I think lecture we define the notion of a homogeneous space, which is a smooth manifold diffeomorphic to a quotient space $G/H$ where $G$ is a Lie group and $H$ is a Lie subgroup. (For this to make the sense, the quotient better be a smooth manifold—we prove this first!)

We then 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:$$S^n \cong \mathrm{O}(n)\big/ \mathrm{O}(n-1),$$and$$S^{2n-1} \cong \mathrm{SU}(n)\big/ \mathrm{SU}(n-1),$$

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

Next lecture we move onto the second part of the course, and begin our discussion of vector bundles.

Comments and questions?