This book on geometry has been written by the physicist -- theorist. For him the geometry is a science on space-time properties of the environment. In this sense the geometry is a foundation of any physical theory. The presented monograph is a result of a development and attempts to find a sense of original formal mathematical papers having been written by the author in 1959--1992. Nondegenerate geometry is the metric geometry presented in terms of natural geometric objects (NGO). Appearance of this geometry is connected with overcoming a serious psychlogical barrier. Appearance of the term "natural geometrical object" and realization of its value for presentation of the geometry is the principal author's contribution, although presentation of geometry in terms of points, straights and planes (this are natural geometric objects of the Euclidean geometry) goes back to the time of Euclidus. For instance, in the Riemannian geometry a geodesic plays the role of a straight line. In the nondegenerate geometries the analog of the straight appears to be a hallow tube (i.e. non-one-dimensional geometric object, but not a line). This was the psychlogical barrier for development of the nondegenerate geometry. The author can judge this by his own experience. In 1959 author had understood that the world function (half of the square of distance), given on a manifold, describes the geometry completely. Nevertheless he needed about thirty years to realize a possibility of existence of nondegenerate geometries (where straights are replaced by tubes), although the author succeeded to derive mathematical formalism leading to construction of nondegenerate geometries without efforts. These geometries describe the general (nondegenerate) case, whereas the Euclidean (and Riemannian) geometry, where these tubes degenerate into lines, represents a special (degenerate) case. It is very difficult to imagine existence of nondegenerate geometries, if one knows only degenerate geometries, although arguments in favour of nondegenerate geometries are very simple.
Let us manifest these arguments in the example of NGO, determined by two points, which represents in Euclidean geometry the straight line, passing through these points. In the Euclidean geometry three points lie on one straight, if and only if the area of the triangle with vertices at these points vanishes. The trangle area can be expressed via distances between vertices by means of the Hero's formula. Fixing position of two points and equating the triangle area to zero, one obtains the equation for determination of the set of points lying on the straight line, passing through two fixed points. This definition of the straight in Euclidean geometry contains only distances between pairs of points. It may be used in metric geometry for determination of the natural geometric object (NGO), which is an analog of the Euclidean straight. As far as there is only one equation determining NGO, the dimension of NGO is n-1, if the dimension of the space is n. Degeneration into one-dimensional line is possible also, as it takes place in Euclidean geometry, where it is conditoned by a specific form of the distance function. The nondegenerate geometry is very simple. It does not contain any enigmatic suppositions like hypothesis on stochastic or quantal origin of geometry. It simply carries out one of reasonable possibilities of the geometry construction which was passed for some reason in the process of the mathematics development.
The nodegenerate geometry is very much promising direction in the geometry development. First and foremost it is connected with the fact that the geometry of real space-time is nondegenerate, with characteristic thickness of tubes being microscopic and unessential at description of macroscopic phenomena. In modern natural sciences one extrapolates the degenerate geometry into the field of microscopic phenomena, where it does not work. This leads necessarily to difficulties, because the space-time geometry is a foundation of all natural sciences. But scientists always were inventive and could remove consequences of incorrect statements by means of special compensative constructions. Well known example of such a construction is the Ptolemaic doctrine in celestial mechanics, compensating negative consequences of the incorrect hypothesis on fixed Earth and predicting correctly solar eclipses and motion of planets. But such a compensation never is complete. In the scope of the Ptolemaic doctrine it is impossible to discover the gravitation law and laws of the mechanics which are known now as the Newton mechanics.
In the mordern physics it is the quantum theory that plays the role of the conception compensating our incorrect notion on the space-time geometry at small scales. Application of nondegenerate geometry where thickness of tubes is connected with the quantum constant admits to explain quantum effects without using the quantum principles. It leads to understanding the fact that the field of application of quantum principles is restricted and opens the door for more deeper understanding of the quantum phenomena and other phenomena of microphysics.
There is a short author's review (in English) of the monograph. There 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.