At the metric conception of geometry (MCG) the geometry is determined
on any point set **M **by setting real function
*G(P,P') *with the properties

*G(P,P')=G(P',P),
G(P,P)=0*

The function *G(P,P') *is called the world
function, and the geometry determined by the world function is called tubular
geometry, or shortly T-geometry. All relations of T-geometry are derived
as relations of proper Euclidean geometry written in terms of the world
function.

Vector ** PQ **of T-geometry is the ordered
set of points

In the special case, when the points *P* and
*R* coincide, one has

(2)

which means the cosine theorem for the triangle with vortices at the
points *P,Q,S.*

Two vectors ** P**_

Taking into account that |** P**_

(4)

In general, the condition (4) is not so strong, as the condition (3),
it means that vectors ** P**_

In general, in the proper Euclidean space
*n*
vectors ** P**_

(5)

Thus, in virtue of (2) the necessary and sufficient condition of the
linear dependence of *n* vectors
** P**_

The proper Euclidean geometry can be described completely in terms of
the world function and finite sets of points *P^n={P_0,P_1,...P_n}*.
Such a description of geometry will be referred to as sigma-immanent description.
At such a description of the Euclidean geometry the world function is to
satisfy some sigma-immanent conditions I - III.

One can prove the following

This theorem means that information, concluded in the world function
is sufficient for construction of the Euclidean geometry. This information
is sufficient also for construction of a geometry, if conditions I
- III, specific for Euclidiean geometry are not satisfied.
Any world function assiciates with some geometry (T-geometry).

In particular, in any geometry *n* points
*P^n={P_0,P_1,...P_n}*,
which do not satisfy the condition* *(5) determines
the *n*th order natural geometric object (NGO).
This object, called the *n*th order tube, is
the set *T*(*P*^n*)
*of points, defined by the relation

(6)

In the case of the proper Euclidean geometry the *n*th
order tube *T*(*P*^n*)
*is the *n*-dimensional plane. In the
case of two points *P_0,
P_1 *the first order tube *T(P_0P_1
) *is the straight line, passing through
points *P_0,
P_1*. Changing the Euclidean world function
*G*_E, one obtains destorted
Euclidean space with the world function *G*_D.
In this more general case, when conditions I - III are not fulfilled, and
T-geometry is not the Euclidean one, the tube *T(P_0P_1
) *is a hallow *(m-1)*-dimensional
tube (*m* is the dimension of the space), but
not a line. The T-geometry with world function

(7)

where *D *is some function, is nondegenerate
T-geometry. In such a geometry the segment *T*([*P**_0P_1
*])of the tube*T(P_0P_1)
*between points *P_0,
P_1 *has the cigarlike shape. This
"cigar" is a set of ends *R* of the vector * P_0R*,
parallel to the vector

Natural geometrical objects (NGO) are sets of points determined by the
geometry and several parameters (basic points of NGO). Points of the space
are parameters of NGOs. Any geometry is described by their natural geometric
objects (NGO). The order of NGO is determined by the number of basic points.
In the proper Euclidean geometry the point *P *is the zeroth order
NGO defined by one parameter (the same point *P*). The straight *L(P,Q)
*is the first order NGO defined by two different points *P,Q*.
The plane *L(P,Q,R) *is the second order* *NGO, defined by three
different points *P,Q,R* which do not lee on one straight. In other
geometries NGOs have another form, but they always are defined by corresponding
geometry. For instance, in the Riemannian geometry the first order NGO
is a geodesic. Description of a geometry in terms of its NGOs is the most
adequate way of construction of the geometry. For instance, the proper
Euclidean geometry is described conventionally in terms of points, straights,
planes,....(i.e. in terms of its NGOs). In the proper Euclidean geometry
NGOs are defined by their properties by means of axioms. On the other hand,
if distances *S=S(P,Q) *between all pairs of points *P,Q *are
given, the NGOs of Euclidean geometry can be defined via distances. In
this case the axioms describing properties of NGOs turn to theorems, and
geometry is defined by the form of the distance function *S=S(P,Q) .*
Practically it is more convenient to use instead of *S=S(P,Q) *the
so called world function

*G(P,Q) = S(P,Q) S(P,Q)/2*

which is always real, even so the geometry is the Minkowski geometry.

Geometry of any space may be considered as a result of a deformation
of the proper Euclidean geometry, when one changes distances *S=S(P,Q)
*between the space points *P, Q*. Any such deformation changes
the shape of NGOs. On the other hand, any geometry is some kind of generalization
of the proper Euclidean geometry. The way of generalization of the Euclidean
geometry depends essentially on the way of definition of the Euclidean
straight (the first order NGO). If the Euclidean straight *L(P,Q)*
is defined as shortest **line** between the points *P,Q ,* determining
*L(P,Q)*, one obtains the Riemannian geometry. If the Euclidean straight
*L(P,Q)* is defined by its collimetric property as a **set of points**,
one obtains the T-geometry (or tubular geometry, or geometry of tubes).
The term "collimetric" is constructed of two terms "collinear" and "metric".
Collimetric property means the collinearity of two vectors **RP** and
**RQ** expressed via distances between points *P,Q,R*. Mathematically
collinearity **RP**||**RQ **is expressed by means of the relation

(**RP**.**RQ**)(**RP**.**RQ**) = (**RP**.**RP**)(**RQ**.**RQ**)

where according to the cosine theorem the scalar product (**RP**.**RQ**)
of vectors **RP** and **RQ
**can** **be expressed via their lengths
|**RP|** and |**RQ|**, or via** **world function *G(P,Q) = S(P,Q)
S(P,Q)/2 *by the relation

(**RP**.**RQ**) = *G(R,P) + G(R,Q) -- G(P,Q)*

The Riemannian geometry is such a generalization of the proper Euclidean
geometry which uses definition of the first order NGO as the shortest **line .
**This generalization to the Euclidean geometry is

The T-geometry is a generalization of the proper Euclidean geometry
which uses the definition of the first order NGO as a **set of points**
having the collimetric property. The set of points does not need an additional
definition. As a result the **T-geometry is a geometry based on only metric
structure**. The T-geometry is more general and simpler than the Riemannian
geometry. T-geometry contains the Riemannian geometry as a special case.
On the other hand the first order NGO defined as a set of points is **not
a line in general**. The first order NGO in T-geometry appears to be
a hallow tube. In some cases these tubes can degenerate into one-dimensional
lines. Then one obtains degenerate geometries (Riemannian and Euclidean).
Nondegenerate geometry is another name for the T-geometry.

In the T-geometry the first order tube *T_{P,Q}* (the first order
NGO) is determined by the points *P* and *Q
*as the set of running
points *R*

*T_{PQ} =
*(*R|***RP**||**RQ
**)*
= *(*R|*(**RP**.**RQ**)(**RP**.**RQ**)
= (**RP**.**RP**)(**RQ**.**RQ**))

where all scalar products can be expressed via the world function by means of the relation

(**RP**.**RQ**) = *G(R,P) + G(R,Q) -- G(P,Q).*

It is clear intuitively that the whole geometry can be constructed,
if one can construct NGO. It is clear in the case of the proper Euclidean
geometry. It is so in the case of a general geometry defined by an arbitrary
world function *G(P,Q)*.

For degenerate geometries (Euclidean, Riemannian, Minkowski) the first
order tube *T_{P,Q}* degenerates into one-dimensional line *L_{P,Q}*
due to extremal properties of the world function. In this case it is possible
to construct a manifold, to determine the dimensionality of the space and
introduce a coordinate system. (see the monograph
and corresponding reference in the list of papers).
Construction of the manifold on the basis of the metric means that in some
cases it is possible to construct the topological structure on the basis
of the metric one.

In the general case it is impossible to construct a manifold. A mathematical technique of work with a nondegenerate geometry is not yet developed. But in the case, when the real world function is distinguished slightly from that of the Minkowski space, one can use approximately the relation

*G=G*_M*+D*,

where *G_*M is the world
function of the Minkowski space and *D* is a distortion function describing
a degree of nondegeneracy of the space-time.

In this case it is possible to construct a manifold on the base of *G*_M,
to introduce a coordinate system and to consider the distortion function
*D=D(P,Q)* as some two-point field in the Minkowski space-time. If
the real space-time is uniform and isotropic, the distortion function is
a function of only *G*_M

*D=D(G*_M*)*

In the space-time with the nonvanishing distortion* D* a world
tube of a real particle of the mass *m* is defined as a broken tube
consisting of segments *T_[P,P'] *of the same
length. The length of the segments is proportional to the mass *m*
of the particle. Thus, the mass of a particle becomes a geometrical quantity.
As a result of the nonvanishing distortion the relative position of two
adjoining segments is not fixed, (although the hyperbolic cosine of the
angle between them for a free particle is equal to *1*). In this sense
the world tube of a real particle is stochastic. The stochasticity is the
more intensive the less the mass of the particle.

Experiments with a single *S_*st
are irreproducible, whereas experiments with *E[S_*st*]*
are reproducible. In other words, a result of a single experiment is irreproducible,
but distributions of results are reproducible,

The principal goal of the statistical description is a construction
of *E[S_*st*]*
as a dynamic system. If such a dynamic system *E[S_*st*]*
has been constructed, then, investigating this dynamic system, one can
calculate and explain all results of reproducible experiments with *S_*st
(in reality with ensembles of *S_*st
). There is no general way of the construction of the dynamic system
*E[S_*st*],*
corresponding to the stochastic system *S_*st.

The statistical description in itself does not need any probabilistic
constructions. These probabilistic constructions (the statistical ensemble
as a tool for calculation of average values) are needed only for interpretation
of the *E[S_*st*]
*dynamcis and experiments with *E[S_*st*]*
in terms of the mean behaviour of the stochastic system *S_*st.
Such an interpretaition has also a heuristic meaning.

If the distortion function has asymptotic value
*D =
h/b*, where *h* is the quantum constant
and *b* is some universal constant connecting
the usual mass of the particle with the length of the world tube segment
*T_[P,P']*, the statistical
description of stochastic world tubes coincides with the quantum description
(Schroedinger equation).

Updated 25/12/2001