Skip to main content
View expanded cover

Riemannian Geometry pp 275–311Cite as

Ricci Curvature Comparison

  • 9787 Accesses

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

Abstract

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.

Keywords

  • 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.

This is a preview of subscription content, access via your institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-319-26654-1_7
  • Chapter length: 37 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   59.99
Price excludes VAT (USA)
  • ISBN: 978-3-319-26654-1
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   79.99
Price excludes VAT (USA)
Hardcover Book
USD   79.99
Price excludes VAT (USA)
Learn about institutional subscriptions
Fig. 7.1
Fig. 7.2
Fig. 7.3
Fig. 7.4
Fig. 7.5

Bibliography

  1. M.T. Anderson, Short geodesics and gravitational instantons. J. Differ. Geom. 31, 265–275 (1990)

    MathSciNet  MATH  Google Scholar 

  2. M.T. Anderson, Metrics of positive Ricci curvature with large diameter. Manuscripta Math. 68, 405–415 (1990)

    MathSciNet  CrossRef  MATH  Google Scholar 

  3. R.L. Bishop, R.J. Crittenden, Geometry of Manifolds (Academic Press, New York, 1964)

    MATH  Google Scholar 

  4. J. Cheeger, D. Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature. J. Differ. Geom. 6, 119–128 (1971)

    MathSciNet  MATH  Google Scholar 

  5. P. Eberlein, Geometry of Nonpositively Curved Manifolds (The University of Chicago Press, Chicago 1996)

    MATH  Google Scholar 

  6. J.H. Eschenburg, E. Heintze,  An elementary proof of the Cheeger-Gromoll splitting theorem. Ann. Glob. Anal. Geom. 2, 141–151 (1984)

    MathSciNet  CrossRef  MATH  Google Scholar 

  7. M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces (Birkhäuser, Boston 1999)

    MATH  Google Scholar 

  8. K. Grove, P. Petersen (eds.), Comparison Geometry, vol. 30 (MSRI publications, New York; Cambridge University Press, Cambridge, 1997)

    Google Scholar 

  9. P. Hajłasz, P. Koskela, Sobolev Met Poincaré, vol. 688 (Memoirs of the AMS, New York, 2000)

    MATH  Google Scholar 

  10. Y. Otsu, On manifolds of positive Ricci curvature with large diameters. Math. Z. 206, 255–264 (1991)

    MathSciNet  CrossRef  MATH  Google Scholar 

  11. Y. Shen, S.-h. Zhu, A sphere theorem for 3-manifolds with positive Ricci curvature and large diameter, preprint, Dartmouth

    Google Scholar 

  12. B. Wilking, On fundamental groups of manifolds of nonnegative curvature. Differ. Geom. Appl. 13(2), 129–165 (2000)

    MathSciNet  CrossRef  MATH  Google Scholar 

  13. J. Wolf, Spaces of Constant Curvature (Publish or Perish, Wilmington, 1984)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, Los Angeles, Los Angeles, CA, USA

    Peter Petersen

Authors
  1. Peter Petersen
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and Permissions

Copyright information

© 2016 Springer International Publishing AG

About this chapter

Cite this chapter

Petersen, P. (2016). Ricci Curvature Comparison. In: Riemannian Geometry. Graduate Texts in Mathematics, vol 171 . Springer, Cham. https://doi.org/10.1007/978-3-319-26654-1_7

Download citation