ISBN-10: 0203643844

ISBN-13: 9780203643846

ISBN-10: 0415298024

ISBN-13: 9780415298025

A.D. Alexandrov is taken into account through many to be the daddy of intrinsic geometry, moment purely to Gauss in floor idea. That appraisal stems essentially from this masterpiece--now to be had in its totally for the 1st time considering that its 1948 ebook in Russian. Alexandrov's treatise  starts with an summary of the elemental strategies, definitions, and effects suitable to intrinsic geometry. It reports the final thought, then offers the considered necessary normal 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 learn then splits into nearly self sufficient 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 reports the generalization of the full idea to convex surfaces within the Lobachevskii area 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 e-book 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 unreal 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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Additional info for A.D. Alexandrov: Selected Works Part II: Intrinsic Geometry of Convex Surfaces

Sample text

Also, we can show that the tangent cone can be defined as the locus of points lying on all half-lines that are the limits of the half-lines traveling from the point O to some points X of the surface F as these points X converge to O. Let a curve L be drawn from the point O of the surface F . The limit of the half-line traveling from O to the variable points X of the curve O as X tends to O is called the half-tangent to L at O. The half-tangent is a generator of the tangent cone whenever this half-tangent exists.

Be a sequence of convex polyhedra converging to this surface. Take a point O inside F and also inside all Pn; this is possible whenever Pn are sufficiently close to F . Circumscribe a sphere S around O, and project the surface F and all polyhedra Pn to this sphere from the point O. Then the metrics ρF (XY ) and ρPn (XY ) are transferred from F and Pn to the sphere S if the distance between the corresponding points on the surface F and on the polyhedra Pn, respectively, is put in correspondence to a pair of points A and B on S as the distance between A and B.

At first glance, this definition seems absurd, but it does make sense. Indeed, let Xt = X(t) be a curve emanating from a point O = X(t0 ) in a manifold with intrinsic metric ρ. We shall consider this curve as two coinciding curves Xt = X(t) and Xt = X (T ); the angle between these curves is defined by the formula α = lim γ(t, t ). t,t →0 It is absolutely obvious that for some curves X(t) this limit fails to exist for all ways in which t and t vanish. Considering this question for a curve on the plane, we easily see that the limit limt,t →0 γ(t, t ) exists if the curve X(t) has tangent at the point X(0).