2
$\begingroup$

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.

$\endgroup$
2
  • 2
    $\begingroup$ Your quadric is affine? projective? $\endgroup$
    – abx
    Jul 31 at 19:37
  • $\begingroup$ 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. $\endgroup$
    – Nick L
    Aug 1 at 20:50

1 Answer 1

2
$\begingroup$

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 $G/P$.

Indeed in this way any flag manifold can be given a Riemannian structure and the ones for which this gives a Riemannian symmetric space are called the symmetric R-spaces. These basically comprise the compact Hermitian symmetric spaces and their real forms.

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.

$\endgroup$
1
  • $\begingroup$ +1 indeed I had confused the terms symmetric space and homogeneous space in my comment. $\endgroup$
    – Nick L
    Sep 13 at 19:12

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy