Ricci Curvature Comparison

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


In this chapter we shall introduce some of the fundamental theorems 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. With these techniques we shall show numerous results on restrictions of fundamental groups of such spaces and also present a different proof of the estimate for the first Betti number by Bochner. The proof of the splitting theorem is self-contained. It uses the generalized maximum principle, but we show how one can get around the regularity issue for harmonic distance functions using some of our previous work on distance functions.


Riemannian Manifold Distance Function Fundamental Group Support Function Betti Number 
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Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Peter Petersen
    • 1
  1. 1.Department of MathematicsUniversity of California, Los AngelesLos AngelesUSA

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