The Quotient Manifold Theorem

In this section we prove that smooth, free, and proper group actions always yield smooth manifolds as orbit spaces. The basic idea of the proof is that if G acts smoothly, freely and properly on M, the set of orbits form a foliation of M whose leaves are embedded submanifolds diffeomorphic to G. Flat charts for the foliation can then be used to construct coordinates on the orbit space.

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