By S.S. Kutateladze
A.D. Alexandrov is taken into account by means of many to be the daddy of intrinsic geometry, moment simply to Gauss in floor conception. That appraisal stems basically from this masterpiece--now to be had in its totally for the 1st time when you consider that its 1948 e-book in Russian. Alexandrov's treatise starts off with an overview of the elemental techniques, definitions, and effects correct to intrinsic geometry. It experiences the overall thought, then offers the considered necessary basic theorems on rectifiable curves and curves of minimal size. evidence of a few of the overall houses of the intrinsic metric of convex surfaces follows. The examine then splits into nearly autonomous strains: extra exploration of the intrinsic geometry of convex surfaces and facts of the lifestyles of a floor with a given metric. the ultimate bankruptcy experiences the generalization of the total conception to convex surfaces within the Lobachevskii house and within the round area, concluding with an overview of the idea of nonconvex surfaces. Alexandrov's paintings was once either unique and intensely influential. This publication gave upward push to learning surfaces "in the large," rejecting the constraints of smoothness, and reviving the fashion of Euclid. development in geometry in contemporary many years correlates with the resurrection of the substitute tools of geometry and brings the tips of Alexandrov once more into concentration. this article is a vintage that continues to be unsurpassed in its readability and scope.
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Extra resources for A.D. Alexandrov: Selected Works Part II: Intrinsic Geometry of Convex Surfaces
18 Consequently, we have proved that the metric of each surface is intrinsic in the sense that the distance between two points is equal to the greatest lower bound of the lengths of connecting curves which are measured in the metric of this surface. 19 Some general properties of manifolds with intrinsic metric will be considered in Chapter II. Convex surfaces are a particular class of these manifolds. 7. Basic Concepts of Intrinsic Geometry We give here the definitions of some basic concepts of intrinsic geometry.
As is known, this property © 2006 by Taylor & Francis Group, LLC 36 Ch. I. Basic Concepts and Results distinguishes regular convex surfaces from all regular surfaces in general; we shall prove that this property is also characteristic of all convex surfaces. Roughly speaking, we shall prove that each point of a manifold with intrinsic metric in which the curvature of every set is nonnegative has a neighborhood isometric to a convex surface. Here, we are rushing ahead; a precise formulation of this result will be given in the next section.
E = m + n; therefore, f − k + m + n = 1. (3) Finally, each triangle has three sides, but there are only n sides lying on the boundary, each of which belongs to one triangle, and each of the sides belongs to two triangles. Therefore, 3f = 2k − n. (4) Multiplying equation (3) by equation (2) and putting 2k = 3f + n in (3), we obtain f = 2m + n − 2. (5) Substituting this expression for f into formula (2), we obtain n m 2πm + πn − 2π = αi + i=1 or n m αi − (n − 2)π = i=1 θj j=1 (2π − θj ); j=1 as required.