Download PDF by S.S. Kutateladze: A.D. Alexandrov: Selected Works Part II: Intrinsic Geometry
By S.S. Kutateladze
A.D. Alexandrov is taken into account by means of many to be the daddy of intrinsic geometry, moment in simple terms to Gauss in floor thought. That appraisal stems basically from this masterpiece--now on hand in its totally for the 1st time on account that its 1948 ebook in Russian. Alexandrov's treatise starts off with an overview of the elemental techniques, definitions, and effects appropriate to intrinsic geometry. It stories the final thought, then offers the needful basic theorems on rectifiable curves and curves of minimal size. facts of a few of the overall houses of the intrinsic metric of convex surfaces follows. The examine then splits into nearly self sustaining strains: additional exploration of the intrinsic geometry of convex surfaces and facts of the life of a floor with a given metric. the ultimate bankruptcy stories the generalization of the complete thought to convex surfaces within the Lobachevskii area and within the round area, concluding with an overview of the speculation of nonconvex surfaces. Alexandrov's paintings was once either unique and intensely influential. This publication gave upward thrust to learning surfaces "in the large," rejecting the constraints of smoothness, and reviving the fashion of Euclid. development in geometry in fresh a long time correlates with the resurrection of the artificial tools of geometry and brings the information of Alexandrov once more into concentration. this article is a vintage that is still unsurpassed in its readability and scope.
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Additional info for A.D. Alexandrov: Selected Works Part II: Intrinsic Geometry of Convex Surfaces
Tn emanate from a point O on a convex surface. Then the angles of the sectors into which these shortest arcs divide a neighborhood of the point O are equal to the angles of the sectors into which their half-tangents divide the tangent cone at the point O. (The proof is given in Sec. ) In particular, this implies that the angles of sectors on a convex surface arc adjoined exactly in the same manner as the angles of sectors on the cone, and the sum of the angles of sectors which form a full neighborhood of the point O does not depend on these sectors and is equal to the complete angle of the tangent cone at this point.
7, one plane lies entirely over the other. The edges of the cuts along which the gluing was carried out are slightly moved apart. © 2006 by Taylor & Francis Group, LLC 32 Ch. I. Basic Concepts and Results Let L and M be two shortest arcs emanating from a point O in a manifold with intrinsic metric, which have no other common points near O. (As we show in Sec. ” Let U be one of these sectors; we shall draw shortest arcs N1 , N2 , . , Nn in this sector and number them in the order of their location between L and M .
The figure is compact in common parlance). A sphere, a half-sphere, a band on a cylinder bounded by two base lines can serve as examples of polygons. Also, we can define a disk and a circle as follows: a disk is the locus of points lying at a distance less than or equal to a given number from a given point and a circle is the locus of points equidistant from a given point. We now define the concept of angle between two curves emanating from a common point of a manifold with intrinsic metric. , X0 = Y0 .
A.D. Alexandrov: Selected Works Part II: Intrinsic Geometry of Convex Surfaces by S.S. Kutateladze