Advertisement

Annals of Global Analysis and Geometry

, Volume 50, Issue 1, pp 65–84 | Cite as

The evolution of positively curved invariant Riemannian metrics on the Wallach spaces under the Ricci flow

  • Nurlan Abievich Abiev
  • Yuriĭ Gennadievich Nikonorov
Article

Abstract

This paper is devoted to the study of the evolution of positively curved metrics on the Wallach spaces \(\mathrm{SU}(3)/T_{\max }\), \(\mathrm{Sp}(3)/\mathrm{Sp}(1)\times \mathrm{Sp}(1)\times \mathrm{Sp}(1)\), and \(F_4/\mathrm{Spin}(8)\). We prove that for all Wallach spaces, the normalized Ricci flow evolves all generic invariant Riemannian metrics with positive sectional curvature into metrics with mixed sectional curvature. Moreover, we prove that for the spaces \(\mathrm{Sp}(3)/\mathrm{Sp}(1)\times \mathrm{Sp}(1)\times \mathrm{Sp}(1)\) and \(F_4/\mathrm{Spin}(8)\), the normalized Ricci flow evolves all generic invariant Riemannian metrics with positive Ricci curvature into metrics with mixed Ricci curvature. We also get similar results for some more general homogeneous spaces.

Keywords

Wallach space Generalized Wallach space Riemannian metric Ricci curvature Ricci flow Scalar curvature Sectional curvature Planar dynamical system Singular point 

Mathematics Subject Classification

Primary 53C30 Secondary 53C44 37C10 34C05 

Notes

Acknowledgments

The authors are grateful to the anonymous referee for helpful comments and suggestions that improved the presentation of this paper. The authors are indebted to Prof. Christoph Böhm, to Prof. Nolan R. Wallach, and to Prof. Wolfgang Ziller for helpful discussions concerning this paper. The project was supported by Grant 1452/GF4 of Ministry of Education and Sciences of the Republic of Kazakhstan for 2015–2017.

References

  1. 1.
    Abiev, N.A., Nikonorov, Y.G.: The evolution of positively curved invariant Riemannian metrics on the Wallach spaces under the Ricci flow. arXiv:1509.09263. (Preprint)
  2. 2.
    Abiev, N.A., Arvanitoyeorgos, A., Nikonorov, Y.G., Siasos, P.: The dynamics of the Ricci flow on generalized Wallach spaces. Differ. Geom. Appl. 35(Suppl.), 26–43 (2014)Google Scholar
  3. 3.
    Abiev, N.A., Arvanitoyeorgos, A., Nikonorov, Y.G., Siasos, P.: The Ricci flow on some generalized Wallach spaces. In: Rovenski, P., Walczak, V. (eds.) Geometry and its Applications. Springer Proceedings in Mathematics and Statistics, vol. 72, pp. 3–37. Springer, Switzerland (2014). (VIII+243 p)Google Scholar
  4. 4.
    Aloff, S., Wallach, N.: An infinite family of 7-manifolds admitting positively curved Riemannian structures. Bull. Am. Math. Soc. 81, 93–97 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Amann, H.: Ordinary differential equations. An introduction to nonlinear analysis. In: de Gruyter Studies in Mathematics, vol. 13, pp. xiv+458. Walter de Gruyter & Co., Berlin (1990). (Translated from the German by Gerhard Metzen)Google Scholar
  6. 6.
    Andronov, A.A., Leontovich, E.A., Gordon, I.I., Maier, A.G.: Qualitative theory of second-order dynamic systems. In: A Halsted Press Book. Wiley, New York (1973)Google Scholar
  7. 7.
    Bando, S.: On the classification of three-dimensional compact Kaehler manifolds of nonnegative bisectional curvature. J. Differ. Geom. 19(2), 283–297 (1984)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bérard, B.L.: Les variétés riemanniennes homogènes simplement connexes de dimension impaire à courbure strictement positive. J. Math. Pure Appl. 55, 47–68 (1976)Google Scholar
  9. 9.
    Berger, M.: Les varietes riemanniennes homogenes normales simplement connexes a courbure strictment positive. Ann Scuola Norm. Sup. Pisa 15, 191–240 (1961)Google Scholar
  10. 10.
    Besse, A.L.: Einstein Manifolds, pp. XII+510. Springer, Berlin (1987)Google Scholar
  11. 11.
    Böhm, C.: On the long time behavior of homogeneous Ricci flows. Comment. Math. Helv. 90, 543–571 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Böhm, C., Wang, M., Ziller, W.: A variational approach for compact homogeneous Einstein manifolds. GAFA Geom. Funct. Anal. 14, 681–733 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Böhm, C., Wilking, B.: Nonnegatively curved manifolds with finite fundamental groups admit metrics with positive Ricci curvature. GAFA Geom. Funct. Anal. 17, 665–681 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Buzano, M.: Ricci flow on homogeneous spaces with two isotropy summands. Ann. Glob. Anal. Geom. 45(1), 25–45 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Chen, Z., Kang, Y., Liang, K.: Invariant Einstein metrics on three-locally-symmetric spaces. Commun. Anal. Geom. arXiv:1411.2694. (to appear)
  16. 16.
    Cheung, M.W., Wallach, N.R.: Ricci flow and curvature on the variety of flags on the two dimensional projective space over the complexes, quaternions and octonions. Proc. Am. Math. Soc. 143(1), 369–378 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hamilton, R.S.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17, 255–306 (1982)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Hamilton, R.S.: Non-singular solutions of the Ricci flow on three-manifolds. Commun. Anal. Geom. 7(4), 695–729 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Jablonski, M.: Homogeneous Ricci solitons. J. Reine Angew. Math. 699, 159–182 (2015)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Lafuente, R.: Scalar curvature behavior of homogeneous Ricci flows. J. Geom. Anal. 25(4), 2313–2322 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Lauret, J.: Ricci flow on homogeneous manifolds. Math. Z. 274(12), 373–403 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lomshakov, A.M., Nikonorov, Y.G., Firsov, E.V.: Invariant Einstein metrics on three-locally-symmetric spaces. Mat. Tr. 6(2), 80–101 (2003) (Russian). [English translation. In: Sib. Adv. Math. 14(3), 43–62 (2004)]Google Scholar
  23. 23.
    Ni, L.: Ricci flow and manifolds with positive curvature. In: Howe, R., Hunziker, M., Willenbring, J.F. (eds.) Symmetry: Representation Theory and Its Applications. In honor of Nolan R. Wallach. Progress in Mathematics, vol. 257, pp. 491–504. Birkhäuser/Springer, New York (2014) (XXVIII+538 p)Google Scholar
  24. 24.
    Nikonorov, Y.G.: On a class of homogeneous compact Einstein manifolds. Sib. Mat. Zh. 41(1), 200–205 (2000) (Russian). [English translation. In: Sib. Math. J. 41(1), 168–172 (2000)]Google Scholar
  25. 25.
    Nikonorov, Y.G.: Classification of generalized Wallach spaces. Geom. Dedicata (2016). doi: 10.1007/s10711-015-0119-z MathSciNetzbMATHGoogle Scholar
  26. 26.
    Nikonorov, Y.G., Rodionov, E.D., Slavskii, V.V.: Geometry of homogeneous Riemannian manifolds. J. Math. Sci. (N. Y.) 146(7), 6313–6390 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Payne, T.L.: The Ricci flow for nilmanifolds. J. Mod. Dyn. 4(1), 65–90 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Püttmann, T.: Optimal pinching constants of odd dimensional homogeneous spaces. Invent. Math. 138(3), 631–684 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Rodionov, E.D.: Einstein metrics on even-dimensional homogeneous spaces admitting a homogeneous Riemannian metric of positive sectional curvature. Sib. Mat. Zh. 32(3), 126–131 (1991) (Russian). [English translation. In: Sib. Math. J. 32(3), 455–459 (1991)]Google Scholar
  30. 30.
    Shankar, K.: Isometry groups of homogeneous, positively curved manifolds. Differ. Geom. Appl. 14, 57–78 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Valiev, F.M.: Precise estimates for the sectional curvature of homogeneous Riemannian metrics on Wallach spaces. Sib. Mat. Zh. 20, 248–262 (1979) (Russian). [English translation. In: Sib. Math. J. 20, 176–187 (1979)]Google Scholar
  32. 32.
    Verdiani, L., Ziller, W.: Positively curved homogeneous metrics on spheres. Math. Z. 261, 473–488 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Vol’per, D.E.: Sectional curvatures of a diagonal family of \(Sp(n+1)\)-invariant metrics on \((4n+3)\)-dimensional spheres. Sib. Mat. Zh. 35(6), 1230–1242 (1994) (Russian). [English translation. In: Sib. Math. J. 35(6), 1089–1100 (1994)]Google Scholar
  34. 34.
    Vol’per, D.E.: A family of metrics on the 15-dimensional sphere. Sib. Mat. Zh. 38(2), 263–275 (1997) (Russian). [English translation. In: Sib. Math. J. 38(2), 223–234 (1997)]Google Scholar
  35. 35.
    Vol’per, D.E.: Sectional curvatures of nonstandard metrics on \(\mathbf{{CP}}^{2n+1}\). Sib. Mat. Zh. 40(1), 49–56 (1999) (Russian). [English translation. In: Sib. Math. J. 40(1), 39–45 (1999)]Google Scholar
  36. 36.
    Wallach, N.R.: Compact homogeneous Riemannian manifolds with strictly positive curvature. Ann. Math. Second Ser. 96, 277–295 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Wilking, B., Ziller, W.: Revisiting homogeneous spaces with positive curvature. J. die Reine und Angew. Math. (2015). doi: 10.1515/crelle-2015-0053 Google Scholar
  38. 38.
    Xu, M., Wolf, J.A.: \(Sp(2)/U(1)\) and a positive curvature problem. Differ. Geom. Appl. 42, 115–124 (2015)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Zhifen, Z., Tongren, D., Wenzao, H., Zhenxi, D.: Qualitative Theory of Differential Equations. American Mathematical Society, Providence (1992)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.M. Kh. Dulaty Taraz State UniversityTarazKazakhstan
  2. 2.Southern Mathematical Institute of VSC RASVladikavkazRussia

Personalised recommendations