Construction: quotient manifold P/G for a free proper action

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:

1. The orbit space $P/G$ carries a unique smooth manifold structure such that the projection

   $$\pi:P\to P/G$$

   is a smooth submersion.

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

3. 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

1. 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 $P/G\cong M$.

2. 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.

3. 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 .