Finite Topological Type

In this paper we show that a complete noncompact n-dimennian Riemannian manifold M with k-th Ricci curvature Ric_(k)(M)≥0 has finite topological type with some volume conditions.And there are three sections in this paper.The first section is the introduction of this paper.The second section is the preliminary knowledge.The third section has induced the definition of k-th Pdcci curvature of Riemannian manifold M.And we have that for m＞k,if Ric_(k)(M)≥kc,then Ric_(m)(M)≥mc. We also show that an n-dimensional complete noncompact Riemannian manifold M,for any r＞0 let K_p(r)=inf_M\B(p,r)K,where K denotes the sectional curvature of M,and the infimum is taken over,all the sections at all points on M\B(p,r).We will show that K_p(r)≤0 and that K_p(r) is a monotone function of r.At last we recited and showed the finite topological type theorem of a kind of Riemannian manifolds.