By S.S. Kutateladze
A.D. Alexandrov is taken into account by way of many to be the daddy of intrinsic geometry, moment purely to Gauss in floor thought. That appraisal stems essentially from this masterpiece--now on hand in its solely for the 1st time considering that its 1948 booklet in Russian. Alexandrov's treatise starts off with an overview of the fundamental recommendations, definitions, and effects suitable to intrinsic geometry. It studies the overall concept, then offers the considered necessary common theorems on rectifiable curves and curves of minimal size. evidence of a few of the final houses of the intrinsic metric of convex surfaces follows. The examine then splits into virtually autonomous traces: additional exploration of the intrinsic geometry of convex surfaces and evidence of the lifestyles of a floor with a given metric. the ultimate bankruptcy reports the generalization of the complete 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 very influential. This booklet 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 unreal tools of geometry and brings the guidelines 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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Additional resources for A.D. Alexandrov: Selected Works Part II: Intrinsic Geometry of Convex Surfaces
Hence, in contrast to the case of regular surfaces, it is not possible in general to draw a shortest arc in any direction on a convex surface. However, we shall show below in Sec. 5 of Chapter V that the set of such specific directions at each point has angle measure zero. In view of the fact that there are no shortest arcs tangent to the base circle on a cone of revolution, it is not convenient to introduce the intrinsic concept of tangent to a curve as the limit of shortest secants. Instead, we introduce the concept of direction of a curve at a given point.
These polygons cover the whole sphere, and since each of them can be developed onto the plane, these polygons form the development that represents the metric given on S. The edges of the development form a net of segments. , we can pass from one segment to another proceeding only along the segments of this net. Indeed, let a and b be two edges, and let P and Q be the faces containing these edges. By the connectedness property of the development, we can pass from P to Q going along the faces, which are glued with each other.
Inequalities (5) and (7) yield s0 (L) = s(L); as required. Since the lengths of curves on a surface coincide for both definitions, the distance on a surface is also equal to the greatest lower bound of the lengths of curves in the sense of the definition of length via the intrinsic metric of this surface. This formulation is free of any reference to the distance and length in the space; it belongs entirely to intrinsic geometry and, therefore, can be abstracted to metric spaces. The metric of an arbitrary metric space is called intrinsic if the distance between each two points of this space is equal to the greatest lower bound of the lengths of curves connecting these points; moreover, the length of each curve is defined in terms of the metric by formula (2).
A.D. Alexandrov: Selected Works Part II: Intrinsic Geometry of Convex Surfaces by S.S. Kutateladze