Invariant generalized almost complex structures on real flag manifolds

TL;DR

This paper characterizes real flag manifolds with invariant generalized almost complex structures, showing non-existence of integrable structures on certain maximal flags and describing the geometry on specific types.

Contribution

It provides a classification of invariant generalized almost complex structures on real flag manifolds, including explicit descriptions and homotopy equivalences for various types.

Findings

No $GM_2$-maximal real flag manifolds admit integrable structures.

The space of invariant structures on certain flags is homotopy equivalent to a torus.

Complete classification of invariant generalized almost Hermitian structures on specific flag types.

Abstract

We characterize those real flag manifolds that can be endowed with invariant generalized almost complex structures. We show that no $GM_2$-maximal real flag manifolds admit integrable invariant generalized almost complex structures. We give a concrete description of the generalized complex geometry on the maximal real flags of type $B_2$, $G_2$, $A_3$, and $D_l$ with $l\geq 5$, where we prove that the space of invariant generalized almost complex structures under invariant $B$-transformations is homotopy equivalent to a torus and we classify all invariant generalized almost Hermitian structures on them.