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Teoria fondamentale degli spazii di curvatura constante.Milano: Giuseppe Bernardoni, 1868. First edition. A key paper in the history of non-Euclidean geometry; in which Beltrami established the fundamental connection between Riemann's differential geometry and the hyperbolic geometries of Lobachevsky and Bolyai. In this paper Betrami demonstrated that the hyperbolic spaces of Lobachevsky and Bolyai are just particular Riemannian manifolds with constant negative curvature - a turning point in the general acceptance of non-Euclidean geometry. Beltrami's work set the scheme for the later unifications of geometries by Felix Klein and Henri Poincaré. Two papers appeared in 1868 by Beltrami on the interpretation of non-Euclidean geometry, i.e., first his 'Saggio di interpretazione della geometria non-euclidea' and then the offered paper on 'The fundamental theory of spaces of constant curvature'. In his 'Saggio' Beltrami gave an important model of Lobachevsky's imaginary plane; His discoveries originated from the classical cartographic problem of how to make a plane map of a curved surface. More specifically how does one construct a one-to-one correspondence between geodesics of the surface and straight lines in the plane (geodesics are lines on the surface which constitute the shortest path between two points, e.g., geodesics in the plane are straight lines). Beltrami showed that such mappings are possible only when the curvature of the surface is constant. His main discovery in his 'Saggio' was that when mapping geodesics of a surface with constant negative curvature (a surface which Beltrami coined a 'pseudo-sphere') onto the plane one does not get the ordinary Euclidean geometry in two dimensions but Lobachevsky's non-Euclidean 'imaginary' plane. Beltrami initially interpretated this result as a justification of the validity of Lobachevsky's geometry, however as he noted himself, the case of Lobachevsky's three dimensional space could not be treated in the same manner, and as Helmholtz pointed out Beltrami's model only models a finite region of Lobachevsky's plane, and not the whole infinite plane. Beltrami found the solution to this problem after having read Riemann's recently published habilitations paper 'Ueber die Hypothesen welche der Geometrie zu Grunde liegen' (1867). In this famous lecture Riemann generalized Gauss' non-axiomatic description of surfaces and their intrinsic geometry into general n-dimensional space (the Riemannian manifold). Riemann had in his lecture himself discussed manifolds with constant positive curvature (i.e., elliptic geometry) but had not seen the link between Lobachevsky's hyperbolic geometry and his theory of manifolds. This key insight was first established by Beltrami in his 'Teoria fondamentale degli spazii di curvatura constante' (the offered paper) in which Beltrami was the first to identify Lobachevsky's plane and Lobachevsky's space with Riemannian manifolds of constant negative curvature of dimensions 2 and 3 correspondingly. Beltrami's result was an important factor in the rapid developments which took place, during the next decades, on the geometries of Lobachevky and Bolyai, and formed the foundation for Felix Klein's important unifications of these with projective and affine geometry.. In: Annali di matematica pura ed aplicata, series 2, volume 2, pp. 232-255. The complete volume offered in contemporary half calf. (4), 348 pp [Bookseller: Sophia Rare Books]
Last Found On: 2013-01-08
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