Invariant almost complex structures on real flag manifolds

TL;DR

This paper investigates the existence and integrability of invariant almost complex structures on real flag manifolds, revealing that only certain cases admit complex structures, unlike the well-known complex case.

Contribution

It characterizes which real flag manifolds admit invariant almost complex structures and identifies those with integrable structures, especially within the Lie algebra C_l.

Findings

Some real flag manifolds do not admit invariant almost complex structures.

Only certain flag manifolds of Lie algebra C_l admit complex structures.

The paper distinguishes between existence and integrability of these structures.

Abstract

In this work we study the existence of invariant almost complex structures on real flag manifolds associated to split real forms of complex simple Lie algebras. We show that, contrary to the complex case where the invariant almost complex structures are well known, some real flag manifolds do not admit such structures. We check which invariant almost complex structures are integrable and prove that only some flag manifolds of the Lie algebra $ C_{l}$ admit complex structures.