-hausdorff Distance, Gromov Convergence Theorem, Compact Riemannian Manifold

Abstract:This paper tries to give a detailed proof of this convergence theorem which appears in [1]:Let {(M_iⁿ, g_i)} be a sequence of Riemannian manifolds satisfying(M_iⁿ,g_i) (?) (X,d). ({X,d) is a compact metric space) andthen for k∈N large ,there are diffeomorphisms:f_l : M_lâ†'M_k(l≥k) and there is a subsequence {(M_jⁿ,g_i)} of {(M_iⁿ,g_i)} such that the pull-back metric g_i on M := M_k converge(in C¹ norm) to a C^1,α metric g on M .