Grechneva M. The geometry of the two-dimensional surface of Minkowski space and its grassman image

The thesis deals with the research of the links between the differential geometries of the surfaces and their Grassman images in Minkowski space and with the use of these links for the investigation of the surfaces. The concept of the stationary angles of the couple of the l-planes has been defined on the set of the l-planes in Minkowski space, the relative positions of the l-planes have been investigated. The smooth manifold structure has been constructed on the subsets of the timelike, the spacelike and the isotropic l-planes. Analogues of Cartan formula for the calculation of Gauss curvature have been obtained. The submanifolds of Grassman manifold are mapped by the algebraic surfaces in the six-dimensional space; it is proved that on these surfaces the pseudo-Riemannian metric is induced and the form of this metric has been obtained. Some properties of the submanifolds of Grassman manifold and of its point images are studied. The tensor of the curvature of the submanifolds of the non-isotropic planes has been constructed. The formulas for the calculation of the sectional curvature of the submanifolds along the different types of the tangential planes have been obtained and it has been proved that the sectional curvature of this manifold can take on any real values. The concept of Grassman image of the surface and its determination by Plucker coordinates has been considered. A link is found between the curvature of the submanifolds of the non-isotropic planes along the domains which are tangential to Grassman image of the two-dimensional surface in Minkowski space and the intrinsic geometry of the surface. The classes of the surfaces in Minkowski space with the stationary values of the sectional curvature of Grassman manifold along the domains which are tangential to Grassman image have been found. The affine classification of the points of the non-isotropic surface in the four-dimensional Minkowski space and the classification with the help of Glassman image are provided. Conditions of the equivalence of these classifications have been found. The problem connected with the finding of the surface in Minkowski space with the given Grassman image has been solved. The procedure of the reconstruction of the surface, Grassman image of which coincides with the given two-dimensional surface on Grassman manifold is presented. Theorems on the existence of the surface with boundary which has the given Grassman image are proved.