Gorkavyy V. Transformations of surfaces in spaces of constant curvature

This work is devoted to development of a theory of generalised Backlund-Bianci transformations for pseudo-spherical submanifolds in many-dimensional spaces of constant curvature and in more general Riemannian spaces, to solution of problems of restoring of surfaces in Euclidean spaces from a given Grassmann image and to characterization of submanifolds in Eclidean and semi-Euclidean spaces by means of transformations preserving the Grassmann image, to construction of isometric transformations with increasing volume for closed surfaces in Е3. In the theory of Backlund-Bianci transformations it is shown that two-dimensional surfaces in En ( Sn, Hn) , n>3, relied by pseudo-spherical congruences are pseudo-spherical. The possibility for constructing Backlund-Bianci transformations for pseudo-spherical surfaces in En is discussed. Pseudo-spherical surfaces in E4, which allow Bianci transformations, are described. The problem of constructing pseudo-spherical congruences and finding analogues of Backlund-Bianci transformations for two-dimensional surfaces in Riemannian products SnxR1, HnxR 1 is initiated. Many-dimensional pseudo-spherical submanifolds with degenerated Bianci transformation in Sn and in Hn are described. In the theory of Grassmann image the problem of restoring of a closed surface in E4 from its Grassmann image of special type in G+(2,4) is solved. Besides, general, conformal, isometric, equiareal transformations preserving the Grassmann image are studied for surfaces in Euclidean space and for light-like surfaces in Minkowski space. A geometrically grounded theory of strong-isotropic and l-minimal surfaces in En,1, which are proposed to be light-like analogues of minimal surfaces in En, is developed. In the theory of isometric tranformations special linear bendings with increasing volume are constructed for right pyramids and prisms in E3 . Moreover, iterative linear bendings with increasing volume for right convex polyhedra in E3 are constructed: in particular, the reached relative increase of volume makes more than 44 % for the tetrahedron and more than 24 % for the cube. Besides, special short transformations of Paulsen type for piecewise smooth closed convex surface of revolution are studied: a criterion is proved for a surface of revolution to admit a short deformation with increasing volume.