nLab invariant metric

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.

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

Examples

In discrete metric spaces:

• Cayley distance, Kendall distance

Related concepts

• invariant differential form

References

On invariant metrics on Lie groups:

• John Milnor, Curvatures of left invariant metrics on Lie groups, Advances in Mathematics, Volume 21, Issue 3, September 1976, Pages 293-329 (doi:10.1016/S0001-8708(76)80002-3)

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:

• Glen Bredon, Introduction to compact transformation groups, Academic Press 1972 (pdf, ISBN:9780080873596)

Review:

• Wolfgang Ziller, Group actions, 2013 (pdf, pdf)