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Classification of generalized Wallach spaces

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Abstract

In this paper, we present the classification of generalized Wallach spaces and discuss some related problems.

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References

  1. Abiev, N.A., Arvanitoyeorgos, A., Nikonorov, Yu.G, Siasos, P.: The dynamics of the Ricci flow on generalized Wallach spaces. Differ. Geom. Appl. 35(Supplement), 26–43 (2014)

  2. Abiev, N.A., Arvanitoyeorgos, A., Nikonorov, Yu.G., Siasos, P.: The Ricci flow on some generalized Wallach spaces. In: Rovenski, V., Walczak, P. (eds.). Geometry and Its Applications. Springer Proceedings in Mathematics & Statistics, V. 72, Switzerland, Springer, VIII+243 p., pp. 3–37 (2014)

  3. Bahturin, Y., Goze, M.: \(\mathbb{Z}_2\times \mathbb{Z}_2\)-symmetric spaces. Pacific J. Math. 236(1), 1–21 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Besse, A.L.: Einstein Manifolds. Springer, Berlin, etc., XII+510 p. (1987)

  5. Böhm, C.: Homogeneous Einstein metrics and simplicial complexes. J. Differ. Geom. 67, 79–165 (2004)

    MathSciNet  MATH  Google Scholar 

  6. Böhm, C., Wang, M., Ziller, W.: A variational approach for compact homogeneous Einstein manifolds. Geom. Funct. Anal. 14, 681–733 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, Z., Kang, Y., Liang, K.: Invariant Einstein metrics on three-locally-symmetric spaces. Commun. Anal. Geom., arXiv:1411.2694 (2014, to appear)

  8. D’Atri, J.E., Ziller, W.: Naturally reductive metrics and Einstein metrics on compact Lie groups. Mem. Amer. Math. Soc. 215, 1–72 (1979)

    MathSciNet  MATH  Google Scholar 

  9. Dynkin, E.B.: Semisimple subalgebras of semisimple Lie algebras. Mat. Sb. 30(2), 349–462 (1952). (Russian); English translation in: Am. Math. Soc. Transl. Ser 2., v. 6, pp. 111–244 (1957)

    MathSciNet  MATH  Google Scholar 

  10. Helgason, S.: Differential geometry, Lie groups, and symmetric spaces. Pure Appl. Math. 80, Academic Press. New York, XV+628 p. (1978)

  11. Horn, R.A., Johnson, C.R.: Matrix Analysis, 2nd edn, Cambridge University Press, Cambridge, XVIII+643 p. (2013)

  12. Jing-Song, H., Jun, Yu.: Klein four-subgroups of Lie algebra automorphisms. Pac. J. Math. 262(2), 397–420 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kollross, A.: Exceptional \(\mathbb{Z}_2\times \mathbb{Z}_2\)-symmetric spaces. Pac. J. Math. 242(1), 113–130 (2009)

    Article  MathSciNet  Google Scholar 

  14. Ledger, A.J., Obata, M.: Affine and Riemannian s-manifolds. J. Differ. Geom. 2(4), 451–459 (1968)

    MathSciNet  MATH  Google Scholar 

  15. Lomshakov, A.M., Nikonorov, Yu.G., Firsov, E.V.: Invariant Einstein metrics on three-locally-symmetric spaces. Mat. Tr. 6(2), 80–101 (2003). (Russian); English translation. In: Siberian Adv. Math., V. 14, No. 3, P. 43–62 (2004)

  16. Lutz, R.: Sur la géométrie des espaces \(\Gamma \)-symétriques. C. R. Acad.Sci. Paris Sér. I Math. 293(1), 55–58 (1981)

    MathSciNet  MATH  Google Scholar 

  17. Minchenko, A.N.: Semisimple subalgebras of exceptional Lie algebras. Tr. Mosk. Mat. Obs. 67, 256–293 (2006). (Russian); English translation in: Trans. Moscow Math. Soc. 2006, P. 225–259 (2006)

    MathSciNet  MATH  Google Scholar 

  18. Nikonorov, Yu.G.: On a class of homogeneous compact Einstein manifolds. Sib. Mat. Zhurnal 41(1), 200–205 (2000). (Russian); English translation in: Siberian Math. J., V. 41, No. 1, P. 168–172 (2000)

  19. Nikonorov, Yu.G.: Invariant Einstein metrics on the Ledger-Obata spaces. Algebra I Anal. 14(3), 169–185 (2002). (Russian); English translation in: St. Petersburg Math. J., V. 14, No. 3, P. 487–497 (2003)

  20. Nikonorov, Yu.G., Rodionov, E.D., Slavskii, V.V.: Geometry of homogeneous Riemannian manifolds. J. Math. Sci. (N. Y.) 146(7), 6313–6390 (2007)

  21. Sánchez, C. U.: Algebraic set associated to isoparametric submanifolds. In: Gordon, C.S., Tirao, J., Vargas, J.A., Wolf J.A. (eds.). New Developments in Lie Theory and Geometry. Contemp. Math., 491, Amer. Math. Soc., Providence, RI, X+348 p., P. 37–56 (2009)

  22. Wallach, N.: Compact homogeneous Riemannian manifolds with strictly positive curvature. Ann Math Second Ser 96, 277–295 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  23. Wang, M.: Einstein metrics from symmetry and bundle constructions. In: LeBrun, C., Wang, M. (eds.). Surveys in Differential Geometry, vol. VI: Essays on Einstein manifolds. Surveys in Differential Geometry. Suppl. to J. Differential Geom. 6. Cambridge, MA: International Press, X+423 p. (1999), pp. 287–325

  24. Wang, M.: Einstein metrics from symmetry and bundle constructions: a sequel. In Differential geometry. Adv. Lect. Math. (ALM). 22. International Press, Somerville, MA, pp. 253–309 (2012)

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Wolf, J.A.: Spaces of constant curvature, 6th edn. AMS Chelsea Publishing, Providence, RI, XVIII+424 pp (2011)

  27. Wolf, J.A., Gray, A.: Homogeneous spaces defined by Lie group automorphisms. I, II. J. Differ. Geom., 2, 77–114, 115–159 (1968)

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Acknowledgments

The project was supported in part by Grant 1452/GF4 of Ministry of Education and Sciences of the Republic of Kazakhstan for 2015-2017 (Agreement 299, February 12, 2015). The author thanks Christoph Böhm, Vicente Cortés, and Yuri Nikolayevsky for interesting discussions concerning this project. The author is grateful to the anonymous referee for helpful comments and suggestions that improved the presentation of this paper.

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  1. Southern Mathematical Institute of Vladikavkaz Scientific Centre of the Russian Academy of Sciences, Markus St. 22, Vladikavkaz, Russia, 362027

    Yu. G. Nikonorov

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  1. Yu. G. Nikonorov
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Correspondence to Yu. G. Nikonorov.

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Nikonorov, Y.G. Classification of generalized Wallach spaces. Geom Dedicata 181, 193–212 (2016). https://doi.org/10.1007/s10711-015-0119-z

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