Construction: quotient manifold P/G for a free proper action - Imported Knowlpedia Content

Let $G$ be a Lie group acting smoothly on a smooth manifold $P$ . Write the action as a right action $(p,g)\mapsto p\cdot g$ .

Construction (Quotient manifold and principal bundle projection)

Assume the action is:

Free: $p\cdot g=p$ implies $g=e$ .
Proper: the map $P\times G\to P\times P$ , $(p,g)\mapsto (p,p\cdot g)$ is proper.

Then:

The orbit space $P/G$ carries a unique smooth manifold structure such that the projection $\pi:P\to P/G$ is a smooth submersion.

With this smooth structure, $\pi:P\to P/G$ is a principal G-bundle with the given right action.

For each $p\in P$ , the vertical tangent space is $\ker(d\pi)_p \;=\; \{X^\#_p: X\in\mathfrak g\},$ where $X^\#$ is the fundamental vector field defined using the right-action convention.

This construction is the standard way principal bundles arise from symmetry.

Examples

Trivial example. If $P=M\times G$ with right action $(x,h)\cdot g=(x,hg)$ , the action is free and proper and the quotient is canonically $P/G\cong M$ .

Hopf fibration viewpoint. The standard free action of $S^1$ on $S^{2n+1}\subset\mathbb C^{n+1}$ by scalar multiplication is free and proper; the quotient is complex projective space $S^{2n+1}/S^1\cong \mathbb{CP}^n$ , and the projection is a principal $S^1$ -bundle.

Non-example (failure of properness). If a noncompact group acts freely but not properly, the orbit space may fail to be Hausdorff, violating the manifold convention in the manifold hypotheses.