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Nearly Kähler and Hermitian f-structures on homogeneous Φ-spaces of order k with special metrics

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Abstract

We consider arbitrary homogeneous Φ-spaces of order k ≥ 3 of semisimple compact Lie groups G in the case of a series of special metrics. We give formulas for the Nomizu function of the Levi-Civita connection of these metrics. Using these formulas and other relations for Φ-spaces of order k, we prove necessary and sufficient conditions for the canonical f-structures on these spaces to lie in some generalized Hermitian geometry classes of f-structures: nearly Kähler (NKf-structures) and Hermitian (Hf-structures).

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References

  1. Yano K., “On a structure defined by a tensor field f of type (1, 1) satisfying f 3 + f = 0,” Tensor, 14, 99–109 (1963).

    MATH  MathSciNet  Google Scholar 

  2. Yano K. and Kon M., CR Submanifolds of Kaehlerian and Sasakian Manifolds, Birkhäuser, Boston, Basel, and Stuttgart (1983).

    MATH  Google Scholar 

  3. Kirichenko V. F., “The methods of the generalized Hermitian geometry in the theory of almost contact manifolds,” in: Problems of Geometry [in Russian], VINITI, Moscow, 1986, Vol. 18, pp. 25–71 (Itogi Nauki i Tekhniki).

    Google Scholar 

  4. Fedenko A. S., Spaces with Symmetries [in Russian], Izdat. Belorussk. Gos. Univ., Minsk (1977).

    Google Scholar 

  5. Wolf J. A. and Gray A., “Homogeneous spaces defined by Lie group automorphisms,” J. Differential Geometry, 2, No. 1–2, 77–159 (1968).

    MATH  MathSciNet  Google Scholar 

  6. Gray A., “Riemannian manifolds with geodesic symmetries of order 3,” J. Differential Geometry, 7, No. 3–4, 343–369 (1972).

    MATH  MathSciNet  Google Scholar 

  7. Stepanov N. A., “Homogeneous 3-cyclic spaces,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 12, 65–74 (1967).

  8. Balashchenko V. V. and Stepanov N. A., “Canonical affinor structures of classical type on regular Φ-spaces,” Sb.: Math., 186, No. 11, 1551–1580 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  9. Balashchenko V. V., “Homogeneous Hermitian f-manifolds,” Russian Math. Surveys, 56, No. 3, 575–577 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  10. Balashchenko V. V., “Invariant nearly Kähler f-structures on homogeneous spaces,” Contemp. Math., 288, 263–267 (2001) (Global Differential Geometry: The Mathematical Legacy of Alfred Gray).

    MathSciNet  Google Scholar 

  11. Samsonov A. S., “Homogeneous Φ-spaces of pseudoorthogonal groups O(2, k),” Vestn. Beloruss. Gos. Univ. Ser. 1 Fiz. Mat. Inform., No. 3, 112–118 (2007).

  12. Balashchenko V. V. and Samsonov A. S., “Canonical f-structures on naturally reductive Φ-spaces of order 6,” Dokl. Nats. Akad. Nauk Belarusi, 54, No. 3, 26–31 (2010).

    MathSciNet  Google Scholar 

  13. Balashchenko V. V. and Samsonov A. S., “Nearly Kähler and Hermitian f-structures on homogeneous k-symmetric spaces,” Dokl. Math., 81, No. 3, 386–389 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  14. Kirichenko V. F., “Quasihomogeneous manifolds and generalized almost-Hermitian structures,” Math. USSR-Izv., 23, No. 3, 473–486 (1984).

    Article  MATH  Google Scholar 

  15. Kirichenko V. F., Differential Geometric Structures on Manifolds [in Russian], MPGU, Moscow (2003).

    Google Scholar 

  16. Balashchenko V. V., “Homogeneous nearly Kähler f-manifolds,” Dokl. Math., 63, No. 1, 56–58 (2001).

    MathSciNet  Google Scholar 

  17. Stepanov N. A., “Basic facts of the theory of φ-spaces,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 3, 88–95 (1967).

  18. Kowalski O., Generalized Symmetric Spaces, Springer-Verlag, Berlin, Heidelberg, and New York (1980).

    MATH  Google Scholar 

  19. Kobayashi Sh. and Nomizu K., Foundations of Differential Geometry. Vol. 2 [Russian translation], Nauka, Moscow (1981).

    Google Scholar 

  20. Churbanov Yu. D., “Integrability of canonical affinor structures of homogeneous periodic Φ-spaces,” Russian Math. (Izv. VUZ), 52, No. 8, 35–47 (2008).

    MATH  MathSciNet  Google Scholar 

  21. Kobayashi Sh. and Nomizu K., Foundations of Differential Geometry. Vol. 1 [Russian translation], Nauka, Moscow (1981).

    Google Scholar 

  22. Wang M. and Ziller W., “Existence and non-existence of homogeneous Einstein metrics,” Invent. Math., 84, 177–194 (1986).

    Article  MATH  MathSciNet  Google Scholar 

  23. Rodionov E. D., “Einstein metrics on even-dimensional homogeneous spaces admitting a homogeneous Riemannian metric of positive sectional curvature,” Siberian Math. J., 32, No. 3, 455–459 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  24. Balashchenko V. V., “Invariant f-structures in the generalized Hermitian geometry,” in: Proc. Conf. Contemp. Geometry and Related Topics, Univ. Belgrade, Belgrade, 2006, pp. 5–27.

    Google Scholar 

  25. Balashchenko V. V., Nikonorov Yu. G., Rodionov E. D., and Slavskiĭ V. V., Homogeneous Spaces: Theory and Applications [in Russian], Poligrafist, Khanty-Mansiĭsk (2008).

    Google Scholar 

  26. Sakovich A., “Invariant metric f-structures on specific homogeneous reductive spaces,” Extracta Math., 23, No. 1, 85–102 (2008).

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Belarus State University, Minsk, Belarus

    A. S. Samsonov

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  1. A. S. Samsonov
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Correspondence to A. S. Samsonov.

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Original Russian Text Copyright © 2011 Samsonov A. S.

The author was supported by the Belarussian Republic Foundation of Fundamental Researches (Grant F10R-132) in the framework of a joint project with the RFBR.

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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 52, No. 6, pp. 1373–1388, November–December, 2011.

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Samsonov, A.S. Nearly Kähler and Hermitian f-structures on homogeneous Φ-spaces of order k with special metrics. Sib Math J 52, 1092–1103 (2011). https://doi.org/10.1134/S0037446611060140

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