Lobachevsky Geometry Research Articles

On the nonexistence of a Lobachevsky geometry model of an isotropic and homogeneous universe

According to the Einstein cosmological principle, our universe is homogeneous and isotropic, i.e. its curvature is constant at any point and in any direction. On large scales, when all local irregularities are ignored, this assumption has been confirmed by astronomers. We show that there is no reasonable hyperbolic geometry model in R 4 of a homogeneous and isotropic universe for a fixed time which would fit the cosmological principle. Hence, there does not exist any model in R 4 of an isotropic universe which would be represented by a three-dimensional hypersurface with the Lobachevsky geometry.

An unexpected source of Russian neo-Kantianism: Alexander Vvedensky and Lobachevsky's Geometry

It has long been a commonplace that Kant's philosophy was rela tively unimportant in Russia, the land of Schelling and Hegel par excellence. Kant's ideas, it seems, did not keep the youth of Moscow and St. Petersburg in the 1840s up until all hours, delaying dinner for the sake of discussing the possibility of synthetic a priori judg ments. The importance of Kant for Russian thought at the close of the nineteenth century, in contrast, is both undeniable and the object of increasing scholarly attention. Kant's influence, it turns out, can be felt not only among those who explicitly identified themselves as Kantians or Neo-Kantians, but among writers, poets, social thinkers, and religious philosophers. All the same, it seems peculiar that the reception of Kant, whose ideas Vladimir Solov'?v considered "the single main turning point in the history of human thought,"1 should occur so late in Russia almost a century after the philosopher's death. In the present essay, I would like to look at the ideas of the leading exponent of Neo-Kantianism at St. Petersburg University, Alexander Vvedensky, in the larger context of Russian Kantian ism. Vvedensky (1856-1925) is practically identified with Russian academic Neo-Kantianism at the turn of the century; his interpre tation of Kant was immensely influential for an entire generation of students. I would like to suggest that, indeed, the story of Kant in Russia is less simple, and rather more engaging, than the above sketch indicates; Vvedensky's philosophy points to a more compli cated, "homegrown" context of the renewed interest in Kant in the 1890s.

Lobachevsky geometry and unsolved problems of solar cosmogony

An axiomatic method of constructing physics in Lobachevsky space is proposed. Dynamical equations are applied to explain the principal properties of the solar system, its planets, their satellites, and rings.

Bending and stamping deformations of hyperbolic manifolds

The paper deals with some geometric approaches (from the viewpoint of (n + 1)- dimentional Lobachevsky geometry) to the deformation theory for uniformized conformal (i.e., flat conformal) structures on a hyperbolic n-manifold M with finite volume. Namely, two kinds of deformations are studied: bendings and stampings along totally geodesic submanifolds of M. The construction of the last deformation disproves a conjecture of C. Kourouniotis.

A special-relativistic theory of the locally anisotropic space-time

A special-relativistic theory of the locally anisotropic space-time has been developed. The fundamentals of the theory are set forth in this work. An expression for the interval in the anisotropic space of events and the 8-parameter group of linear transformations relating the co-ordinates of events in physically equivalent inertial frames of reference have been found. It has been shown that, as in the usual theory, the 3-velocity space has the Lobachevsky geometry. In other words, the 3-velocities are added by the Einsteinian law.

The Thomas precession

The equation for the Thomas precession is derived by using simple matrix calculation. No knowledge of group theory of Lobachevski geometry is assumed.

Twenty-eight years of soviet science

The Soviet Union received considerable scientific legacy of the prerevolutionary Russia, included the physical chemistry, Lobachevs ky's geometry, Mendeleyev's periodic system, Lebedev's light pressure, Sechenov's physiological researches, Jacobis Yablochkov's and Popov's technical experiments, Krashenninnikov's, Myklukha-Maklai's and Przhevalsky's remarkable journeys of exploration, the Russian philo logy, orientology and well developed science of history. The Soviet Union inherited the two hundred years Academy of Sciences, fine Moscow and Leningrad State Universities, Pulkovo ob servatory which twice in its time was "the astronomical capital of the world", the chamber of weights and measures and many other insti tutions.

Lobachevsky’s geometry

Lobachevsky’s geometry