RECOVERING A SURFACE IN ISOTROPIC SPACE USING DUAL MAPPING ACCORDING TO CURVATURE INVARIANTS

Abstract

The problem of recovering a surface according to its curvature is one of the fundamental problems of differential geometry. Problems of recovering surfaces in various spaces by their total or mean curvature have been widely studied in many works.Recovering of a surface by its total curvature is equivalent to solving the Monge-Ampere equation of elliptic type; such problems are solved in special cases. When the right part is given concretely.The Monge-Ampere equation is solved using a dual mapping of isotropic space, in which the dual surface is a transfer surface. Also, some special cases are used to find the surface equation.The connection between dual mean curvature and amalgamatic curvature is studied.The equivalence of the problem of recovering by dual mean and amalgamatic curvature is shown. In particular, the problem of recovering surfaces with total negative constant curvature, the mean curvature of which is a function of one variable, is solved. Furthermore, the problems of the recovering surfaces are solved according to their dual mean curvature, amalgamatic and Casorati curvatures.