Generalized normal homogeneous spheres

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• 作者：V. N. Berestovski?
• 关键词：geodesic orbit space ; geodesic vector ; δ ; homogeneous space ; δ ; vector ; naturally reductive space ; (generalized) normal homogeneous Riemannian space ; Riemannian submersion ; weakly symmetric space ; submetry
• 刊名：Siberian Mathematical Journal
• 出版年：2013
• 出版时间：July 2013
• 年：2013
• 卷：54
• 期：4
• 页码：588-603
• 全文大小：262KB
• 参考文献：1. / Berestovskii V. and Plaut C., “Homogeneous spaces of curvature bounded below,-J. Geom. Anal., 9, No. 2, 203-19 (1999). CrossRef
  2. / Berestovskii V. N. and Nikonorov Yu. G., “On / δ-homogeneous Riemannian manifolds,-Differential Geom. Appl., 26, No. 5, 514-35 (2008). CrossRef
  3. / Berestovskii V. N., Nikitenko E. V., and Nikonorov Yu. G., “Classification of generalized normal homogeneous Riemannian manifolds of positive Euler characteristic,-Differential Geom. Appl., 29, No. 4, 533-46 (2011). CrossRef
  4. / Berestovski? V. N., “Homogeneous manifolds with intrinsic metric. I,-Siberian Math. J., 29, No. 6, 887-97 (1988). CrossRef
  5. / Berestovski? V. N. and Nikonorov Yu. G., “The Chebyshev norm on the Lie algebra of the motion group of a compact homogeneous Finsler manifold,-J. Math. Sci., 161, No. 1, 97-21 (2008). CrossRef
  6. / Berger M., “Les variétés Riemanniennes homogènes normales simplement connexes à Courbure strictement positive,-Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 15, No. 3, 179-46 (1961).
  7. / Berestovskii V. N. and Guijarro L., “A metric characterization of Riemannian submersions,-Ann. Global Anal. Geom., 18, 577-88 (2000). CrossRef
  8. / Hopf H. and Samelson H., “Ein Satz über die Wirkungr?ume geschlossener Liescher Gruppen,-Comment. Math. Helv., 13, No. 1, 240-51 (1940-941). CrossRef
  9. / Selberg A., “Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces, with applications to Dirichlet series,-J. Indian Math. Soc., 20, 47-7 (1956).
  10. / Nomizu K., “Invariant affine connections on homogeneous spaces,-Amer. J. Math., 76, No. 1, 33-5 (1954). CrossRef
  11. / Kostant B., “On differential geometry and homogeneous spaces. I and II,-Proc. Nat. Acad. Sci. USA, 42, 258-61; 354-57 (1956). CrossRef
  12. / Besse A. L., Einstein Manifolds. Vol. 1 and 2, Springer-Verlag, Berlin (2008).
  13. / Berestovski? V. N. and Nikonorov Yu. G., “On / δ-homogeneous Riemannian manifolds. II,-Siberian Math. J., 50, No. 2, 214-22 (2009). CrossRef
  14. / Kowalski O. and Vanhecke L., “Riemannian manifolds with homogeneous geodesics,-Boll. Un. Mat. Ital. B, 5, No. 1, 189-46 (1991).
  15. / Berndt J., Kowalski O., and Vanhecke L., “Geodesics in weakly symmetric spaces,-Ann. Global Anal. Geom., 15, No. 2, 153-56 (1997). CrossRef
  16. / Montgomery D. and Samelson H., “Transformation groups of spheres,-Ann. Math., 44, No. 6, 454-70 (1943). CrossRef
  17. / Borel A., “Some remarks about transformation groups transitive on spheres and tori,-Bull. Amer. Math. Soc., 55, No. 6, 580-87 (1949). CrossRef
  18. / Borel A., “Le plan projectif des octaves et les sphères comme espaces homogènes,-C. R. Acad. Sci., 230, No. 15, 1378-380 (1950).
  19. / Onishchik A. L., “Transitive compact transformation groups,-Mat. Sb., 60, No. 4, 447-85 (1963).
  20. / Ziller W., “Homogeneous Einstein metrics on spheres and projective spaces,-Math. Ann., 259, No. 3, 351-58 (1982). CrossRef
  21. / Vol’per D. E., “Sectional curvatures of a diagonal family of Sp( / n + 1)-invariant metrics on (4 / n + 3)-dimensional spheres,-Siberian Math. J., 35, No. 6, 1089-100 (1994). CrossRef
  22. / Verdiani L. and Ziller W., “Positively curved homogeneous metrics on spheres,-Math. Z., Bd 261,Ht 3, 473-88 (2009). CrossRef
  23. / Vol’per D. E., “Sectional curvatures of nonstandard metrics on CP ^{2 / n+1},-Siberian Math. J., 40, No. 1, 39-5 (1999). CrossRef
  24. / Nikonorov Yu. G., “Geodesic orbit Riemannian metrics on spheres,-Vladikavkazsk. Mat. Zh. (2013) (to be published).
  25. / Ziller W., “Weakly symmetric spaces,-in: Topics in Geometry: in Memory of Joseph D’Atri, Boston, Birkh?user, 1996, pp. 355-68 (Prog. Nonlinear Differ. Equations; V. 20). CrossRef
  26. / Ziller W., “The Jacobi equation on naturally reductive compact Riemannian homogeneous spaces,-Comment. Math. Helv., 52, 573-90 (1977). CrossRef
  27. / Grove K. and Ziller W., “Cohomogeneity one manifolds with positive Ricci curvature,-Invent. Math., 149, No. 3, 619-46 (2002). CrossRef
  28. / Tamaru H., “Riemannian geodesic orbit metrics on fiber bundles,-Algebra Groups Geom., 15, 55-7 (1998).
  29. / Berestovski? V. N. and Nikonorov Yu. G., “Clifford-Wolf restrictively homogeneous Riemannian manifolds,-in: Papers of the International School-Workshop “Lomonosov Reading on Alta?-(Barnaul, November 8-1, 2011). In the Four Parts, AltGPA, Barnaul, 2011. Part 1, pp. 51-7.
  30. / Wolf J. A., Spaces of Constant Curvature, Wilmington, Delaware (1984).
  31. / Bourguignon J. P. and Karcher H., “Curvature operators. Pinching estimates and geometric examples,-Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 11, No. 1, 71-2 (1978).
  32. / Vol’per D. E., “A family of metrics on a 15-dimensional sphere,-Siberian Math. J., 38, No. 2, 223-34 (1997). CrossRef
  
• 作者单位：V. N. Berestovski? (1)
  
  1. Omsk Branch of the Sobolev Institute of Mathematics, Omsk, Russia
  
• ISSN：1573-9260

文摘

We find new generalized normal homogeneous but not normal homogeneous Riemannian metrics on spheres of dimensions 4n+3, n ?1, and all homogeneous space forms covered by them; all these spaces have zero Euler characteristic. Deriving consequences, alongside some other new results we obtain new proofs for analogous known results for all complex projective spaces of odd complex dimension starting from three.