Orbit space

Let $G$ be a Lie group acting smoothly on a manifold $M$ (see smooth actions ).

Definition

The orbit space (or quotient space) is the set

of all orbits , equipped with the quotient topology for the canonical projection

Smooth structure in the free and proper case

In general, $M/G$ need not be a manifold. A standard sufficient condition is:

• If the action is free and proper , then $M/G$ carries a unique smooth manifold structure such that $\pi$ is a smooth submersion.

In this situation, each orbit is embedded and diffeomorphic to $G$, and $\pi$ exhibits $M$ as a principal homogeneous object for $G$ along the fibers (compare principal homogeneous spaces ).

Basic examples

• If the action is transitive , then $M/G$ is a single point.
• If $M=G$ acts on itself by left translation, then all orbits are all of $G$ and again $M/G$ is a point; if $G$ acts by conjugation, orbit spaces encode conjugacy classes and are typically singular.