Evolutes Of Surfaces In 4 Dimensional Non-flat Space

We propose a way to study the differential geometry of surfaces in four dimensional non-flat space in this paper from the singularities viewpoint.Firstly,we investigate the differential geometry of surfaces in four dimensional hyperbolic space and sphere space,respectively.The notion of evolutes of surfaces in hyperbolic space and sphere space are given.As an application of the theory of Lagrangian singularities,we study the contact of surfaces with the families of hyperspheres and equidistant hypersurfaces,which is measured by the singularities of height functions on surfaces.At last,we summarize the corresponding conclusions in four dimensional non-flat space form.