nLab isometry group

Context

Riemannian geometry

Riemannian geometry

Applications

Differential geometry

differential geometry

synthetic differential geometry

Contents

Definition

For $(X,g)$ a Riemannian manifold or pseudo-Riemannian manifold its isometry group

$Iso(X,g) \subset Diff(X)$

is that subgroup of the group of all diffeomorphisms $\phi : X \to X$ that are isometries: which preserve the metric $g$ in that

$\phi^* g = g \,.$

Properties

Killing vectors

The Lie algebra of $Iso(X,g)$ is spanned by the Killing vectors of $(X,g)$.

Examples

Revised on March 25, 2015 12:42:29 by Urs Schreiber (195.113.30.252)