The intersection of two real flag manifolds in a complex flag manifold

TL;DR

This paper characterizes when two real flag manifolds intersect transversally within complex flag manifolds, showing their intersections are antipodal and establishing that real flag manifolds are globally tight Lagrangian submanifolds.

Contribution

It provides a necessary and sufficient condition for transversal intersection of real flag manifolds and proves their antipodal nature and tight Lagrangian property.

Findings

Intersection is antipodal.

Real flag manifolds are globally tight Lagrangian.

Condition for transversal intersection based on symmetric triad.

Abstract

We give a necessary and sufficient condition for two real flag manifolds, which are not necessarily congruent, in a complex flag manifold to intersect transversally in terms of the symmetric triad. Then we show that the intersection of two real flag manifolds is antipodal. As an application, we prove that any real flag manifold in a complex flag manifold is a globally tight Lagrangian submanifold.