Global Differential Geometry Of Hyperbolic Manifolds

The conic sections are represented in non real planes using a new real plane called the Entire plane. We proved by using two methods (Cartan structure equations and Christoffel Symbol of the Second Kind) that the non real planes and with suitable metrics are hyperbolic planes and the geodesics in these planes are hyperbolic straight lines. The seven non real spaces of three dimensions and are distinguished. A proof is introduced showing that these non real spaces with suitable metrics are hyperbolic spaces. The surfaces of second degree are represented in these non real spaces using a new real space called the Entire space. In addition, the stability on a surface M (hyperbolic space) is studied by using the function , where H is the mean curvature and f is the normal deformation at each point . Thus, this study provides new techniques and proofs in the field of hyperbolic geometry and it is essential for further study in hyperbolic geometry.