nLab
invariant metric

Contents

Context

Riemannian geometry

Representation theory

Contents

Idea

For Lie groups

A Riemannian metric on a Lie group GG is called

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

  • right invariant if for each element gGg \in G the right multiplication diffeomorphism ()g:GG(-) \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 GG-action (a G-manifold) is called invariant if all elements of GG 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 GG-Manifolds

Proposition

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

Let XX be a smooth manifold, GG a compact Lie group and ρ:G×XX\rho \;\colon\; G \times X \to X a proper action by diffeomorphisms.

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

(Bredon 72, VI Theorem 2.1, see also Ziller 13, Theorem 3.0.2)

Examples

In discrete metric spaces:

References

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:

Last revised on April 18, 2021 at 13:36:22. See the history of this page for a list of all contributions to it.