Different representations of Euclidean geometry and their application to the space-time geometry

Abstract

Three different representation of the proper Euclidean geometry are considered. They differ in the number of basic elements, from which the geometrical objects are constructed. In E-representation there are three basic elements (point, segment, angle) and no additional structures. V-representation contains two basic elements (point, vector) and additional structure: linear vector space. In sigma-representation there is only one basic element and additional structure: world function \sigma =\rho^{2}/2, where \rho is the distance. The concept of distance appears in all representations. However, as a structure, determining the geometry, the distance appears only in the sigma-representation. The sigma-representation is most appropriate for modification of the proper Euclidean geometry. Practically any modification of the proper Euclidean geometry turns it into multivariant geometry, where there are many vectors Q_0Q_1, Q_0Q_1^{\prime},..., which are equal to the vector P_0P_1, but they are not equal between themselves, in general. Concept of multivariance is very important in application to the space-time geometry. The real space-time geometry is multivariant. Multivariance of the space-time geometry is responsible for quantum effects.