# Definition:Bi-Invariant Riemannian Metric

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## Definition

Let $G$ be a Lie group.

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

Suppose $g$ is both left-invariant and right-invariant.

Then $g$ is said to be **bi-invariant**.

## Sources

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