Geometriae Dedicata

, Volume 162, Issue 1, pp 337–344 | Cite as

On the fundamental group of Riemannian manifolds with nonnegative Ricci curvature

  • Bing Ye Wu
Original Paper


In 1968 Milnor conjectured that the fundamental group of any complete Riemannian manifold with nonnegative Ricci curvature is finitely generated. In this paper we obtain two results concerning Milnor’s conjecture. We first prove that the generators of fundamental group can be chosen so that it has at most logarithmic growth. Secondly we prove that the conjecture is true if additional the volume growth satisfies certain condition.


Riemannian manifold Ricci curvature Fundamental group Finitely generated 

Mathematics Subject Classification

53C23 53B40 58B20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anderson M.: On the topology of complete manifolds of nonnegative Ricci curvature. Topology 29, 41–55 (1990)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Chavel I.: Riemannian Geometry: A Modern Introduction. Cambridge University Press, Cambridge (1993)MATHGoogle Scholar
  3. 3.
    Gromov M.: Groups of polynomial growth and expanding maps. Inst. Hautes Etudes Sci. Publ. Math. 53, C53–C73 (1981)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Gromov, M., Lafontaine, J., Pansu, P.: Structures Métriques pour les Variétés Riemanniennes(French), Textes Mathématiques, 1. CEDIC, Paris (1981)Google Scholar
  5. 5.
    Li P.: Large time behavior of the heat equation on complete manifolds with nonnegative Ricci curvature. Ann. Math. 124, 1–21 (1986)MATHCrossRefGoogle Scholar
  6. 6.
    Milnor J.: A note on curvature and fundamental group. J. Diff. Geom. 2, 1–7 (1968)MathSciNetMATHGoogle Scholar
  7. 7.
    Petersen P.: Riemannian Geometry, 2nd edn. Springer, New York (2006)MATHGoogle Scholar
  8. 8.
    Sormani C.: Nonnegative Ricci curvature, small linear diameter growth and finite generation of fundamental groups. J. Diff. Geom. 54, 547–559 (2000)MathSciNetMATHGoogle Scholar
  9. 9.
    Yeganefar N.: On the fundamental group of some open manifolds. Diff. Geom. Appl. 25, 251–257 (2007)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Department of MathematicsMinjiang UniversityFuzhouChina

Personalised recommendations