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.

Autorentext

Dr. Rania Bahgat amer Lecturer at the Physics & Mathematics Department faculty of engineering, Zagazig University. B.Sc. in (Civil engineering), M.Sc and Ph.D in differential geometry from Faculty of engineering, Zagazig University, Egypt.

Klappentext

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.