I am learning some symmetric spaces, and I tried to look at some example, including Grassmannian (say, $p$-dimensional subspaces in $\mathbb{C}^{p+q}$) and the projective space $\mathbb{CP}^n$. I follow Helgason, therefore I wish to describe them in terms of Lie groups. My problem is:
How to determine the isometry group of a manifold whose geometric description is known?
Let me elaborate more. For $\mathbb{CP}^n$, I understand that the special unitary group $SU(n+1)$ acts transitively on it. Also, I know the isotropy group for a certain point is $U(n)$. It is tempting now to claim $\mathbb{CP}^n = SU(n+1)/U(n)$. But I have not checked that $SU(n+1)$ is the isometry group!
Similar problems arise when I try Grassmannians, Lagrangian Grassmannians, Orthogonal Grassmannians, etc. I usually can find a group acting well on these spaces but I cannot determine whether that group is the isometry group.
Anyone can help? Thanks a lot!
It usually goes the other way around.
Take the real Grassmannian space $Gr_k(n)$ for example. The group $O(n)$ acts on it transitively by diffeomorphisms, and the isotropy group is $O(k)\times O(n-k)$. This indeed shows that $$Gr_k(n)=O(n)/(O(k)\times O(n-k)),$$as a diffeomorphism of smooth manifolds. Then, it is natural to equip $Gr_k(n)$ with the unique Riemannian metric which makes the quotient map $O(n)\to Gr_k(n)$ a Riemannian submersion.
The same reasoning applies for all the other examples. If you deal with complex and Hermitian structures, just make sure that the group acts by biholomorphisms.
It is true that, if your symmetric space already has a metric, you need to verify that the group acts on it by isometries. This may be the case with $\mathbb{C}P^n$, as the Fubini-Study metric can be constructed by a few different approaches.