The tragedy for the pioneers of non-Euclidean geometry (N. Lobachevsky and J. Boyai) was their quarrel with the scientific tradition. Figuratively speaking, in the judgment of the scientific world they could not provide proof of their views, and substantive law of science was not on their side despite the efforts of such an influential advocate as Karl Friedrich Gauss. They lost the civil process to the scientific layman, who sincerely believed that the earth is flat. Traditionally mathematical logic considers a new idea proven, if it is derived by inference from already proven ones, or recognized as obvious, or recognized without proof (postulates). Yet the founders of non-Euclidean geometry could not imagine such traditional evidence at all desire, because it had not yet been developed, and most importantly respective starting points (axioms, postulates, and theorems) had not been recognized by mathematicians. The paper outlines the original concept of non-Euclidean geometries. Hyperbolic geometry of Lobachevsky is considered based on viewing the sphere as a surface of zero curvature. In this case, the plane will have a real curvature properties of hyperboloid or a pseudosphere depending on the absolute and space anisotropy index, which replaces the concept of curvature of space; i.e. the notion of the curvature of the surface is converted to purely analytical attributes. Parabolic geometry of Euclid with degenerate absolute becomes a special case of geometries with non-degenerate absolute. The geometry of Riemann having the absolute of imaginary surface with negative Gaussian curvature at all points is declared not real but imaginary, since, according to the authors, it is impossible for plotting. References to textbooks of mechanics and mathematics departments of universities.
Some differential complexes within and beyond parabolic geometry
