Definition:Left-Invariant Riemannian Metric

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.