On geometry on a two-dimensional plane in a five-dimensional pseudo-Euclidean space of index two

Abstract

The study of the geometry of surfaces having a codimension greater than one in multidimensional spaces is one of the most difficult problems in geometry. When the multidimensional geometry under consideration has a pseudo-Euclidean metric, its complexity increases. Two-dimensional surfaces in a five-dimensional pseudo-Euclidean space of index two are considered in the article. Geometry on two-dimensional planes of this space can be of three types, Euclidean, Minkowski, and Galilean. Therefore, two-dimensional surfaces are also divided into three types according to the geometry on the tangent plane. A special class of two-dimensional surfaces given by a vector equation is considered. Using the dual space, the geometry of a two-dimensional surface is studied, reduced to a Euclidean or pseudo-Euclidean surface of a three-dimensional space. Conditions are revealed and theorems are proved on the existence of a surface that does not lie in a four-dimensional hyperplane and has tangent planes with one internal geometry.