I want to obtain the Fubini-Study metric of $\mathbb{H}P^n$ in affine coordinates, similar to this question but here of the quaternionic projective space.
How exactly is the Fubini-Study metric defined on $\mathbb{H}P^n$? Every literature I consult says: "We do it simlarly as in the complex case". What does that even mean?
How do we get the Fubini-Study metric in terms of affine coordinates, where $\mathbb{H}P^n$ is of course considered as a real manifold?
I appreciate both, literature and/or explaination which might help me to get on the right track. Thank you! :)
