Advertisement

Mathematische Annalen

, Volume 330, Issue 1, pp 59–91 | Cite as

Metrics of positive Ricci curvature on quotient spaces

  • Lorenz J. Schwachhöfer
  • Wilderich TuschmannEmail author
Article
First Online:

Keywords

Quotient Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Atiyah, M., Bott, R.: A Lefschetz fixed point formula for elliptic complexes. II. Applications. Ann. Math. 2(88), 451–491 (1968)zbMATHGoogle Scholar
  2. 2.
    Aubin, T.: Métriques riemanniennes et courbure. J. Differ. Geom. 4, 383–424 (1970)zbMATHGoogle Scholar
  3. 3.
    Back, A., Hsiang, W.Y.: Equivariant geometry and Kervaire spheres. Trans. Am. Math. Soc 304, 207–227 (1987)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bérard Bergery, L.: Certains fibrés à courbure de Ricci positive. C. R. Acad. Sci. Paris 286, 929–931 (1978)zbMATHGoogle Scholar
  5. 5.
    Boyer, C., Galicki, K., Nakamaye, M.: Sasakian Geometry, Homotopy Spheres and Positive Ricci Curvature. Preprint, 2002, http://arxiv.org/math.DG/0201147Google Scholar
  6. 6.
    Bredon, G.: Introduction to Compact Transformation Groups. Academic Press, New York–London, 1972Google Scholar
  7. 7.
    Brieskorn, E.: Beispiele zur Differentialtopologie von Singularitäten. Invent. math. 2, 1–14 (1966)zbMATHGoogle Scholar
  8. 8.
    Browder, W.: The Kervaire invariant of framed manifolds and its generalization. Ann. Math. 2(90), 157–186 (1969)zbMATHGoogle Scholar
  9. 9.
    Browder, W.: Cobordism invariants, the Kervaire invariant and fixed point free involutions. Trans. Am. Math. Soc. 178, 193–225 (1973)zbMATHGoogle Scholar
  10. 10.
    Cheeger, J.: Some examples of manifolds of non-negative curvature. J. Diff. Geom. 8, 623–628 (1973)zbMATHGoogle Scholar
  11. 11.
    Dickinson, W.: Curvature properties of the positively curved Eschenburg spaces. Preprint 2000, to appear in Diff. Geom. Appl.Google Scholar
  12. 12.
    Ehrlich, P.E.: Metric deformations of curvature, I. Local convex deformations. Geom. Dedicata 5, 1–23 (1976)zbMATHGoogle Scholar
  13. 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, 1984Google Scholar
  14. 14.
    Fang, W.: Left invariant metrics on simple compact Lie groups and metrics on spheres, thesis, University of Pennsylavania, Philadelphia, USA, 2000Google Scholar
  15. 15.
    Fukaya, K., Yamaguchi, T.: The fundamental groups of almost nonnegatively curved manifolds. Ann. Math. 136, 253–333 (1992)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Giffen, C.: Desuspendability of free involutions on Brieskorn spheres. Bull. Am. Math. Soc. 75, 426–429 (1969)zbMATHGoogle Scholar
  17. 17.
    Giffen, C.: Smooth homotopy projective spaces. Bull. Am. Math. Soc. 75, 509–513 (1969)zbMATHGoogle Scholar
  18. 18.
    Giffen, C.: Weakly complex involutions and cobordism of projective spaces. Ann. Math. 2(90), 418–432 (1969)zbMATHGoogle Scholar
  19. 19.
    Gilkey, P., Park, J.-H., Tuschmann, W.: Invariant metrics of positive Ricci curvature on principal bundles. Math. Z. 227, 455–463 (1998)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Gromoll, D., Meyer, W.: Examples of complete manifolds with positive Ricci curvature. J. Diff. Geom. 21, 195–211 (1985)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Gromov, M.: Curvature, diameter and Betti numbers. Comment. Math. Helv. 56, 179–195 (1981)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Gromov, M., Lawson, H.B.: Spin and scalar curvature in the presence of a fundamental group I. Ann. Math. 111, 209–230 (1980)MathSciNetzbMATHGoogle Scholar
  23. 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)zbMATHGoogle Scholar
  24. 24.
    Grove, K., Verdiani, L., Wilking, B., Ziller, W.: Obstructions to nonnegatively curved metrics in cohomogeneity one. In preparationGoogle Scholar
  25. 25.
    Grove, K., Ziller, W.: Curvature and symmetry of Milnor spheres. Ann. Math. 2(152), 331–36 (2000)zbMATHGoogle Scholar
  26. 26.
    Grove, K., Ziller, W.: Cohomogeneity one manifolds with positive Ricci curvature. Invent. math. 149, 619–646 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  27. 27.
    Grove, K., Ziller, W.: Vector bundles over ℂP2 with nonnegative curvature. PreprintGoogle Scholar
  28. 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. 73–87Google Scholar
  29. 29.
    Hirzebruch, F., Mayer, K.: O(n)-Mannigfaltigkeiten, exotische Sphären und Singu- laritäten. Lecture Notes in Mathematics 57, Springer-Verlag, Berlin-New York, 1968Google Scholar
  30. 30.
    Hitchin, N.: Harmonic spinors. Adv. Math. 14, 1–55 (1974)zbMATHGoogle Scholar
  31. 31.
    Hsiang, W.-C., Hsiang, W.-Y.: On compact subgroups of the diffeomorphism groups of Kervaire spheres. Ann. Math. 2(85), 359–369 (1967)zbMATHGoogle Scholar
  32. 32.
    Iwata, K.: Classifiaction of compact transformation groups on cohomology quaternion projective spaces with codimension one orbits. Osaka J. Math. 15, 475–508 (1978)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Kapovitch, V., Ziller, W.: Biquotients with singly generated rational cohomology. Preprint 2002, http://arXiv: math.DG/0210231, to appear in Geometriae DedicataGoogle Scholar
  34. 34.
    Lichnerowicz, A.: Spineurs harmoniques. C.R. Acad. Sci. Paris 257, 7–9 (1963)zbMATHGoogle Scholar
  35. 35.
    Lopez de Medrano, S.: Involutions on manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete 59, Springer-Verlag, New York-Heidelberg, 1971Google Scholar
  36. 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, 1968Google Scholar
  37. 37.
    Montgomery, D., Samelson, H.: Transformation groups of spheres. Ann. Math 44, 454–470 (1943)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Müter, M.: Krümmungserhöhende Deformationen mittels Gruppenaktionen, thesis, Westfälische Wilhelms-Universität Münster, Münster, Germany, 1987Google Scholar
  39. 39.
    Nash, J.C.: Positive Ricci curvature on fiber bundles. J. Differ Geom 14, 241–254 (1979)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Orlik, P.: Smooth homotopy lens spaces. Michigan Math. J. 16, 245–255 (1969)CrossRefzbMATHGoogle Scholar
  41. 41.
    Poor, W.A.: Some exotic spheres with positive Ricci curvature. Math. Annalen 216, 245–252 (1975)zbMATHGoogle Scholar
  42. 42.
    Rosenberg, J.: C*-algebras, positive scalar curvature, and the Novikov Conjecture, III. Topology 25, 319–336 (1986)CrossRefMathSciNetzbMATHGoogle Scholar
  43. 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)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Schoen, R., Yau, S.-T.: On the structure of manifolds with positive scalar curvature. Manuscripta Math. 28, 159–183 (1979)MathSciNetzbMATHGoogle Scholar
  45. 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.deGoogle Scholar
  46. 46.
    Sha, J.-P., Yang, D.-G.: Positive Ricci curvature on the connected sums of Sn× Sm. J. Diff. Geom. 33, 127–137 (1991)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Uchida, F.: Classification of compact transformation groups on cohomology complex projective spaces with codimension one orbits. Japan. J. Math. 3, 141–189 (1977)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Wall, C.T.C.: Surgery on compact manifolds. London Mathematical Society Monographs 1, Academic Press, London-New York, 1970Google Scholar
  49. 49.
    Weinstein, A.: Positively curved deformations of invariant Riemannian metrics. Proc. Am. Math. Soc. 26, 151–152 (1970)zbMATHGoogle Scholar
  50. 50.
    Wilking, B.: On fundamental groups of manifolds of nonnegative curvature. Diff. Geom. Appl. 13, 129–165 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  51. 51.
    Witten, E.: Monopoles and four-manifolds. Math. Res. Let. 1, 769–796 (1994)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Wraith, D.: Exotic spheres with positive Ricci curvature. J. Diff. Geom. 45, 638–649 (1997)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Yau, S.-T.: Calabi’s conjecture and some new results in algebraic geometry. Proc. Nat. Acad. Sci. USA 74, 1798–1799 (1977)zbMATHGoogle Scholar
  54. 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)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Université Libre de BruxellesBruxellesBelgium
  2. 2.Westfälische Wilhelms-Universität MünsterMünsterGermany
  3. 3.Mathematics InstitutUniversität DortmundDortmundGermany

Personalised recommendations

Cite article

Buy options