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Invariant Generalized Almost Complex Structures on Real Flag Manifolds

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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 \(l\ge 5\), 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.

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Acknowledgements

We would like to express our more sincere gratitude to Luiz San Martin, Viviana del Barco, Cristián Ortiz, and Sebastián Herrera for their valuable comments and suggestions during several stages of the present work. Varea thanks Instituto de Matemática e Estatística - Universidade de São Paulo for the support provided while this work was carried out. Valencia was supported by Grant 2020/07704-7 São Paulo Research Foundation - FAPESP. Varea was supported by Grant 2020/12018-5 São Paulo Research Foundation - FAPESP.

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Authors and Affiliations

  1. Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, Cidade Universitária, São Paulo, 05508-090, Brazil

    Fabricio Valencia

  2. Departamento de Matemática, Universidade Tecnológica Federal do Paraná, Av. Alberto Carazzai 1640, Centro Cornélio Procópio, Paraná, 86300-000, Brazil

    Carlos Varea

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  1. Fabricio Valencia
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Correspondence to Carlos Varea.

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Valencia, F., Varea, C. Invariant Generalized Almost Complex Structures on Real Flag Manifolds. J Geom Anal 32, 296 (2022). https://doi.org/10.1007/s12220-022-01039-2

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