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Different representations of Euclidean geometry
and their application to the space-time geometry
Institute for Problems in Mechanics, Russian Academy of Sciences,
101-1, Vernadskii Ave., Moscow, 119526, Russia.

Web site:

*http *:

*//rsf q*1

*.physics.sunysb.edu/*˜

*rylov/yrylov.htm*
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*//gasdyn − ipm.ipmnet.ru/*˜

*rylov/yrylov.htm*
Three different representation of the proper Euclidean geometry are con-
sidered. They differ in the number of basic elements, from which the geomet-rical objects are constructed. In E-representation there are three basic ele-ments (point, segment, angle) and no additional structures. V-representationcontains two basic elements (point, vector) and additional structure: linearvector space. In

*σ*-representation there is only one basic element and ad-ditional structure: world function

*σ *=

*ρ*2

*/*2, where

*ρ *is the distance. Theconcept of distance appears in all representations. However, as a structure,determining the geometry, the distance appears only in the

*σ*-representation.

The

*σ*-representation is most appropriate for modification of the proper Eu-clidean geometry. Practically any modification of the proper Euclidean ge-ometry turns it into multivariant geometry, where there are many vectorsQ0Q1

*, *Q0Q1

*, .*, which are equal to the vector P0P1, but they are not equal
between themselves, in general. Concept of multivariance is very importantin application to the space-time geometry. The real space-time geometryis multivariant. Multivariance of the space-time geometry is responsible forquantum effects.

The proper Euclidean geometry studies mutual dispositions of geometrical objects(figures) in the space (in the set Ω of points). Any geometrical object

*O *is a subset

*O ⊂*Ω of points. Relations between different objects are relations of equivalence,when two different objects

*O*1 and

*O*2 are considered to be equivalent (

*O*1eqv

*O*2).

The geometrical object

*O*1 is considered to be equivalent to the geometrical object

*O*2, if after corresponding displacement the geometrical object

*O*1 coincides withthe geometrical object

*O*2.

The geometrical object is considered to be constructed of basic elements (blocks).

There are at least three representations of Euclidean geometry, which differ in thenumber and in the choice of basic elements (primary concepts).

The first representation (Euclidean representation, or E-representation) of the
Euclidean geometry was obtained by Euclid many years ago. The basic elementsin the E-representation are point, segment and angle. The segment is a segmentof the straight line. It consists of infinite number of points. The segment is deter-mined uniquely by its end points. The angle is a figure, formed by two segmentsprovided the end of one segment coincides with the end of other one. Properties ofbasic elements are described by a system of axioms. Any geometrical object may beconsidered to be some composition of blocks (

*point, segment, angle*). The numberof the geometric object constituents may be infinite. The segments determine dis-tances. The angles determine the mutual orientation of segments. Comparison ofdifferent geometrical objects (figures)

*O*1 and

*O*2 is produced by their displacementand superposition. If two figures coincide at superposition, they are considered tobe equivalent (equal). The process of displacement in itself is not formalized inE-representation. However, the law of the geometric objects displacement is used atthe formulation of the Euclidean geometry propositions and at their proofs.

The second representation (vector representation, or V-representation) of the
Euclidean geometry contains two basic elements (

*point, vector *). From the view-point of E-representation the vector is a directed segment of straight, determinedby two points. One point is the origin of the vector, another point is the end of thevector. But such a definition of vector takes place only in E-representation, wherethe vector is a secondary (derivative) concept. In V-representation the vector isdefined axiomatically as an element of a linear vector space, where two operationsunder vectors are defined. These operations (addition of two vectors and multipli-cation of a vector by a real number) formalize the law of displacement of vectors.

Strictly, these operations are simply some formal operations in the linear vectorspace. They begin to describe the law of displacement only after introduction ofthe scalar product of vectors in the vector space. After introduction of the scalarproduct the linear vector space becomes to be the Euclidean space, and the ab-stract vector may be considered as a directed segment of straight, determined bytwo points. However, it is only interpretation of a vector in the Euclidean represen-tation. In the V-representation the vector is a primary object. It is an element ofthe linear vector space. Nevertheless interpretation of a vector as a directed segmentof straight is very important at construction of geometrical objects (figures) frompoints and vectors.

The E-representation contains three basic elements:

*point, segment *and

*angle*.

The V-representation contains only two basic elements:

*point *and

*vector*. The angleof the E-representation is replaced by the linear vector space, which is a structure,describing interrelation of two vectors (directed segments). The vector has someproperties as an element of the linear vector space. Any geometrical figure may
be constructed of points and vectors. It means that the method of construction ofany figure may be described in terms of points and vectors. The properties of avector as an element of the linear vector space admit one to describe properties ofdisplacement of figures and their compositions.

In V-representation the angle appears to be an derivative element. It is deter-
mined by two vectors (by their lengths and by the scalar product between thesevectors). In the V-representation the angle

*θ *between two vectors P0P1 and P0P2is defined by the relation

*|*P0P1

*| · |*P0P2

*| *cos

*θ *= (P0P1

*.*P0P2)
where (P0P1

*.*P0P2) is the scalar product of vectors P0P1 and P0P2. The quantities

*|*P0P1

*| *and

*|*P0P2

*| *are their lengths

*|*P0P1

*|*2 = (P0P1

*.*P0P1)

*,*
*|*P0P2

*|*2 = (P0P2

*.*P0P2)
Thus, transition from the E-representation with three basic elements to the V-
representation with two basic elements is possible, provided the properties of onebasic element (vector) are determined by the fact, that the vector is an element ofthe linear vector space with the scalar product, given on it.

Is it possible a further reduction of the number of basic elements in the represen-
tation of the Euclidean geometry? Yes, it is possible. The representation (in termsworld function, or

*σ*-representation) of the Euclidean geometry may contain onlyone basic element (point), provided there are some constraints, imposed on any twopoints of the Euclidean space.

The transition from the E-representation to the V-representation reduces the
number of basic elements. Simultaneously this transition generates a new structure(linear vector space), which determine properties of new basic element (vector).

The transition from the V-representation to

*σ*-representation also reduces the
number of basic elements. Only one basic element (

*point*) remains. Simultaneouslythis transition replaces the linear vector space by a new two-point structure (

*worldfunction*), which describes interrelation of two points instead of two vectors. Theworld function

*σ *is defined by the relation

*σ *(

*P, Q*) =

*σ *(

*Q, P *)

*,*
The world function

*σ *is connected with the distance

*ρ *by the relation

*σ *(

*P, Q*) =

*ρ*2 (

*P, Q*)

*,*
The world function of the proper Euclidean space is constrained by a series of re-strictions (see below).

There is a difference between the structures of the V-representation and that
of

*σ*-representation. Linear vector space as a structure of the V-representation re-flects the symmetry properties of the Euclidean geometry. These symmetries of the
Euclidean space admit the motion group of the Euclidean space (translations, rota-tions). These motion groups admit one to move blocks without deformations andto construct geometric objects from blocks. The motion groups admit one also tocompare different geometrical objects, moving them in the space.

The world function

*σ*E as a structure of the proper Euclidean space reflects all
properties of the proper Euclidean space, but not only its symmetries. The worldfunction

*σ*E describes the properties of the proper Euclidean geometry completely.

Any change of the world function

*σ*E is a deformation of the proper Euclidean space,which changes its properties.

It should note in this connection, that there are different points of view on
that, what is a geometry. Well known mathematician Felix Klein supposed thatsymmetries of a geometry are the most essential properties of the geometry, andthere exist no geometries without a symmetry. For instance, he wrote that theRiemannian geometry is not a geometry. It is rather a geography or topography.

Such a viewpoint is characteristic for mathematicians. Most of them believe, that ageometry is a logical construction, and there exist no nonaxiomatizable geometries.

Alternative viewpoint is characteristic for physicists, who believe, that a geom-
etry is a science on a shape of geometrical objects and on their mutual disposition.

At such an approach it is of no importance, whether or not the geometry has anysymmetries and whether or not it is axiomatizable. If a geometry is science onthe disposition of geometrical objects, the geometry is described completely by theworld function

*σ*. Approach of physicists seems to be more realistic, because thematter distribution influences on the space-time geometry, and one cannot be sure,that the space-time geometry is uniform, and it has some symmetries.

Existence of the

*σ*-representation, containing only one basic element, means that
all geometrical objects and all relations between them may be recalculated to the

*σ*-representation, i.e. expressed in terms of the world function and only in terms ofthe world function.

If we want to construct a generalized geometry, we are to modify properties of
the proper Euclidean geometry. It means that we are to modify properties of basicelements of the proper Euclidean geometry. In the E-representation we have threebasic elements (

*point, segment, angle*). Their properties are connected, becausefinally the segment and the angle are simply sets of points. Modification the threebasic elements cannot be independent. It is very difficult to preserve connectionbetween the modified basic elements of a generalized geometry. Nobody does modifythe proper Euclidean geometry in the E-representation.

In V-representation there are two basic elements of geometry (

*point, vector *),
and in some cases the modification of the proper Euclidean geometry is possible.

However, the V-representation contains such a structure, as the linear vector space.

One cannot avoid this structure, because it is not clear, what structure may be usedinstead. Modification of the proper Euclidean geometry in the framework of thelinear vector space is restricted rather strong. (It leads to the pseudo-Euclideangeometries and to the Riemannian geometries). Besides, it appears, that there existsuch modifications of the proper Euclidean geometry, which are incompatible with
the statement that any geometrical object may be constructed of points and vectors.

The

*σ*-representation of the proper Euclidean geometry is most appropriate for
modification, because it contains only one basic element (

*point*). Any modificationof the proper Euclidean geometry is accompanied by a modification of the structure,associated with the

*σ*-representation. This structure is distance (world function),and any modification of the proper Euclidean geometry is accompanied by a mod-ification of world function (distance), and vice versa. The meaning of distance isquite clear. This concept appears in all representations of the proper Euclidean ge-ometry, but only in the

*σ*-representation the distance (world function) plays therole of a structure, determining the geometry. Modification of the distance means adeformation of the proper Euclidean geometry.

The V-representation appeared in the nineteenth century, and most of contem-
porary mathematicians and physicists use this representation. The

*σ*-representationappeared recently in the end of the twentieth century. Besides, it appears in im-plicit form in the papers, devoted to construction of T-geometry [1, 2]. In thesepapers the term ”

*σ*-representation” was not mentioned, and the

*σ*-representationwas considered as an evident possibility of the geometry description in terms of theworld function. Apparently, such a possibility was evident only for the author ofthe papers, but not for readers. Now we try to correct our default and to discussproperties of the

*σ*-representation of the proper Euclidean geometry.

*σ*-representation of the proper Euclideangeometry
Let Ω be the set of points of the Euclidean space. The distance

*ρ *=

*ρ *(

*P*0

*, P*1)between two points

*P*0 and

*P*1 of the Euclidean space is known as the Euclideanmetric. In E-representation as well as in V-representation we have

*ρ*2 (

*P*0

*, P*1) =

*|*P0P1

*|*2 = (P0P1

*.*P0P1)

*,*
It is convenient to use the world function

*σ *(

*P*0

*, P*1) = 1

*ρ*2 (

*P*
teristic of the Euclidean geometry. To approach this, one needs to describe propertiesof any vector P0P1 as an element of the linear vector space in terms of the worldfunction

*σ *(

*P*0

*, P*1), which is associated with the vector P0P1. In

*σ*-representationthe vector P0P1 is defined as the ordered set of two points P0P1 =

*{P*0

*, P*1

*}*.

We can to add two vectors, when the end of one vector is the origin of the other
P0Q1 = P0Q0 + Q0Q1 or Q0Q1 = P0Q1

*− *P0Q0

*,*
Then according to properties of the scalar product in the Euclidean space we obtainfrom the second relation (2.2)
(P0P1

*.*Q0Q1) = (P0P1

*.*P0Q1)

*− *(P0P1

*.*P0Q0)
Besides, according to the properties of the scalar product we have

*|*P1Q1

*|*2 = (P0Q1

*− *P0P1

*.*P0Q1

*− *P0P1) =

*|*P0Q1

*|*2

*− *2 (P0P1

*.*P0Q1) +

*|*P0P1

*|*2

*σ *(

*P*0

*, P*1) =

*σ *(

*P*1

*, P*0) =
(P0P1

*.*P0Q1) =

*σ *(

*P*0

*, P*1) +

*σ *(

*P*0

*, Q*1)

*− σ *(

*P*1

*, Q*1)

*,*
*∀P*0

*, P*1

*, Q*1

*∈ *Ω
It follows from (2.4) and (2.7), that for any two vectors P0P1 and Q0Q1 the
(P0P1

*.*Q0Q1) =

*σ *(

*P*0

*, Q*1)+

*σ *(

*P*1

*, Q*0)

*−σ *(

*P*0

*, Q*0)

*−σ *(

*P*1

*, Q*1)

*,*
*∀P*0

*, P*1

*, Q*0

*, Q*1

*∈ *Ω
Setting

*Q*0 =

*P*0 in (2.8) and comparing with (2.7), we obtain

*n *vectors P0P1

*, *P0P2

*, . *P0P

*n *are linear dependent, if and only if the Gram’s
determinant

*Fn *(

*Pn*),

*Pn ≡ {P*0

*, P*1

*, .Pn} *vanishes

*Fn *(

*Pn*)

*≡ *det

*||*(P0P

*i.*P0P

*k*)

*|| *= 0

*,*
In the

*σ*-representation the condition (2.10) is written in the developed form

*Fn *(

*Pn*)

*≡ *det

*||σ *(

*P*0

*, Pi*) +

*σ *(

*P*0

*, Pk*)

*− σ *(

*Pi, Pk*)

*|| *= 0

*,*
*i, k *= 1

*, *2

*, .n *(2.11)
If

*σ *is the world function of

*n*-dimensional Euclidean space, it satisfies the fol-
I. Definition of the dimension and introduction of the rectilinear coordinate sys-

*∃Pn ≡ {P*0

*, P*1

*, .Pn} ⊂ *Ω

*,*
where

*Fn *(

*Pn*) is the Gram’s determinant (2.11). Vectors

*P*0

*Pi*,

*i *= 1

*, *2

*, .n *arebasic vectors of the rectilinear coordinate system

*Kn *with the origin at the point

*P*0. The covariant metric tensor

*gik *(

*Pn*),

*i, k *= 1

*, *2

*, .n *and the contravariant one

*gik *(

*Pn*),

*i, k *= 1

*, *2

*, .n *in

*Kn *are defined by the relations

*gik *(

*Pn*)

*glk *(

*Pn*) =

*δi,*
*il *(

*P n*) = (P0P

*i.*P0P

*l*)

*,*
*i, l *= 1

*, *2

*, .n *(2.13)

*Fn *(

*Pn*) = det

*||gik *(

*Pn*)

*|| *= 0

*,*
II. Linear structure of the Euclidean space:

*i *(

*P *)

*− xi *(

*Q*)) (

*xk *(

*P *)

*− xk *(

*Q*))

*,*
where coordinates

*xi *(

*P *)

*, i *= 1

*, *2

*, .n *of the point

*P *are covariant coordinates ofthe vector P0P, defined by the relation

*xi *(

*P *) = (P0P

*i.*P0P)

*,*
III: The metric tensor matrix

*glk *(

*Pn*) has only positive eigenvalues
IV. The continuity condition: the system of equations
(P0P

*i.*P0P) =

*yi ∈ *R

*,*
considered to be equations for determination of the point

*P *as a function of coordi-nates

*y *=

*{yi}*,

*i *= 1

*, *2

*, .n *has always one and only one solution. Conditions I

*÷*IV contain a reference to the dimension

*n *of the Euclidean space.

One can show that conditions I

*÷ *IV are the necessary and sufficient conditions
of the fact that the set Ω together with the world function

*σ*, given on Ω, describesthe

*n*-dimensional Euclidean space [1].

Thus, in the

*σ*-representation the Euclidean geometry contains only one primary
geometrical object: the point. Any two points are described by the world function

*σ*, which satisfies conditions I

*÷ *IV. Any geometrical figure and any relation can bedescribed in terms of the world function and only in terms of the world function.

In the

*σ*-representation the vector P0P1 is a ordered set

*{P*0

*, P*1

*} *of two points.

Scalar product of two vectors is defined by the relation (2.8).

Two vectors P0P1 and Q0Q1 are collinear (linear dependent), if
(P0P1

*.*Q0Q1)2 =

*|*P0P1

*|*2

*|*Q0Q1

*|*2
Two vectors P0P1 and Q0Q1 are equivalent (equal), if
(P0P1

*.*Q0Q1) =

*|*P0P1

*| · |*Q0Q1

*| ∧ |*P0P1

*| *=

*|*Q0Q1

*| *(2.20)
Vector S0S1 with the origin at the given point

*S*0 is the sum of two vectors P0P1
if the points

*S*1 and

*R *satisfy the relations
In the developed form it means that the points

*S*1 and

*R *satisfy the relations
(S0R

*.*P0P1) =

*|*S0R

*| · |*P0P1

*| ,*
*|*S0R

*| *=

*|*P0P1

*|*
(RS1

*.*Q0Q1) =

*|*RS1

*| · |*Q0Q1

*| ,*
*|*RS1

*| *=

*|*Q0Q1

*|*
where scalar products are expressed via corresponding world functions by the re-lation (2.8). The points

*P*0

*, P*1

*, Q*0

*, Q*1

*, S*0 are given. One can determine the point

*R *from two equations (2.23). As far as the world function satisfy the conditions I

*÷ *IV, the geometry is the Euclidean one, and the equations (2.23) have one andonly one solution for the point

*R*. When the point

*R *has been determined, one candetermine the point

*S*1, solving two equations (2.24). They also have one and onlyone solution for the point

*S*1.

Result of summation does not depend on the choice of the origin

*S*0 in the
following sense. Let the sum of vectors P0P1 and Q0Q1 be defined with the originat the point

*S *by means of the conditions
where the points

*S *and

*R *satisfy the relations
Then in force of conditions I

*÷ *IV the geometry is the Euclidean one, and there isone and only one solution of equations (2.26) and
Let us stress that in the

*σ*-representation the sum of two vectors does not de-
pend on the choice of the origin of the resulting vector, because the world functionsatisfies the Euclideaness conditions I

*÷ *IV. If the world function does not satisfythe conditions I

*÷ *IV, the result of summation may depend on the origin

*S*0, as wellas on the order of vectors P0P1, Q0Q1 at summation. Besides, the result may bemultivariant even for a fixed point

*S*0, because the solution of equations (2.22) maybe not unique.

Multiplication of the vector P0P1 by a real number

*α *gives the vector S0S1 with
Here the points

*P*0

*, P*1

*, S*0 are given, and the point

*S*1 is determined by the relations
(S0S1

*.*P0P1) = sgn (

*α*)

*|*S0S1

*| · |*P0P1

*| ,*
*|*S0S1

*| *=

*|α| · |*P0P1

*|*
Because of conditions I

*÷ *IV there is one and only one solution of equations (2.30)and the solution does not depend on the point

*S*0.

Uniqueness of operations in the properEuclidean geometry
To define operations under vectors (equality, summation, multiplication) in the

*σ*-representation, one needs to solve algebraic equations (2.20), (2.26), (2.29), whichare reduced finally to equations (2.20), defining the equality operation.

Equality of two vectors P0P1 and Q0Q1, (P0P1 = Q0Q1) is defined by two
(P0P1

*.*Q0Q1) =

*|*P0P1

*| · |*Q0Q1

*| ,*
*|*P0P1

*| *=

*|*Q0Q1

*|*
The number of equations does not depend on the dimension of the proper Euclideanspace.

In V-representation the number of equations, determining the equality of vec-
tors, is equal to the dimension of the Euclidean space. To define equality of P0P1and Q0Q1, one introduces the rectilinear coordinate system

*Kn *with basic vectorse0

*, *e1

*, ., *e

*n−*1 and the origin at the point

*O*. Covariant coordinates

*xk *= (P0P1)

*k*
and

*yk *= (Q0Q1) are defined by the relations
The equality equations of vectors P0P1 and Q0Q1 in V-representation have the form

*k *= 0

*, *1

*, ., n − *1
According to the linear structure of the Euclidean space (2.15) and due to defi-
nition of the scalar product (2.8) we obtain

*il *= (e

*i.*e

*l*)

*,*
*i, k *= 0

*, *1

*, ., n − *1
A summation 0

*÷ *(

*n − *1) is produced on repeated indices.

Due to relations (3.4) two equality relations (3.1) take the form
By means of the second equation (3.6) the first equation (3.6) may be written in theform

*gkl *(

*xk − yk*) (

*xl − yl*) = 0
According to the III condition (2.17) the matrix of the metric tensor

*gkl *has onlypositive eigenvalues. In this case the equation (3.7) has only trivial solution for

*xk − yk*,

*k *= 0

*, *1

*, .n − *1. Then the equation (3.7) is equivalent to

*n *equations

*l *= 0

*, *1

*, ., n − *1
Thus, in the

*σ*-representation two equations (3.1) of the vector equality are equiv-
alent to

*n *equations (3.3) of vector equality in V-representation. This equivalency isconditioned by the linear structure (2.15) of the Euclidean space and by the positivedistinctness of the Euclidean metric. In the pseudo-Euclidean space the matrix ofthe metric tensor has eigenvalues of different sign. In this case the relations (3.7)and (3.8) cease to be equivalent.

Generalization of the proper Euclideangeometry
Let us consider a simple example of the proper Euclidean geometry modification.

The matrix

*gik *of the metric tensor in the rectilinear coordinate system

*Kn *with basicvectors e0

*, *e1

*, .*e

*n−*1 is modified in such a way that its eigenvalues have differentsigns. We obtain pseudo-Euclidean geometry. For simplicity we set the dimension

*n *= 4 and eigenvalues

*g*0 = 1

*, g*1 =

*g*2 =

*g*3 =

*−*1. It is the well known spaceof Minkowski (pseudo-Euclidean space of index 1). In the space of Minkowski thedefinition (3.1) of two vectors equality in

*σ*-representation does not coincide, ingeneral, with the two vectors equality (3.3) in V-representation. The two definitionsare equivalent for timelike vectors (

*gklxkxl > *0), and they are not equivalent forspacelike vectors (

*gklxkxl < *0).

Indeed, using definition (3.1), we obtain the relation (3.7) for any vectors of the
same length. It means that the vector with coordinates

*xk −yk *is an isotropic vector.

However the sum, as well as the difference of two timelike vectors of the same lengthin the Minkowski space is either a timelike vector, or a zeroth vector. Hence, therelations (3.8) take place. Thus, definitions of equality (3.6) and (3.3) coincide fortimelike vectors.

In the case of spacelike vectors

*xk *and

*yk *their difference

*xk − yk *may be an
isotropic vector. The relation (3.7) states this. It means that equality of two space-like vectors of the same length in the

*σ*-representation of the Minkowski space ismultivariant, i.e. there are many spacelike vectors Q0Q1

*, *Q0Q

*, . *which are equiv-
alent to the spacelike vector P0P1, but the vectors Q0Q1

*, *Q0Q

*, . *are not equivalent
The result is rather unexpected. Firstly, the definition of equality appears to
be different in V-representation and in

*σ*-representation. Secondly, the equalitydefinition in

*σ*-representation appears to be multivariant, what is very unusual.

In the V-representation the equality of two vectors is single-variant by definition
of a vector as an element of the linear vector space. In the

*σ*-representation theequality of two vectors is defined by the world-function. The pseudo-Euclidean spaceis a result of a deformation of the proper Euclidean space, when the Euclidean worldfunction is replaced by the pseudo-Euclidean world function. The definition of thevector equality via the world function remains to be the equality definition in allEuclidean spaces.

For brevity we shall use different names for geometries with different definition
of the vectors equality. Let geometry with the vectors equality definition (3.3), (3.2)be the Minkowskian geometry, whereas the geometry with the vectors equality defi-nition (3.1) will be referred to as the

*σ*-Minkowskian geometry. The world functionis the same in both geometries. The geometries differ in the structure of the linearvector space and, in particular, in definition of the vector equality. Strictly, there isno linear vector space in the

*σ*-Minkowskian geometry. The linear vector space is notnecessary for formulation of the Euclidean geometry, as well as for formulation ofthe

*σ*-Minkowskian geometry. The Minkowskian geometry and the

*σ*-Minkowskianone are constructed in different representations. The geometry, constructed on thebasis of the world function is a more general geometry, because the world functionexists in both geometries.

*Important remark. *In application of the space-time geometry of Minkowski to
the space-time the spacelike world lines and spacelike vectors are not used. Thedifference between the geometry of Minkowski and

*σ*-minkowskian geometry appearsto be unessential from this viewpoint. However, this remark does not concern moregeneral space-time geometries.

What of the two equality definitions (2.20), or (3.3) are valid? Maybe, both? Ap-
parently, both geometries are possible as abstract constructions. The Minkowskiangeometry and the

*σ*-Minkowskian geometry differ in such a property as the mul-tivariance of spacelike vectors equality. In the geometry, constructed on the basisof the linear vector space, the multivariance is absent in principle (by definition).

On the contrary, in the

*σ*-representation the equivalent vector is determined as asolution of the equations (2.20). For arbitrary world function one can guaranteeneither existence, nor uniqueness of the solution. They may be guaranteed, only ifthe world function satisfies the conditions I

*÷ *IV. It means, that the multivarianceis a general property of the generalized geometry, whereas the single-variance of theproper Euclidean geometry is a special property of the proper Euclidean geome-try. The special properties of the proper Euclidean geometry are described by theconditions I

*÷*IV. All these conditions contain a reference to the dimension of theEuclidean space. The single-variance of the proper Euclidean geometry is a specialproperty, which is determined by the form of the world function, but it does notcontain a reference to the dimension, because it is valid for proper Euclidean spaceof any dimension.

From formal viewpoint the

*σ*-Minkowskian geometry is more consistent, because
the definition (3.1) does not depend on the choice of the coordinate system. In theMinkowskian geometry the two vectors equality definition contains a reference to thecoordinate system, which may be considered as an additional structure introducedto the geometry of Minkowski. After this the geometry of Minkowski should bequalified as a fortified geometry. It means that a physical geometry is equipped bysome additional geometric structure. This structure suppresses multivariance of thespacelike vectors equality.

We may dislike this fact, because we habituated ourselves to single-variance
of the two vectors equivalence. However, we may not ignore the fact, that themultivariance is a natural property of geometry. It means that we are to use the
two vector equivalence in the form (2.20).

Let us discuss corollaries of the new definition of the vector equivalence for the
real space-time. In the Euclidean space the straight line, passing through the points

*P*0

*, P*1 is defined by the relation
In the

*σ*-representation the parallelism P0R
defined by the relation (2.19) in the

*σ*-Minkowskian space-time. In the Minkowskianspace-time the vector parallelism P0R P0P1 is defined by four equations, describ-ing proportionality of components of vectors P0R and P0P1. In the

*σ*-Minkowskianspace-time the straight (4.1) is a one-dimensional line for the timelike vector P0P1,whereas the straight (4.1) is a three-dimensional surface (two planes) for the space-like vector P0P1. In the Minkowskian space-time all straights (timelike and space-like) are one-dimensional lines. Thus, the straights, generated by the spacelike vec-tor, are described differently in the Minkowskian geometry and in the

*σ*-Minkowskianone.

The timelike straight in the geometry of Minkowski describes the world line of
a free particle, whereas the spacelike straight is believed to describe a hypotheticalparticle taxyon. The taxyon is not yet discovered. In the

*σ*-Minkowskian space-timegeometry this fact is explained as follows. The taxyon, if it exists, is described bytwo isotropic three-dimensional planes, which is an envelope to a set of light cones,having its vertices on some one-dimensional straight line. Such a taxyon has not beendiscovered, because one looks for it in the form of the one-dimensional straight line.

In the conventional Minkowskian space-time the taxyon has not been discovered,because there exist no particles, moving with the speed more than the speed ofthe light. Thus, the absence of taxyon is explained on the geometrical level in the

*σ*-Minkowskian space-time geometry, whereas the absence of taxyon is explained onthe dynamic level in the conventional Minkowskian space-time geometry.

Let us consider the non-Riemannian space-time geometry

*G*d, described by the

*σ*d =

*σ*M + sgn (

*σ*M)

*d,*

*−*1

*, *if

*x < *0
where

*σ*M is the world function of the geometry of Minkowski,
constant,

*c *is the speed of the light and

*b *is some universal constant [

*b*] =g/cm. Theconstant

*λ*0 is some universal length.

The space-time geometry (4.2) is discrete , because in this space-time geometry
there are no vectors P0P1, whose length

*|*P0P1

*| *be small enough, i.e.

*∈ *0

*, *4

*λ*4

*,*
In other words, the space-time geometry (4.2) has no close points. The space-timegeometry, where there are no close points should be qualified as a discrete geometry.

It is rather unexpected that a discrete space-time geometry may be uniform andisotropic as the geometry (4.2). It is unexpected, that a discrete geometry may begiven on a continuous manifold. But this puzzle is connected with the fact, thatusually one uses V-representation, where a discrete geometry is given on a latticespace. Another unexpected properties of a discrete space-time geometry can befound in [5].

In the space-time geometry (4.2) a free pointlike particle motion is described by
a chain

*T*br of connected segments

*T*[

*PkPk*+1] of straight

*T*[

*PkPk*+1] =

*R| *2

*σ*d (

*Pk, R*) +
The particle 4-momentum p is described by the vector P

*k*P

*k*+1
p =

*bc*P

*k*P

*k*+1

*,*
*m *=

*b |*P

*k*P

*k*+1

*| *=

*bµ,*
Here

*m *is the particle mass,

*µ *is the geometrical particle mass, i.e. the particle massexpressed in units of length. Description of

*G*d is produced in the

*σ*-representation,because description of non-Riemannian geometry in the V-representation is impos-sible. In the geometry

*G*d segments

*T*[

*PkPk*+1] of the timelike straight are multivariant
in the sense, that

*T*[

*PkPk*+1] is a cigar-shaped three-dimensional surface, but not a
For the free particle the adjacent links (4-momenta)

*T*[

*PkPk*+1] and

*T*[

*Pk*+1

*Pk*+2] are
P

*k*P

*k*+1eqvP

*k*+1P

*k*+2 :
(P

*k*P

*k*+1

*.*P

*k*+1P

*k*+2) =

*|*P

*k*P

*k*+1

*| · |*P

*k*+1P

*k*+2

*| ,*
*|*P

*k*P

*k*+1

*| *=

*|*P

*k*+1P

*k*+2

*|*
This equivalence is multivariant in the sense that at fixed link P

*k*P

*k*+1 the adjacentlink P

*k*+1P

*k*+2 wobbles with the characteristic angle

*θ *=
shape of the chain with wobbling links appears to be random. Statistical descriptionof the random world chain appears to be equivalent to the quantum description interms of the Schr¨odinger equation [3]. Thus, quantum effects are a corollary ofthe multivariance. Here the multivariance is taken into account on the level of thespace-time geometry. In conventional quantum theory the space-time geometry issingle-variant, whereas the dynamics is multivariant, because, when one replacesthe conventional dynamic variable by a matrix or by operator, one introduces themultivariance in dynamics.

Thus, the multivariance is a natural property of the real world (especially of
microcosm). At the conventional approach one avoids the multivariance ”by hand”from geometry and introduces it ”by hand” in dynamics, to explain quantum ef-fects. It would be more reasonable to remain the multivariance in the space-time
geometry, because it appears there naturally. Besides, multivariance of the space-time geometry has another corollaries, other than quantum effects. In particular,multivariance of the space-time geometry is a reason of the restricted divisibility ofreal bodies into parts (atomism) [4].

Multivariance and possibility of the geometryaxiomatization
In the E-representation of the proper Euclidean geometry one supposes that any ge-ometrical object can be constructed of basic elements (points, segments, angles). Inthe V-representation any geometrical object is supposed to be constructed of basicelements (points, vectors). Deduction of propositions of the proper Euclidean ge-ometry from the system of axioms reproduces the process of the geometrical objectconstruction. There are different ways of the basic elements application for a con-struction of some geometrical object, because such basic elements as segments andvectors are simply sets of points. In the analogical way a proposition of the properEuclidean geometry may be obtained by different proofs, based on the system ofaxioms.

Let us consider a simple example of the three-dimensional proper Euclidean
space. In the Cartesian coordinates x =

*{x, y, z} *the world function has the form
(

*x − x *)2 + (

*y − y *)2 + (

*z − z *)2
We consider a ball

*B *with the boundary

*x*2 +

*y*2 +

*z*2 =

*R*2
where

*R *is the radius of the ball. The ball

*B *may by considered as constructed onlyof the set

*S *of the straight segments

*T *(

*−x, y, z*;

*x, y, z*),

*x*2 +

*y*2 +

*z*2 =

*R*2

*T *(

*−x, y, z*;

*x, y, z*)
Any segment

*T *(

*−x, y, z*;

*x, y, z*) is a segment of the length

*R*2

*− y*2

*− z*2. Its center
is placed at the point

*{*0

*, y, z}*. The segment

*T *(

*x, y, z*;

*−x, y, z*) is the set of points

*P *with coordinates (

*x , y, z*)

*T *(

*−x, y, z*;

*x, y, z*) =

*x |x *2

*≤ x*2

*, y, z*
Different segments

*T *(

*−x, y, z*;

*x, y, z*) have no common points, and any point

*P *ofthe ball

*B *belongs to one and only one of segments of the set

*S*.

Let us consider the ball

*B *with the boundary

*x*2 +

*y*2 +

*z*2 =

*R*2

*− d/*2
But now we consider non-Riemannian geometry

*G*d, described by the world function

*σ*d
where

*σ*E is the proper Euclidean world function (5.1). Thus, if

*σ*d
function

*σ*d distinguishes slightly from

*σ*E.

The straight segment

*T*[

*P*0

*P*1] between the points

*P*0 and

*P*1 is the set of points

*T*[

*P*0

*P*1] =

*R| *2

*σ*d (

*P*0

*, R*) +
The segment

*T*[

*P*0

*P*1] is cigar-shaped two-dimensional surface. Let the length of thesegment be

*l*
*d *and

*τ *, 0

*< τ < l *be a parameter along the segment. The radius

*ρ *of the hollow segment tube as a function of

*τ *has the form
(2

*τ − d*) (2

*l − *3

*d*) (2

*l − *2

*τ − d*)

*,*
2

*d < τ < l − *2

*d *(5.8)
The maximal radius of the segment tube

*ρ*max =

*ld/*2 at

*τ *=

*l/*2.

It is clear that the ball

*B *cannot be constructed only of such tube segments.

These hollow segments could not fill the ball completely. The problem of construc-tion of the ball, consisting of basic elements (points and segments), appears to bea very complicated problem in the modified geometry

*G*d. This problem of the ballconstruction associates with the impossibility of deducing the geometry

*G*d from asystem of axioms. It seems that the geometry, deduced from the system of axioms,cannot be multivariant, because any proposition, obtained from axioms by meansof the formal logic is to be definite. It cannot contain different versions.

On the other hand, the multivariance is an essential property of the real micro-
cosm. It is a reason of quantum effects and atomism. The space-time geometry is abasis of dynamics in microcosm.

In the multivariant geometry one constructs geometrical objects by means of
the deformation principle. Geometrical object

*O*E is constructed in some region

*S*1 of the proper Euclidean space. In means that all blocks of

*O*E as well as thegeometrical object

*O*E itself are expressed in terms of the world function

*σ*E ofthe proper Euclidean geometry. Let us imagine that we need to shift

*O*E in otherregion

*S*2 of the space. We may move all blocks of

*O*E from

*S*1 to

*S*2 and construct ageometrical object

*O *in

*S*
2 from blocks, using the same prescription of construction,
which has been used at construction of

*O*E. This prescription, written in terms ofthe world function, has the same form in any geometry, if one uses only points asthe basic concept of the geometry. Points considered as blocks are not deformed atmotion from the region

*S*1 to the region

*S*2, even if the geometry in the region

*S*2
distinguishes from the geometry in the region

*S*1 (different world functions in theregions

*S*1 and

*S*2).

This consideration explains application of the deformation principle, but it is
not its proof. The deformation principle is the principle which admits one to usephysical geometries for description of the nonaxiomatizable space-time geometry
Thus, there are three different representation of the proper Euclidean geometry. Inthe E-representation there are three basic elements: point, segment, angle. Thesegment and the angle are some auxiliary structures. The segment is determinedby two points. The angle is determined by three points, or by two connected seg-ments. In V-representation there are two basic elements: point and vector (directedsegment). The angle is replaced by two segments (vectors). Its value is determinedby the scalar product of two vectors. Reduction of the number of basic elementsis accompanied by appearance of new structure: linear vector space with the scalarproduct, given on it. Information, connected with the angle, is concentrated now inthe scalar product and in the linear vector space. Such a concept as distance is aproperty of a vector (or a property of two points).

In the

*σ*-representation there is only one basic element: point. Interrelation of
two points (segment) is described by the world function (distance). Mutual direc-tivity of segments, (or an angle) is considered as an interrelation of three points.

It is described by means of the scalar product, expressed via distance (world func-tion). In

*σ*-representation the world function (distance) turns into a structure in thesense, that the world function satisfies a series of constraints (conditions I

*÷ *IV).

The concept of distance exists in all representations. But in the E-representationand in the V-representation the distance is not considered as a structure, becausethe conditions I

*÷ *IV are not considered as a constraints, imposed on the distance(world function). Of course, these conditions are fulfilled in all representations, butthey are considered as direct properties of the proper Euclidean space, but not asconstraints, imposed on distance of the proper Euclidean space.

The

*σ*-representation is interesting in the sense, that it contains only one basic
element (point) and only one structure (world function). All other concepts ofEuclidean geometry appear to be expressed via world function. Supposing thatthese expressions have the same form in other geometries, one can easily constructthem, replacing world function. This replacement looks as a deformation of theproper Euclidean space.

Usually a change of a representation is a formal operation, which is not accom-
panied by a change of basic concepts. For instance, representations in differentcoordinate systems differ only in the form of corresponding algebraic expressions.

A change of representation of the proper Euclidean geometry is accompanied by achange of basic concepts, when the primary concepts turns to the secondary onesand vice versa. The procedure of a change of representation may be qualified as
the logical reloading. The logical reloading is rather rare logical procedure. Such achange of primary concepts is unusual and difficult for a perception.

In particular, such difficulties of perception appear, because the linear vector
space is considered as an attribute of the geometry (but not as an attribute of the ge-ometry description). The linear vector space is an attribute of the V-representationof the proper Euclidean geometry. There are geometries (physical geometries), wherethe linear vector space cannot be introduced at all. Physical geometries are describedcompletely by the world function, and the world function is the only characteristic ofthe physical geometry. All other attributes of the physical geometry are derivative.

They can be introduced only via the world function. The physical geometry cannotbe axiomatized, in general. The proper Euclidean geometry

*G*E is an unique knownexample of the physical geometry, which can be axiomatized. Axiomatization ofthe proper Euclidean geometry

*G*E is used for construction of

*G*E. When the properEuclidean geometry

*G*E is constructed (deduced from the Euclidean axiomatics),one uses the fact that the

*G*E is a physical geometry. One expresses all geometricalobjects of the proper Euclidean geometry

*G*E in terms of the Euclidean world func-tion

*σ*E. Replacing

*σ*E in all definitions of the

*G*E by another world function

*σ*, oneobtains all definitions of another physical geometry

*G*. It means that one obtainsanother physical geometry

*G*, which cannot be axiomatized (and deduced from someaxiomatics)
Impossibility of the geometry

*G *axiomatization is conditioned by the fact, that
the equivalence relation is intransitive, in general, in the geometry

*G*. However, inany mathematical model, as well as in any geometry, which can be axiomatized, theequivalence relation is to be transitive. Almost all mathematicians believe, that anygeometry can be axiomatized. Collision of this belief with the physical geometryapplication generates misunderstandings and conflicts [5].

[1] Yu.A.Rylov, Extremal properties of Synge’s world function and discrete geome-
try .

*J.Math. Phys. *31, 2876-2890 (1990).

[2] Yu.A. Rylov, Geometry without topology as a new conception of geometry.

*Int.*
*Jour. Mat. & Mat. Sci. *30, iss. 12, 733-760, (2002), (

*e-print math.MG*/0103002).

[3] Yu.A. Rylov:, Non-Riemannian model of the space-time responsible for quantum
effects.

*Journ. Math. Phys*. 32(8), 2092-2098, (1991).

[4] Yu.A. Rylov, Tubular geometry construction as a reason for new revision of the
space-time conception. Printed in

*What is Geometry? *polimetrica Publisher,Italy, 2006, pp.201-235.

[5] Yu. A. Rylov, Multivariance as a crucial property of microcosm.

*e-print*
Source: http://gasdyn-ipm.ipmnet.ru/~rylov/dreg2e.pdf

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