GLOBAL DIFFERENTIAL GEOMETRY OF SURFACES AND HYPERSURFACES IN FOUR DIMENSIONAL EUCLIDEAN SPACE

Abstract

Studying global differential geometry of surfaces is very important because it deals with the properties and curvatures of surfaces and the necessary conditions imposed on these surfaces or hypersurfaces to characterize the spheres and the hyperspheres among them. This thesis studied the methods of global differential geometry and their applications on surfaces and hypersurfaces in three and four dimensional Euclidean spaces . Chapter(l) : -

This chapter discussed the local differential geometry of surfaces and hypersurfaces in n-dimensional Euclidean space and presented some relations which describe properties of surfaces such as first , second and third fundamental forms and presented the definitions of normal curvatures , principle vectors , umbilical points , principle curvatures and lines of curvatures . Chapter (2) : -

This chapter studied the methods of global differential geometry (Integral formulas , Maximum principle and pseudoanalytic functions) and its applications on theorems in three dimensional Euclidean space .