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

## Yuri A. Rylov

*Institute for Problems in Mechanics, Russian
Academy of Sciences*

* 101-1 ,Vernadskii Ave., Moscow, 119526, Russia*

* email: rylov@ipmnet.ru*

Web site: http://rsfq1.physics.sunysb.edu/~rylov/yrylov.htm

or mirror Web site: http://gasdyn-ipm.ipmnet.ru/~rylov/yrylov.htm

Updated March 3, 2011

#### abstract

Three different
representation of the proper Euclidean geometry are considered. They differ in
the number basic elements, from which the geometrical objects are constructed.
In E-representation there are three basic elements (point, segment, angle) and
no additional structures. Vrepresentation contains two basic elements (point,
vector) and additional structure: linear vector space. In $\sigma
$-representation there are 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
$\mathbf{Q}_{0}\mathbf{Q}_{1},\mathbf{Q}_{0}\mathbf{Q}_{1}^{\prime },...$,
which are equal to the vector \mathbf{P}_{0}\mathbf{P}_{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.

There is
text of the paper in English (pdf, ps) and in Russian (ps, pdf)