Research On The Curvature And The Topology Of Manifolds About Volume Growth

In this thesis, we study the topology of complete noncompact Riemannian manifolds with nonnegative Ricci curvature under some volume growth conditions.Firstly, we consider a complete noncompact Riemannian manifold Mⁿ with nonnegativeRicci curvature and large volume growth. If Mⁿ satisfies that K_p^min≥-C for somepoint p∈M and some positive constant C, we prove that Mⁿ has finite topological typeunder some additional conditions on the large volume growth. We also prove that Mⁿ is diffeomorphic to Rⁿ, if it satisfies some large volume growth conditions and, both the conjugateradius and the critical radius of M are bounded from below by positive constants, that is conj_M≥i₀ > 0 and crit_p≥r₀ > 0. These results generalize some results in [47] and [36]. On the other hand, we prove that Mⁿ must be diffeomorphic to Rⁿ under somelarge volume growth conditions, provided that k_p(r)≥-C/(1+r)^αsome p∈M and allr > 0, where C > 0 and 0≤α≤2. This is a generalization of a main result of C.Xia in [56].Secondly, we apply Gromov-Hausdorff's convergence and Toponogov's comparison theorems to get an estimate of the above bounds of the critical radius C_p. With the relationshipbetween distance function and critical point, we prove that for a complete noncompactRiemannian manifold with nonnegative Ricci curvature andα_M >1/2, if its distance function is finite, then it is diffeomorphic to Rⁿ. This strongly supports P.Petersen's conjecture( [40]).Thirdly, similar to the way of our studying on large volume growth, we study the topology of complete noncompact Riemannian manifolds with nonnegative Ricci curvature and sub-large volume growth. We obtain some results on finite topological type and the fundamental group, which improve a theorem proved by H.Zhan and Z.Shen in [65].