Rigidity of convex divisible domains in flag manifolds

TL;DR

This paper proves the uniqueness of convex divisible domains in certain Grassmannians, showing that only one such domain exists up to projective isomorphism, which models a symmetric space of SO(p,p).

Contribution

It establishes the uniqueness of convex divisible domains in Grassmannians of p-planes in R^{2p} for p > 1, linking them to symmetric spaces of SO(p,p).

Findings

Only one convex divisible domain exists in the Grassmannian of p-planes in R^{2p} for p > 1.

This domain is a model of the symmetric space associated with SO(p,p).

The result contrasts with the abundance of such domains in real projective space.

Abstract

In contrast to the many examples of convex divisible domains in real projective space, we prove that up to projective isomorphism there is only one convex divisible domain in the Grassmannian of $p$-planes in $\mathbb{R}^{2p}$ when $p > 1$. Moreover, this convex divisible domain is a model of the symmetric space associated to the simple Lie group SO$(p, p)$.