^* g = g$

where $*$ denotes the pullback of $g$ by $R_\phi$.

Then $g$ is said to be right-invariant.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 3$: Model Riemannian Manifolds. Invariant Metrics on Lie Groups