# 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

Created on August 22, 2011 16:22:50 by Urs Schreiber (82.113.99.25)