On the Equivalence of Elementary Surfaces with Respect to the Motion Group of Pseudo-Euclidean Space

This paper investigates the conditions for the equivalence of regular surfaces with respect to the action of a certain subgroup of linear transformations. This subgroup is pseudo-orthogonal and preserves a metric structure defined by a matrix with specific sign properties. The study focuses on elementary surfaces, which are considered as mappings from the square of the parameter domain (0,1)×(0,1) into an n-dimensional real vector space. The regularity of a surface is determined by the non-vanishing determinant of a special matrix composed of its partial derivatives. The paper also introduces the concept of surface equivalence. The main theorem establishes necessary and sufficient conditions for the equivalence of regular surfaces under the action of the pseudo-orthogonal group. These conditions are expressed through equalities between products of matrices constructed from the partial derivatives of the surfaces and the pseudo-orthogonal matrix. The obtained results provide a theoretical foundation for understanding the relationships between regular surfaces under the action of the pseudo-orthogonal group and contribute to the further study of their geometric properties and transformations.