Very short author's review of the monograph: ''Introduction to nondegenerate geometry''

Yuri A. Rylov

The monograph is an introduction to the most general type of geometry, which can be defined as the metric geometry described in terms of NGO (natural geometric objects). This new type of geometry (called T-geometry, or nondegenerate geometry) may be interpreted also as an alternative way of generalization of the Euclidean geometry other than the Riemannian geometry. The Riemannian geometry is a generalization of the Euclidean geometry founded on extremal property of the Euclidean straight line, i.e. on the property of the straight to be the shortest (extremal) line connecting two points. The T-geometry is a generalization of the same Euclidean geometry based on collimetric property of the Euclidean straight. The term "collimetric" is formed from two terms "collinear" and "metric" The collimetric property means the condition of collinearity of vectors RP and. RQ expressed via distances between the points P,Q,R. Being applied for description of the space-time, the nondegenerate geometry explains freely quantum effects as geometrical effects. If the nondegenerate geometry is considered to be the space-time geometry, the particle world lines are substituted by stochastic world tubes and the quantum constant appears to be a geometrical quantity (thickness of the tubes). The particle mass becomes a geometrical quantity also. In the nondegenerate space-time geometry one does not need quantum principles for explanation of quantum effects. The nondegenerate space-time geometry admits one to restrict the field of scope of quantum mechanics in the same way as the statistical physics restricts the field of scope of the axiomatic thermodynamics.

Concept of NGO (natural geometric objects) is a crucial concept of T-geometry (T-geometry is another name of nondegenerate geometry ). By definition the natural geometric object (NGO) is a set of points determined by geometry and only by geometry. Any NGO depends on several parameters -- points. For instance, in the Euclidean geometry the NGO of the first order is a straight line which is determined by two parameters -- two different points. The NGO of the second order is a plane which is determined by three parameters -- three different points not lying on one straight line. Description of geometry in terms of NGO is very old idea. For instance, the Euclidean geometry is presented in terms of points, lines and planes (which are NGOs of the Euclidean geometry). NGOs of the Euclidean geometry are defined via their properties by means of the Euclidean axioms, because the NGOs are fundamental geometrical objects which cannot be defined through another more fundamental objects. Practically one needs the Euclidean axiomatics only for definition of NGOs. If one defines NGOs by some nonaxiomatic way, the Euclidean axioms stop to be necessary.

There is a lot of nonuniform geometries, where it is very difficult to define NGO by means of a system of axioms (for instance, the Riemannian geometries), because any geometry has its own specific NGOs. In the T-geometry all NGOs are determined through the metric (distance between any pair points), and it is the main innovation of the T-geometry. As a result the T-geometry needs no axioms like Euclidean axioms, because it is clear intuitionally that the geometry can be formulated in terms of NGOs, and Euclidean geometry is an example of such a kind.

For definition of NGOs via metric one uses the Euclidean geometry, where NGOs may be defined very simply. For instanace, the NGO of the first order (straight line), depending on points A and B can be defined as the set of points R for which the area of the triangle ABR vanishes. Indeed, three points A,B,R lie on one straight line then and only then, when the area of the triangle ABR vanishes. The area of the triangle ABR may be expressed via distances between the points A,B,R by means of the Hero's formula. This admits one to formulate the rule for determination of the first order NGO via metric. NGOs of the highest order can be defined by a like way. In the monogrph it is shown that, indeed, the nondegenerate geometry can be constructed on the basis of NGOs.

Table of contents

1. Introduction. Natural geometric objects and different conception of geometry.

1.1. Axiomatic conception of geometry.

1.2. Conception of differential geometry.

1.3. Metric conception of geometry.

1.4. Difference of geometric conceptions.

1.5. Generalizations of the Euclidean geometry

1.6. On the content of the book.

2. World function of the Riemannian space.

2.1. Riemannian geometry in terms of the world function.

2.2. Two-point formalizm.

2.3. Curvature of the Riemannian space.

3. T-geometry.

3.1. Sigma-space ant its properties.

3.2. Euclidean sigma-space.

3.3. Separation of tube into parts and concept of "inside".

3.4. Euclideness and extremality.

3.5. Extremality and curtailed tubes.

4. Geodesic mapping.

4.1. Vector sigma-space. Geodesic mapping.

4.2. Dense sigma-space.

4.3. Riemannian space.

5. T-geometry on manifold.

5.1. World function on a manifold. The simplest generalizations of the Riemannian space.

5.2. Tubular model of the space-time and free particle in this model.

5.3. Dynamical conception of statistical description.

5.4. Oriented mass.

5.5. Mean statistical dynamic system and reduction of its state.

There more detailed author's review in English with formulas (about 23 pp in PostScript format (205 KB)).

The monogrph is written in Russian (about 170 pp in PostScript format (2000 KB)), Three figures: fig1, fig2, fig3.

It can be downloaded also in the form of pkzipped PostScript file indg.zip (475 KB). Figures: indgf.zip.