The Journal of Geometric Analysis

, Volume 26, Issue 2, pp 996–1010 | Cite as

Geometrically Formal Homogeneous Metrics of Positive Curvature

Article
  • 139 Downloads

Abstract

A Riemannian manifold is called geometrically formal if the wedge product of harmonic forms is again harmonic, which implies in the compact case that the manifold is topologically formal in the sense of rational homotopy theory. A manifold admitting a Riemannian metric of positive sectional curvature is conjectured to be topologically formal. Nonetheless, we show that among the homogeneous Riemannian metrics of positive sectional curvature a geometrically formal metric is either symmetric, or a metric on a rational homology sphere.

Keywords

Geometric formality Positive curvature Homogeneous spaces 

Mathematics Subject Classification

22F30 53C20 57T15 

References

  1. 1.
    Amann, M.: Non-formal homogeneous spaces. Math. Z. 274(3–4), 1299–1325 (2013)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Aloff, S., Wallach, N.: An infinite family of 7-manifolds admitting positively curved Riemannian structures. Bull. Am. Math. Soc. 81, 93–97 (1975)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bär, C.: Geometrically formal 4-manifolds with nonnegative sectional curvature. arXiv:1212.1325v2 (2012)
  4. 4.
    Bazaikin, Y.: On a family of 13-dimensional closed Riemannian manifolds of positive curvature. Siberian Math. J. 37, 1068–1085 (1996)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bérard-Bergery, L.: Les variétés riemanniennes homogènes simplement connexes de dimension impaire à courbure strictement positive. J. Math. Pure et Appl. 55, 47–68 (1976)MATHGoogle Scholar
  6. 6.
    Berger, M.: Les varietes riemanniennes homogenes normales simplement connexes a Courbure strictment positive. Ann. Scuola Norm. Sup. Pisa 15, 191–240 (1961)Google Scholar
  7. 7.
    Borel, A.: Sur la cohomologie des espaces principaux et des espaces homogenes de groupes de Lie compacts. Ann. Math. 57, 115–207 (1953)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Borel, A.: Sur l’homologie et la cohomologie des groupes de Lie compacts connexes. Am. J. Math. 76, 273–342 (1954)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Berger, M.: Les variétés riemanniennes homogènes normales simplement connexes à courbure strictement positive. Ann. Scuola Norm. Sup. Pisa 15, 179–246 (1961)MathSciNetMATHGoogle Scholar
  10. 10.
    Dearricott, O.: A 7-manifold with positive curvature. Duke Math. J. 158, 307–346 (2011)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Eschenburg, J.H.: New examples of manifolds with strictly positive curvature. Invent. Math. 66, 469–480 (1982)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Eschenburg, J.H.: Freie isometrische Aktionen auf kompakten Lie-Gruppen mit positiv gekrümmten Orbiträumen. Schriftenr. Math. Inst. Univ. Münster 32, (1984)Google Scholar
  13. 13.
    Felix, Y., Halperin, S., Thomas, J.-C.: Rational Homotopy Theory. Graduate Texts in Mathematics, vol. 205. Springer, New York (2001)CrossRefGoogle Scholar
  14. 14.
    Grosjean, J.-F., Nagy, P.-A.: On the cohomology algebra of some classes of geometrically formal manifolds. Proc. Lond. Math. Soc. 98, 607–630 (2011)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Grove, K., Verdiani, L., Ziller, W.: An exotic \(T_1{\mathbb{S}}^4\) with positive curvature. Geom. Funct. Anal. 21, 499–524 (2011)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kotschik, D.: On products of harmonic forms. Duke Math. J. 107, 521–531 (2001)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kotschik, D.: Geometric formality and non-negative scalar curvature. arXiv:1212.3317 (2012)
  18. 18.
    Kotschik, D., Terzic, S.: On formality of generalized symmetric spaces. Math. Proc. Cambridge Phil. Soc. 134, 491–505 (2003)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kotschik, D., Terzic, S.: Chern numbers and the geometry of partial flag manifolds. Comm. Math. Helv. 84, 587–616 (2009)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Kotschik, D., Terzic, S.: Geometric formality of homogeneous spaces and biquotients. Pacific J. Math. 249, 157–176 (2011)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Kramer, L.: Homogeneous spaces, Tits buildings, and isoparametric hypersurfaces. Memoirs of the American Mathematical Society, vol. 752. American Mathematical Society, Providence (2002)Google Scholar
  22. 22.
    Nagy, P.-A.: On length and product of harmonic forms in Kähler geometry. Math. Z. 254, 199–218 (2006)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Ornea, L., Pilca, M.: Remarks on the product of harmonic forms. Pac. J. Math. 250, 353–363 (2011)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Prasad, G., Yeung, S.K.: Arithmetic fake projective spaces and arithmetic fake Grassmannians. Am. J. Math. 131, 379–407 (2009)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Püttmann, T.: Optimal pinching constants of odd dimensional homogeneous spaces. Invent. Math. 138, 631–684 (1999)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Valiev, F.M.: Precise estimates for the sectional curvatures of homogeneous Riemannian metrics on Wallach spaces. Sib. Mat. Zhurn. 20, 248–262 (1979)MathSciNetMATHGoogle Scholar
  27. 27.
    Verdiani, L., Ziller, W.: Positively curved homogeneous metrics on spheres. Math. Zeitschrift 261, 473–488 (2009)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Wallach, N.: Compact homogeneous Riemannian manifolds with strictly positive curvature. Ann. Math. 96, 277–295 (1972)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Wang, M., Ziller, W.: On isotropy irreducible Riemannian manifolds. Acta. Math. 166, 223–261 (1991)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Ziller, W.: Homogeneous Einstein metrics on Spheres and projective spaces. Math. Ann. 259, 351–358 (1982)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Ziller, W.: Examples of Riemannian manifolds with nonnegative sectional curvature. In: Grove, K., Cheeger, J. (eds.) Metric and Comparison Geometry. Surveys in Differential Geometry, vol. 11, pp. 63–102 (2007)Google Scholar

Copyright information

© Mathematica Josephina, Inc. 2015

Authors and Affiliations

  1. 1.Karlsruher Institut für TechnologieKarlsruheGermany
  2. 2.University of PennsylvaniaPhiladelphiaUSA

Personalised recommendations