2007/09/18 by Yuri A. Rylov, Rylov, Yuri A.
Mathematics · Physics and Astronomy · #00A35 #97D20 #Advanced Mathematical Theories and Applications #FOS: Mathematics #General Mathematics (math.GM) #Mathematics and Applications #Relativity and Gravitational Theory #math.GM #msc:00A35 #msc:97D20
paper · pdf · doi:10.48550/arxiv.0709.2755
17 pages, 0 figures, some additions
openalex publication_date 2007/09/18 · arxiv created 2011/03/02 · arxiv updated 2011/03/03 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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 σ=ρ2/2, where ρis 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 Q0Q1, Q0Q1′,..., which are equal to the vector P0P1, 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.