Ricci Curvature Comparison

Part of the Graduate Texts in Mathematics book series (GTM, volume 171)


In this chapter we prove some of the fundamental results for manifolds with lower Ricci curvature bounds. Two important techniques will be developed: Relative volume comparison and weak upper bounds for the Laplacian of distance functions. Later some of the analytic estimates we develop here will be used to estimate Betti numbers for manifolds with lower curvature bounds.


Nonnegative Ricci Curvature Relative Volume Comparison Betti Number Estimate Ricci-flat Metrics Decreasing Distance Function 
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.


  1. 2.
    M.T. Anderson, Short geodesics and gravitational instantons. J. Differ. Geom. 31, 265–275 (1990)MathSciNetzbMATHGoogle Scholar
  2. 3.
    M.T. Anderson, Metrics of positive Ricci curvature with large diameter. Manuscripta Math. 68, 405–415 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 14.
    R.L. Bishop, R.J. Crittenden, Geometry of Manifolds (Academic Press, New York, 1964)zbMATHGoogle Scholar
  4. 31.
    J. Cheeger, D. Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature. J. Differ. Geom. 6, 119–128 (1971)MathSciNetzbMATHGoogle Scholar
  5. 38.
    P. Eberlein, Geometry of Nonpositively Curved Manifolds (The University of Chicago Press, Chicago 1996)zbMATHGoogle Scholar
  6. 41.
    J.H. Eschenburg, E. Heintze,  An elementary proof of the Cheeger-Gromoll splitting theorem. Ann. Glob. Anal. Geom. 2, 141–151 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 53.
    M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces (Birkhäuser, Boston 1999)zbMATHGoogle Scholar
  8. 54.
    K. Grove, P. Petersen (eds.), Comparison Geometry, vol. 30 (MSRI publications, New York; Cambridge University Press, Cambridge, 1997)Google Scholar
  9. 60.
    P. Hajłasz, P. Koskela, Sobolev Met Poincaré, vol. 688 (Memoirs of the AMS, New York, 2000)zbMATHGoogle Scholar
  10. 81.
    Y. Otsu, On manifolds of positive Ricci curvature with large diameters. Math. Z. 206, 255–264 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 95.
    Y. Shen, S.-h. Zhu, A sphere theorem for 3-manifolds with positive Ricci curvature and large diameter, preprint, DartmouthGoogle Scholar
  12. 103.
    B. Wilking, On fundamental groups of manifolds of nonnegative curvature. Differ. Geom. Appl. 13(2), 129–165 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 106.
    J. Wolf, Spaces of Constant Curvature (Publish or Perish, Wilmington, 1984)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, Los AngelesLos AngelesUSA

Personalised recommendations