# Complex quadric as a symmetric space

It is known that a smooth complex quadric is a symmetric space. For example, it is $$\operatorname{Spin}(n+2)/G$$ where $$G$$ is the maximal parabolic subgroup.

I want a reference for more details and discussions.

– abx
Jul 31 at 19:37
• In case the quadric is projective there is a standard result (I don't know who it is due to but it is well-known) to represent it as a real Grassmanian of orientable planes $Gr^{+}(2,k)$ where $k$ is the complex dimension plus two. See this answer math.stackexchange.com/questions/2658786/… , it is dimension 4 but the proof is the same regardless of dimension. It follows since the Grassmanian has a transitive action of various Lie groups that it is a symmetric space. Aug 1 at 20:50

So while the projective quadric is indeed a homogeneous space of the form $$G/P$$ for $$P$$ parabolic in $$G$$ (marking it as a generalised flag manifold) this is not the realisation of it as a symmetric space. Indeed this does not even specify a Riemannian (or even pseudo-Riemannian) structure. It does give the quadric a conformal structure so I suppose we could pick a representative metric in that conformal class but still this is not the way to define the symmetric space structure.
To do that we need to take a maximal compact subgroup $$K\subset G$$. Then with $$K_0 := K\cap P$$, $$K/K_0$$ is a symmetric space and is still diffeomorphic to $$G/P$$. Or looking at it another way $$K$$ still acts transitively on $$G/P$$.
Edit: The default reference would be Helgason's "Differential Geometry, Lie Groups and Symmetric Spaces", particularly the sections on Hermitian symmetric spaces of compact type. Other good sources more generally about symmetric R-spaces include "Basic Transformations of symmetric R-spaces" by Takeuchi, "Geometry of Symmetric R-spaces" by Tanaka. My own PhD thesis: "Isothermic surfaces and their generalisations" goes into detail on all the classical examples of these including the projective quadric although it focuses only on the $$G/P$$ realisation and not at all on the $$K/K_0$$ version.