It is known that Grassmannian as a homogeneous space https://en.wikipedia.org/wiki/Grassmannian#The_Grassmannian_as_a_homogeneous_space gives that: $$ {{Gr_{\mathbb{R}}(r, n) {{=}} O(n)/(O(r) \times O(n − r))}}. $$ If the underlying field is $\mathbb{R}$ or $\mathbb{C}$ and $GL(V)$ is considered as a Lie group, then this construction makes the Grassmannian into a smooth manifold. It also becomes possible to use other groups to make this construction. To do this, fix an inner product on $V$. Over $\mathbb{R}$, one replaces $GL(V)$ by the orthogonal group $O(V)$, and by restricting to orthonormal frames, one gets the relation above.
My questions are that:
- do we have: $$ {{Gr_{\mathbb{C}}(r, n) {{=}} U(n)/(U(r) \times U(n − r))}}? $$
- What are some good refs on studying the topological (e.g., cohomology) and geometrical properties of these spaces: $Gr_{\mathbb{R}}(r, n)$ and $Gr_{\mathbb{C}}(r, n)$?