Invariant generalized almost complex structures on real flag manifolds

Abstract

We characterize those real flag manifolds that can be endowed with invariant generalized almost complex structures. We show that no $G{M}_2$-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 $l\ge 5$, 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.

34 pages. Minor changes have been made. Final version to appear in The Journal of Geometric Analysis

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• Invariant generalized complex structures on Lie groups
• Invariant Generalized Complex Structures on Partial Flag Manifolds
• Invariant generalized complex geometry on maximal flag manifolds and their moduli