The space of complex structure compatible with metric

Question

Why the space of all complex structure on a $2n$-dimensional vector space which compatible with a positive definite metric is diffeomorphic to $ O(2n)/U(n) $ ?

Answer 1

$O(2n)$ acts transitively on the space $C$ of compatible complex structures, by sending a complex structure $J$ to $OJO^{-1}$, it's easy to check this is a compatible complex structure. To show transitivity use block-diagonalisation for real matrices to show it any complex structure can be conjugated to a standard one in block-diagonal form. Compatibility just means that the complex structure is skew-symmetric with respect to the metric, so the conjugation can be done by an orthogonal matrix.

Now fix a compatible complex structure $J$ (say the standard one on $\mathbb{R}^{2n}$). The linear maps preserving both the metric and the complex structure will be a copy of $U(n)$ inside $O(2n)$. So the stabiliser of $J$ is $U(n)$ and as $O(2n)$ acts transitively, we have a bijection $O(2n)/U(n) \cong C$ induced by $O \mapsto OJO^{-1}$.

$C$ is a manifold (it's a submanifold of $GL(2n,\mathbb{R})$) and $O(2n)$ acts smoothly and transitively on it, so this bijection is a diffeomorphism.