Ricci Curvature Comparison
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.
KeywordsRiemannian Manifold Distance Function Fundamental Group Support Function Betti Number
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