nLab isometry group

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

• The isometry group of Minkowski spacetime is the Poincaré group.

• The isometry group of anti de Sitter spacetime is the anti de Sitter group.

• The isometry group of de Sitter spacetime is the de Sitter group.

Special subgroups of isometry groups:

• finite rotation group

• wallpaper group

Related concepts

• isometry

• superisometry group