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Annals of Global Analysis and Geometry

, Volume 45, Issue 3, pp 167–196 | Cite as

Generalized normal homogeneous Riemannian metrics on spheres and projective spaces

  • Valeriĭ Nikolaevich Berestovskiĭ
  • Yuriĭ Gennadievich Nikonorov
Article

Abstract

In this paper, we develop new methods to study generalized normal homogeneous Riemannian manifolds. In particular, we obtain a complete classification of generalized normal homogeneous Riemannian metrics on spheres \({S^n}\). We prove that for any connected (almost effective) transitive on \(S^n\) compact Lie group \(G\), the family of \(G\)-invariant Riemannian metrics on \(S^n\) contains generalized normal homogeneous but not normal homogeneous metrics if and only if this family depends on more than one parameters and \(n\ge 5\). Any such family (that exists only for \(n=2k+1\)) contains a metric \(g_\mathrm{can}\) of constant sectional curvature \(1\) on \(S^n\). We also prove that \((S^{2k+1}, g_\mathrm{can})\) is Clifford–Wolf homogeneous, and therefore generalized normal homogeneous, with respect to \(G\) (except the groups \(G={ SU}(k+1)\) with odd \(k+1\)). The space of unit Killing vector fields on \((S^{2k+1}, g_\mathrm{can})\) from Lie algebra \(\mathfrak g \) of Lie group \(G\) is described as some symmetric space (except the case \(G=U(k+1)\) when one obtains the union of all complex Grassmannians in \(\mathbb{C }^{k+1}\)).

Keywords

Clifford algebras Clifford–Wolf homogeneous spaces  Generalized normal homogeneous Riemannian manifolds g.o. spaces Grassmannian algebra Homogeneous spaces Hopf fibrations Normal homogeneous Riemannian manifolds Generalized normal homogeneous but not normal homogeneous Riemannian submersions (weakly) symmetric spaces 

Mathematics Subject Classification (2000)

Primary 53C20 Secondary 53C25 53C35 

Notes

Acknowledgments

We thank Professor Wolfgang Ziller for useful discussions and Natalia Berestovskaya for help in preparation of the text. The authors are indebted to the anonymous referee for helpful comments and suggestions that improved the presentation of this paper.

References

  1. 1.
    Alekseevskii, D.V.: Compact quaternion spaces. Funk. Anal. Pril. 2(2), 11–20 (1968). (Russian)MathSciNetGoogle Scholar
  2. 2.
    Berestovskii, V.N.: Generalized normal homogeneous spheres. Sib. Math. J. 54(4), 588–603 (2013)Google Scholar
  3. 3.
    Berestovskii, V.N.: Generalized normal homogeneous spheres \(S^{4n+3}\) with largest connected motion group \(Sp(n+1)\cdot U(1)\). Sib. Mat. Zh. (2013, in press) (Russian)Google Scholar
  4. 4.
    Berestovskii, V.N., Nikitenko, E.V., Nikonorov, Yu.G.: Classification of generalized normal homogeneous Riemannian manifolds of positive Euler characteristic. Differ. Geom. Appl. 29(4), 533–546 (2011)Google Scholar
  5. 5.
    Berestovskii, V.N., Nikonorov, Yu.G.: Killing vector fields of constant length on Riemannian manifolds. Siber. Math. J. 49(3), 395–407 (2008)Google Scholar
  6. 6.
    Berestovskii, V.N., Nikonorov, Yu.G.: Killing vector fields of constant length on locally symmetric Riemannian manifolds. Transform. Groups 13(1), 25–45 (2008)Google Scholar
  7. 7.
    Berestovskii, V.N., Nikonorov, Yu.G.: On \(\delta \)-homogeneous Riemannian manifolds. Differ. Geom. Appl. 26(5), 514–535 (2008)Google Scholar
  8. 8.
    Berestovskii, V.N., Nikonorov, Yu.G.: On \(\delta \)-homogeneous Riemannian manifolds, II. Siber. Math. J. 50(2), 214–222 (2009)Google Scholar
  9. 9.
    Berestovskii, V.N., Nikonorov, Yu.G.: Clifford–Wolf homogeneous Riemannian manifolds. J. Differ. Geom. 82(3), 467–500 (2009)Google Scholar
  10. 10.
    Berestovskii, V.N., Nikonorov, Yu.G.: The Chebyshev norm on the Lie algebra of the motion group of a compact homogeneous Finsler manifold. Sovrem. Mat. Prilozh. 60, 98–121 (2008) (Russian); English translation in J. Math. Sci. (N. Y.) 161(1), 97–121 (2009)Google Scholar
  11. 11.
    Berestovskii, V.N., Nikonorov, Yu.G.: Riemannian manifolds and homogeneous geodesics. South Mathematical Institute of VSC RAS, Vladikavkaz (2012) (Russian)Google Scholar
  12. 12.
    Berestovskii, V.N., Plaut, C.: Homogeneous spaces of curvature bounded below. J. Geom. Anal. 9(2), 203–219 (1999)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Berger, M.: Les varietes riemanniennes homogenes normales a courbure strictement positive. Ann. Scoula Norm. Sup. Pisa Cl. Sci. Sér. 3 15(3), 179–246 (1961)MATHGoogle Scholar
  14. 14.
    Besse, A.L.: Einstein Manifolds. Springer, Berlin (1987)CrossRefMATHGoogle Scholar
  15. 15.
    Firey, W.J.: \(P\)-means of convex bodies. Math. Scand. 10, 17–24 (1962)MATHMathSciNetGoogle Scholar
  16. 16.
    Firey, W.J.: Some applications of means of convex bodies. Pac. J. Math. 14, 53–60 (1964)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Friedrich, T.: Weak Spin(9)-structures on 16-dimensional Riemannian manifolds. Asian J. Math. 5(1), 129–160 (2001)MATHMathSciNetGoogle Scholar
  18. 18.
    Fuks, D.B., Rokhlin, V.A.: Beginner’s course in topology. Geometric chapters. In: Universitext. Springer series in Soviet Mathematics. Springer, Berlin (1984)Google Scholar
  19. 19.
    Gluck, H., Ziller, W.: The geometry of the Hopf fibrations. L’Enseignement Math. 32, 173–198 (1986)MATHMathSciNetGoogle Scholar
  20. 20.
    Grove, K., Ziller, W.: Cohomogeneity one manifolds with positive Ricci curvature. Inv. Math. 149, 619–646 (2002)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Hopf, H., Samelson, H.: Ein Satz über die Wirkungräume geschlossener Lischer Gruppen. Math. Helv. 13(1), 240–251 (1940/1941)Google Scholar
  22. 22.
    Husemoller, D.: Fibre bundles. In: Graduate Texts in Mathematics, 3rd edn, vol. 20. Springer, New York (1994)Google Scholar
  23. 23.
    James, I.M.: The topology of Stiefel manifolds. In: London Mathematical Society Lecture Note Series, vol. 20. Cambridge University Press, Cambridge (1976)Google Scholar
  24. 24.
    Kozlov, S.E.: Geometry of real Grassmannian manifolds. I, II. J. Math. Sci. (N. Y.) 100(3), 2239–2253 (2000)Google Scholar
  25. 25.
    Nikonorov, Yu.G.: Geodesic orbit Riemannian metrics on spheres. Vladikavkaz. Mat. Zh. (2013, in press)Google Scholar
  26. 26.
    Onishchik, A.L.: Topology of Transitive Transformation Groups. Johann Ambrosius Barth, Leipzig (1994)MATHGoogle Scholar
  27. 27.
    Verdiani, L., Ziller, W.: Positively curved homogeneous metrics on spheres. Math. Zeitschrift. 261(3), 473–488 (2009)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Volper, D.E.: Sectional curvatures of a diagonal family of \(Sp(n+1)\)-invariant metrics on \((4n+3)\)-dimensional spheres. Siber. Math. J. 35(6), 1089–1100 (1994)CrossRefMathSciNetGoogle Scholar
  29. 29.
    Volper, D.E.: A family of metrics on 15-dimensional sphere. Siber. Math. J. 38(2), 223–234 (1997)CrossRefMathSciNetGoogle Scholar
  30. 30.
    Volper, D.E.: Sectional curvatures of nonstandard metrics on \({\mathbb{C}}P^{2n+1}\). Siber. Math. J. 40(1), 39–45 (1999)CrossRefMathSciNetGoogle Scholar
  31. 31.
    Wolf, J.A.: Harmonic Analysis on Commutative Spaces. American Mathematical Society, Providence (2007)CrossRefMATHGoogle Scholar
  32. 32.
    Ziller, W.: The Jacobi equation on naturally reductive compact Riemannian homogeneous spaces. Comment. Math. Helv. 52, 573–590 (1977)CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Ziller, W.: Homogeneous Einstein metrics on spheres and projective spaces. Math. Ann. 259, 351–358 (1982)CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Ziller, W.: Weakly symmetric spaces, pp. 355–368. In: Progress in Nonlinear Differential Equations, vol. 20. Topics in Geometry: in Memory of Joseph D’Atri. Birkhäuser (1996)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Valeriĭ Nikolaevich Berestovskiĭ
    • 1
  • Yuriĭ Gennadievich Nikonorov
    • 2
  1. 1.Omsk Branch of Sobolev Institute of Mathematics of SD RASOmskRussia
  2. 2.South Mathematical Institute of VSC RASVladikavkazRussia

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