geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
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.
Every compact Lie group admits a bi-invariant metric.
(Milnor 76, Cor. 1.4, reviewed in Gallier 18, Prop. 17.6)
(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)
In discrete metric spaces:
On invariant metrics on Lie groups:
Review:
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.