Conventional geometry (Euclidean, Riemannian, Minkowskian)
uses two principal concepts:
(1) manifold, (2) distance (interval).
The concept of a manifold describes affine properties of the space, ordering of space points, linear dependence of vectors, dimensionality, etc.
The distance describes metric properties of the space. Conventionally the distance (interval) is given for infinitesimal close points of the space. Metric properties are connected with affine properties . This fact is displayed as follows. A finite distance satisfies some differential equation. If the distance is given in a finite form, i.e. as a function of any pair of space points and does not satisfy this equation, a contradiction arises. In this case the concept of the manifold is incompatible with the metric properties.
Considering metric properties as a more important properties
and giving the distance in a finite form as a function of any two
points, one can construct the geometry, where affine properties
are described through metric ones. The metric approach to the geometry
is distinguished from the well known metric geometry by
(1) It does not used the triangle rule and can be applied to the space-time.
(2) Metric approach introduces natural geometric objects (tubes) The tube of nth order is determined by n+1 different points of the space.
In the Euclidean space the nth order tube is n-dimensional plane determined by n+1 different points n=0,1,2... Metric relations of the Euclidean geometry can be used for a definition of principal concepts of a more general geometry.
Let vector PQ be an ordered set of two points P and Q of the space. The scalar product (OP.OQ) of two vectors OP and OQ having a common origin O can be defined, basing on the cosine theorem of the Euclidean geometry
This definition can be applied to any space, where the world function is given. For instance, in the Minkowskian space-time the zeroth order tube, determined by the point P, is the light cone with the vertex at P. First order tube, determined by two points P and Q, is a straight line, passing through P and Q, provided the vector PQ is timelike [(PQ.PQ)>0], or null [(PQ.PQ)=0]. If the vector PQ is spacelike [(PQ.PQ)<0], the first order tube is not a line. It is two 3-dimensional planes tangent to all light cones having their vertices on the spacelike straight line containing points P,Q.
The first order tube is different for the conventional approach and for the metric approach. In the first case the natural geometric object is a spacelike straight line. In the case of the metric approach it is a 3-dimensional surface. The difference arises, because passing from the Euclidean space to the Minkowskian one, different properties are generalized. At the conventional approach both metric and affine properties are generalized. At the metric approach only metric properties are used.
Assuming that the first order geometrical object is connected with a free particle, one concludes that the metric approach is preferable, because taxions (particles described by a spacelike world line) are not disovered, whereas spacelike first order tubes are not lines. If they exist and describe some reality, nobody connect them with superlight particles (taxions).
Note that one equation in the n-dimensional space determines, in general, a (n-1)-dimensional surface. In the n-dimensional Euclidean space the first order tube is always one-dimensional straight line (but not a (n-1)-dimensional surface) This fact is connected with specific properties of the world function of the Euclidean space. In the Minkowskian space-time these specific properties do not take place for spacelike vectors PQ. As a result one obtains the general case, when the first order tube is 3-dimensional surface.
(See for details J. Math. Phys. 31,(1990)2876; 32,(1991)2082; 33,(1992)4220)