An Estimate On Betti Numbers For Non-negatively Curved Riemannian Manifolds

The relationships between geometry and topology of Riemannian manifolds are important problems on differential geometry.Among them is the fitness of topology on Riemannian manifolds with non-negative sectional curvature.Due to the development of analysis on manifold,critical point theory and comparison theorems,in 1981,Gromov gave an estimate on Betti numbers of non-negatively curved complete Riemannian manifold in[7].In this thesis,we will give motivations of some definitions and clarify some ambiguous conclusions in[7].Additionally,by using a packing lemma,we will modify Gromov's resulttowhereand bi is the i-th Betti number.