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Metrics of positive Ricci curvature on quotient spaces

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References

  1. Atiyah, M., Bott, R.: A Lefschetz fixed point formula for elliptic complexes. II. Applications. Ann. Math. 2(88), 451–491 (1968)

    MATH  Google Scholar 

  2. Aubin, T.: Métriques riemanniennes et courbure. J. Differ. Geom. 4, 383–424 (1970)

    MATH  Google Scholar 

  3. Back, A., Hsiang, W.Y.: Equivariant geometry and Kervaire spheres. Trans. Am. Math. Soc 304, 207–227 (1987)

    MathSciNet  MATH  Google Scholar 

  4. Bérard Bergery, L.: Certains fibrés à courbure de Ricci positive. C. R. Acad. Sci. Paris 286, 929–931 (1978)

    MATH  Google Scholar 

  5. Boyer, C., Galicki, K., Nakamaye, M.: Sasakian Geometry, Homotopy Spheres and Positive Ricci Curvature. Preprint, 2002, http://arxiv.org/math.DG/0201147

  6. Bredon, G.: Introduction to Compact Transformation Groups. Academic Press, New York–London, 1972

  7. Brieskorn, E.: Beispiele zur Differentialtopologie von Singularitäten. Invent. math. 2, 1–14 (1966)

    MATH  Google Scholar 

  8. Browder, W.: The Kervaire invariant of framed manifolds and its generalization. Ann. Math. 2(90), 157–186 (1969)

    MATH  Google Scholar 

  9. Browder, W.: Cobordism invariants, the Kervaire invariant and fixed point free involutions. Trans. Am. Math. Soc. 178, 193–225 (1973)

    MATH  Google Scholar 

  10. Cheeger, J.: Some examples of manifolds of non-negative curvature. J. Diff. Geom. 8, 623–628 (1973)

    MATH  Google Scholar 

  11. Dickinson, W.: Curvature properties of the positively curved Eschenburg spaces. Preprint 2000, to appear in Diff. Geom. Appl.

  12. Ehrlich, P.E.: Metric deformations of curvature, I. Local convex deformations. Geom. Dedicata 5, 1–23 (1976)

    MATH  Google Scholar 

  13. Eschenburg, J.-H.: Freie isometrische Aktionen auf kompakten Lie-Gruppen mit posi- tiv gekrümmten Orbiträumen. Schriftenreihe des Mathematischen Instituts der Universität Münster (2) 32, Universität Münster, Mathematisches Institut, Münster, 1984

  14. Fang, W.: Left invariant metrics on simple compact Lie groups and metrics on spheres, thesis, University of Pennsylavania, Philadelphia, USA, 2000

  15. Fukaya, K., Yamaguchi, T.: The fundamental groups of almost nonnegatively curved manifolds. Ann. Math. 136, 253–333 (1992)

    MathSciNet  MATH  Google Scholar 

  16. Giffen, C.: Desuspendability of free involutions on Brieskorn spheres. Bull. Am. Math. Soc. 75, 426–429 (1969)

    MATH  Google Scholar 

  17. Giffen, C.: Smooth homotopy projective spaces. Bull. Am. Math. Soc. 75, 509–513 (1969)

    MATH  Google Scholar 

  18. Giffen, C.: Weakly complex involutions and cobordism of projective spaces. Ann. Math. 2(90), 418–432 (1969)

    MATH  Google Scholar 

  19. Gilkey, P., Park, J.-H., Tuschmann, W.: Invariant metrics of positive Ricci curvature on principal bundles. Math. Z. 227, 455–463 (1998)

    MathSciNet  MATH  Google Scholar 

  20. Gromoll, D., Meyer, W.: Examples of complete manifolds with positive Ricci curvature. J. Diff. Geom. 21, 195–211 (1985)

    MathSciNet  MATH  Google Scholar 

  21. Gromov, M.: Curvature, diameter and Betti numbers. Comment. Math. Helv. 56, 179–195 (1981)

    MathSciNet  MATH  Google Scholar 

  22. Gromov, M., Lawson, H.B.: Spin and scalar curvature in the presence of a fundamental group I. Ann. Math. 111, 209–230 (1980)

    MathSciNet  MATH  Google Scholar 

  23. Gromov, M., Lawson, H.B.: Positive scalar curvature and the Dirac operator on complete Riemannian manifolds. Publ. Math. I.H.E.S. 58, 295–408 (1983)

    MATH  Google Scholar 

  24. Grove, K., Verdiani, L., Wilking, B., Ziller, W.: Obstructions to nonnegatively curved metrics in cohomogeneity one. In preparation

  25. Grove, K., Ziller, W.: Curvature and symmetry of Milnor spheres. Ann. Math. 2(152), 331–36 (2000)

    MATH  Google Scholar 

  26. Grove, K., Ziller, W.: Cohomogeneity one manifolds with positive Ricci curvature. Invent. math. 149, 619–646 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  27. Grove, K., Ziller, W.: Vector bundles over ℂP2 with nonnegative curvature. Preprint

  28. Hernandez-Andrade, H.: A class of compact manifolds with positive Ricci curvature. Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 1973), Part 1, Amer. Math. Soc., Providence, RI, 1975, pp. 73–87

  29. Hirzebruch, F., Mayer, K.: O(n)-Mannigfaltigkeiten, exotische Sphären und Singu- laritäten. Lecture Notes in Mathematics 57, Springer-Verlag, Berlin-New York, 1968

  30. Hitchin, N.: Harmonic spinors. Adv. Math. 14, 1–55 (1974)

    MATH  Google Scholar 

  31. Hsiang, W.-C., Hsiang, W.-Y.: On compact subgroups of the diffeomorphism groups of Kervaire spheres. Ann. Math. 2(85), 359–369 (1967)

    MATH  Google Scholar 

  32. Iwata, K.: Classifiaction of compact transformation groups on cohomology quaternion projective spaces with codimension one orbits. Osaka J. Math. 15, 475–508 (1978)

    MathSciNet  MATH  Google Scholar 

  33. Kapovitch, V., Ziller, W.: Biquotients with singly generated rational cohomology. Preprint 2002, http://arXiv: math.DG/0210231, to appear in Geometriae Dedicata

  34. Lichnerowicz, A.: Spineurs harmoniques. C.R. Acad. Sci. Paris 257, 7–9 (1963)

    MATH  Google Scholar 

  35. Lopez de Medrano, S.: Involutions on manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete 59, Springer-Verlag, New York-Heidelberg, 1971

  36. Milnor, J.: Singular points of complex hypersurfaces. Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968

  37. Montgomery, D., Samelson, H.: Transformation groups of spheres. Ann. Math 44, 454–470 (1943)

    MathSciNet  MATH  Google Scholar 

  38. Müter, M.: Krümmungserhöhende Deformationen mittels Gruppenaktionen, thesis, Westfälische Wilhelms-Universität Münster, Münster, Germany, 1987

  39. Nash, J.C.: Positive Ricci curvature on fiber bundles. J. Differ Geom 14, 241–254 (1979)

    MathSciNet  MATH  Google Scholar 

  40. Orlik, P.: Smooth homotopy lens spaces. Michigan Math. J. 16, 245–255 (1969)

    Article  MATH  Google Scholar 

  41. Poor, W.A.: Some exotic spheres with positive Ricci curvature. Math. Annalen 216, 245–252 (1975)

    MATH  Google Scholar 

  42. Rosenberg, J.: C*-algebras, positive scalar curvature, and the Novikov Conjecture, III. Topology 25, 319–336 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  43. Schoen, R., Yau, S.-T.: Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds of non-negative scalar curvature. Ann. Math. 110, 127–142 (1979)

    MathSciNet  MATH  Google Scholar 

  44. Schoen, R., Yau, S.-T.: On the structure of manifolds with positive scalar curvature. Manuscripta Math. 28, 159–183 (1979)

    MathSciNet  MATH  Google Scholar 

  45. Schwachhöfer, L., Tuschmann, W.: Almost nonnegative curvature and cohomogeneity one. Max Planck Institute for Mathematics in the Sciences Preprint Series, Preprint no. 62/2001, http://www.mis.mpg.de

  46. Sha, J.-P., Yang, D.-G.: Positive Ricci curvature on the connected sums of Sn× Sm. J. Diff. Geom. 33, 127–137 (1991)

    MathSciNet  MATH  Google Scholar 

  47. Uchida, F.: Classification of compact transformation groups on cohomology complex projective spaces with codimension one orbits. Japan. J. Math. 3, 141–189 (1977)

    MathSciNet  MATH  Google Scholar 

  48. Wall, C.T.C.: Surgery on compact manifolds. London Mathematical Society Monographs 1, Academic Press, London-New York, 1970

  49. Weinstein, A.: Positively curved deformations of invariant Riemannian metrics. Proc. Am. Math. Soc. 26, 151–152 (1970)

    MATH  Google Scholar 

  50. Wilking, B.: On fundamental groups of manifolds of nonnegative curvature. Diff. Geom. Appl. 13, 129–165 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  51. Witten, E.: Monopoles and four-manifolds. Math. Res. Let. 1, 769–796 (1994)

    MathSciNet  MATH  Google Scholar 

  52. Wraith, D.: Exotic spheres with positive Ricci curvature. J. Diff. Geom. 45, 638–649 (1997)

    MathSciNet  MATH  Google Scholar 

  53. Yau, S.-T.: Calabi’s conjecture and some new results in algebraic geometry. Proc. Nat. Acad. Sci. USA 74, 1798–1799 (1977)

    MATH  Google Scholar 

  54. Yau, S.-T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I. Comm. Pure Appl. Math. 31, 339–41 (1978)

    MathSciNet  MATH  Google Scholar 

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Correspondence to Wilderich Tuschmann.

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Research supported by the Communauté française de Belgique, through an Action de Recherche Concertée de la Direction de la Recherche Scientifique, and by the DFG Priority Program ‘‘Globale Differentialgeometrie’’.

Research supported by a DFG Heisenberg Fellowship and the DFG Priority Program ‘‘Globale Differentialgeometrie’’.

Keywords: Biquotients, cohomogeneity one manifolds, positive Ricci curvature, nonnegative Ricci curvature, almost nonnegative sectional curvature

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Schwachhöfer, L., Tuschmann, W. Metrics of positive Ricci curvature on quotient spaces. Math. Ann. 330, 59–91 (2004). https://doi.org/10.1007/s00208-004-0538-x

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