By S.S. Kutateladze
A.D. Alexandrov is taken into account through many to be the daddy of intrinsic geometry, moment basically to Gauss in floor concept. That appraisal stems basically from this masterpiece--now to be had in its totally for the 1st time due to the fact that its 1948 e-book in Russian. Alexandrov's treatise starts with an overview of the fundamental ideas, definitions, and effects correct to intrinsic geometry. It reports the overall idea, then provides the needful common theorems on rectifiable curves and curves of minimal size. evidence of a few of the overall 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 entire conception to convex surfaces within the Lobachevskii house and within the round area, concluding with an overview of the speculation of nonconvex surfaces. Alexandrov's paintings used to be either unique and intensely influential. This publication gave upward push to learning surfaces "in the large," rejecting the restrictions of smoothness, and reviving the fashion of Euclid. growth in geometry in fresh a long time correlates with the resurrection of the artificial equipment 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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Extra resources for A.D. Alexandrov: Selected Works Part II: Intrinsic Geometry of Convex Surfaces
This diameter is the projection of the arc of the convex curve along which the plane Q intersects the surface F . 16 But the length of OX is at lest the distance ρ(OX) on the surface F , and OX ≤ r and XX ≤ h. Therefore, ρ(OX) ≤ r + h < r0. This means that the point X belongs to the geodesic disk of radius r0. Hence the “extrinsic” neighborhood V lies in the given “intrinsic” neighborhood U , so that U is also an “extrinsic” neighborhood; this is what we have to prove. When we claim that each point of a surface has a neighborhood homeomorphic to a disk, we certainly speak about an “extrinsic” neighborhood since the intrinsic metric was not even mentioned.
For completeness, it is necessary to add the following condition to these two conditions. Its necessity is obvious. 3. The identified sides should be of equal length, and the coinciding segments of sides being glued should be of equal length. Now let X and Y be two points of the development. As is clear from condition 2, we can construct a broken line starting at the point X and ending at the point Y whose links lie subsequently on the faces of the development that have common (identified) sides or vertices; if a segment of this broken line arrives at the boundary of some face at a certain point Z, then we continue this segment in the face that is glued to the first one at the point Z.
Basic Concepts and Results the sense of their intrinsic metrics. 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.