On The Fundamental Group Of Compact Manifolds With Nonpositive Curvature

The topological structure of compact Riemannian manifolds with nonpositive curvature is decided uniquely by their fundamental groups. Can the fundamental groups of any compact topological manifold determin that it admits a non-positively curved Riemannian metric? By the use of group action and normal covering we give an answer in some conditions from topological rigidity theorem in this article.