Proper Lie group action
= Proper Lie group action
= Proper group action
{synonym}
A Lie group action on $M$ is proper if $(g,x)\mapsto(gx,x)$ from $G\times M$ to $M\times M$ is a proper map. A free proper smooth action has a smooth quotient manifold, with the quotient map a submersion and principal bundle. Proper actions with stabilizers can have orbit-type strata and singular quotients. These qualifications are necessary before treating an orbit space as a smooth phase space.