By S.S. Kutateladze
A.D. Alexandrov is taken into account by way of many to be the daddy of intrinsic geometry, moment simply to Gauss in floor conception. That appraisal stems essentially from this masterpiece--now to be had in its completely for the 1st time on the grounds that its 1948 e-book in Russian. Alexandrov's treatise starts off with an summary of the elemental suggestions, definitions, and effects appropriate to intrinsic geometry. It experiences the overall concept, then provides the considered necessary basic theorems on rectifiable curves and curves of minimal size. evidence of a few of the final homes of the intrinsic metric of convex surfaces follows. The research then splits into nearly self sufficient strains: extra 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 idea to convex surfaces within the Lobachevskii house and within the round house, concluding with an overview of the idea of nonconvex surfaces. Alexandrov's paintings was once either unique and very influential. This booklet gave upward push to learning surfaces "in the large," rejecting the restrictions of smoothness, and reviving the fashion of Euclid. development in geometry in contemporary many years correlates with the resurrection of the factitious 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 info for A.D. Alexandrov: Selected Works Part II: Intrinsic Geometry of Convex Surfaces
5 the plane passing through O). The surface F is a domain on the boundary of a certain convex body. Take a point A inside this body and project the surface F to the plane P from this point. The projection of the surface F onto the plane P is bijective and bicontinuous near the point O. Therefore, the projection of F covers a certain neighborhood of the point O on the plane P . Take the disk K of radius r centered at the point O on the plane P such that the projection of the surface F covers this disk.
In exactly the same way, each polyhedron or each regular surface is a manifold from the point of view of the intrinsic metric. In general, it is appropriate to distinguish especially between those surfaces which are manifolds in 17 In topology, it is accepted that a manifold is any topological space that has the properties mentioned above and, in addition, admits a partition into simplexes. In our case, this requirement turns out to be extraneous. © 2006 by Taylor & Francis Group, LLC 28 Ch. I.
In order to connect this definition of the curvature of a polyhedron with the general definition mentioned above, we shall prove the following theorem. Let a polygon P with angles α1, α2, . . , αn which is homeomorphic to a disk be given on a polyhedron, and let θ1 , θ2 , . , θm be the complete angles at the vertices of P lying in the interior of P . Then n 2π − n (π − αi) = i=1 m αi − (n − 2)π = i=1 (2π − θj ). (1) j=1 Briefly, this means that the excess of a polygon is equal to the sum of curvatures of the vertices contained inside this polygon.