Geodesic
Riemannian symmetries in flag manifolds
Archivum mathematicum, Tome 48 (2012) no. 5, pp. 387-398 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Flag manifolds are in general not symmetric spaces. But they are provided with a structure of $\mathbb{Z}_2^k$-symmetric space. We describe the Riemannian metrics adapted to this structure and some properties of reducibility. The conditions for a metric adapted to the $\mathbb{Z}_2^2$-symmetric structure to be naturally reductive are detailed for the flag manifold $SO(5)/SO(2)\times SO(2) \times SO(1)$.
Flag manifolds are in general not symmetric spaces. But they are provided with a structure of $\mathbb{Z}_2^k$-symmetric space. We describe the Riemannian metrics adapted to this structure and some properties of reducibility. The conditions for a metric adapted to the $\mathbb{Z}_2^2$-symmetric structure to be naturally reductive are detailed for the flag manifold $SO(5)/SO(2)\times SO(2) \times SO(1)$.
DOI : 10.5817/AM2012-5-387
Classification : 53C30
Keywords: $\mathbb{Z}_2^k$-symmetric space; flag manifolds; Riemannian metrics 
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     title = {Riemannian symmetries in flag manifolds},
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     url = {http://geodesic.mathdoc.fr/articles/10.5817/AM2012-5-387/}
}
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Piu, Paola; Remm, Elisabeth. Riemannian symmetries in flag manifolds. Archivum mathematicum, Tome 48 (2012) no. 5, pp. 387-398. doi: 10.5817/AM2012-5-387

[1] Arias–Marco, T., Kowalski, O.: Classification of 4–dimensional homogeneous D’Atri spaces. Czechoslovak Math. J. 58 (1) (2008), 203–239. | DOI | MR | Zbl

[2] Bahturin, Y., Goze, M.: $\mathbb{Z}_2^2$–symmetric spaces. Pacific J. Math. 236 (1) (2008), 1–21. | DOI | MR

[3] Bouyakoub, A., Goze, M., Remm, E.: On Riemannian non symmetric spaces and flag manifolds. arXiv:math/0609790. | MR

[4] Goze, M., Remm, E.: $\Gamma $–symmetric spaces. Differential geometry, World Sci. Publ., Hackensack, NJ, 2009, pp. 195–206. | MR | Zbl

[5] Kobayashi, S., Nomizu, K.: Foundations of differential geometry, Vollume II. llume II, Interscience Tracts in Pure and Applied Mathematics, No. 15, Interscience Publishers John Wiley and Sons, Inc., New York–London–Sydney, 1969. | MR

[6] Kobayashi, S., Nomizu, K.: Foundations of differential geometry, volume I. lume I, Interscience Publishers, John Wiley and Sons, New York–London, 1963. | MR

[7] Kollross, A.: Exceptional $Z_2\times Z_2$–symmetric spaces. Pacific J. Math. 242 (1) (2009), 113–130. | DOI | MR

[8] Kowalski, O.: Generalized symmetric spaces, volume II. lume II, Lecture Notes in Math. 805, Springer–Verlag, Berlin–New York, 1980. | MR

[9] Lutz, R.: Sur la géométrie des espaces $\Gamma $–symétriques. C. R. Acad. Sci. Paris Sér. I Math. 293 (1) (1981), 55–58. | MR | Zbl

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