# nLab invariant metric

Contents

### Context

#### Riemannian geometry

Riemannian geometry

## Applications

#### Representation theory

representation theory

geometric representation theory

# Contents

## Idea

### For Lie groups

A Riemannian metric on a Lie group $G$ is called

• left invariant if for each element $g \in G$ the left multiplication diffeomorphism $g \cdot (-) \;\colon\; G \to G$ is an isometry;

• right invariant if for each element $g \in G$ the right multiplication diffeomorphism $(-) \cdot g \;\colon\; G \to G$ is an isometry;

• bi-invariant if it is both left- and right-invariant.

More generally, a Riemannian metric on a differentiable/smooth manifold euqipped with a differentiable/smooth $G$-action (a G-manifold) is called invariant if all elements of $G$ act as isometries.

## Properties

### Existence on Lie groups

###### Proposition

Every compact Lie group admits a bi-invariant metric.

(Milnor 76, Cor. 1.4, reviewed in Gallier 18, Prop. 17.6)

### Existence on $G$-Manifolds

###### Proposition

(existence of $G$-invariant Riemannian metric on G-manifolds)

Let $X$ be a smooth manifold, $G$ a compact Lie group and $\rho \;\colon\; G \times X \to X$ a proper action by diffeomorphisms.

Then there exists a Riemannian metric on $X$ which is invariant with respect to the $G$-action, hence such that all elements of $G$ act by isometries.

## Examples

In discrete metric spaces:

On invariant metrics on Lie groups:

Review:

• Jean Gallier, Section 17.1, 17.2 in: Metrics, Connections, and Curvatureon Lie Groups, Chapter 17 in: Advanced Geometric Methods in Computer Science, Lecture notes 2018 (pdf, web)

On invariant metric on G-manifolds:

Review: