Sectional Curvature Comparison II
In the first section we explain how one can find generalized gradients for distance functions in situations where the function might not be smooth. This critical point technique is used in the proofs of all the big theorems in this chapter. The other important technique comes from Toponogov’s theorem, which we prove in the next section. The first applications of these new ideas are to sphere theorems. We then prove the soul theorem of Cheeger and Gromoll. Next, we discuss Gromov’s finiteness theorem for bounds on Betti numbers and generators for the fundamental group Finally, we show that these techniques can be adapted to prove the Grove-Petersen homotopy finiteness theorem.
KeywordsSectional Curvature Betti Number Ricci Curvature Homotopy Type Integral Curf
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