ON SOME MANIFOLDS OF CONSTANT NEGATIVE CURVATURE

Abstract

Manifold of constant negative curvature a great role in the field of physics, mathematics and engineering because it paves to the knowledge Gaussian curvature , n-sphere(s) is a topological n-manifold and objects of constant negative curvature are less familiar but they do appear in nature in the shape of corals and leave . No surprisingly , its plays an important role in geometric topology. The study aims to explain a generalization manifold of constant  negative curvature . WE followed the analytical induction mathematical method . We found the following some results. Manifold of constant negative curvature indicates to know the behavior of some of the functions and also  it reveals the Cartan-Hadamard theorem which is considered one of the importance aims of simply-connected  manifold of  nonpositive  sectional  curvature.