I'm reading through these lecture notes and trying to apply the following
Theorem: A Lie group $G$ acts globally and transitively on a manifold $M$ if and only if $M \cong G/H$ is isomorphic to the homogeneous space obtained by quotienting $G$ by the isotropy subgroup $H = G_x$ of any designated point $x \in M$.
I've had some success with simple examples, but have come unstuck on others.
In Exercise 2.19, amongst other things, I'm asked to "Determine the homogeneous space structures of the sphere $S^2$ induced by the transitive group action of $\mathrm{GL}(3)$."
The problem is that the general linear transformations do not preserve the sphere. There are lots of general linear transformations that take points from on sphere to points off the sphere. It was easy to prove that $S^2 \cong \mathrm{SO}(3)/\mathrm{SO}(2)$ because when $\mathrm{SO}(3)$ acts on $\mathrm{S}^2$ it send points of the sphere to points on the sphere. I can't see how $\mathrm{GL}(3)$ acts transitively on $S^2$. The subgroup preserving the sphere seems to be $\mathrm O(3)$, and that would just give $S^2 \cong \mathrm{O}(3)/\mathrm{O}(2)$.