nLab
proper action

Definition

The action of a topological group $G$ on a topological space $X$ is called a proper action if the continuous function

$$\array{ G \times X &\longrightarrow& X \times X \\ (g,x) &\mapsto& (g\cdot x, x) }$$

is a proper continuous function.

Examples

Lie groups actions on smooth manifolds

Proposition

Let $X$ be a smooth manifold and let $G$ be a compact Lie group. Then every smooth action of $G$ on $X$ is proper.

(e.g. Lee 12, Cor. 7.2)

For more see at equivariant differential topology.

Related concepts

• properly discontinuous action

• free action, semi-free action, transitive action, regular action

• equivariant differential topology

References

For instance

• Marja Kankaanrinta, Def. 2.1 in Equivariant collaring, tubular neighbourhood and gluing theorems for proper Lie group actions, Algebr. Geom. Topol. Volume 7, Number 1 (2007), 1-27 (euclid:agt/1513796653)

• John Lee, around p. 147 of: Introduction to Smooth Manifolds, Springer 2012 (doi:10.1007/978-1-4419-9982-5, pdf)

See also

• J. Lee, MO comment, Oct 2014