In this lecture we state the Quotient Manifold Theorem, which states that if a Lie group $G$ acts properly and freely on a smooth manifold $M$ then the orbit space $M/G$. The proof of the Quotient Manifold Theorem will come next week, after we have proved the Frobenius Theorem.

A useful application of the Quotient Manifold Theorem is the following: if $H$ is a closed Lie subgroup of $G$ then the space $G/H$ of left cosets is a smooth manifold.

A homogeneous space $M$ is a smooth manifold which is diffeomorphic to such a coset space $G/H$.

We show that “many” manifolds are homogeneous spaces: if $M$ is any smooth manifold admitting a transitive action by a Lie group $G$, then for an appropriate choice of $H$ one has $M \cong G/H$.

Examples of this phenomena include:$$S^m \cong \mathrm{O}(m)\big/ \mathrm{O}(m-1),$$and$$S^{2m-1} \cong \mathrm{SU}(m)\big/ \mathrm{SU}(m-1)$$

(the latter shows that $S^3$ admits the structure of a Lie group).