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.