By S.S. Kutateladze
A.D. Alexandrov is taken into account by way of many to be the daddy of intrinsic geometry, moment in simple terms to Gauss in floor thought. That appraisal stems essentially from this masterpiece--now on hand in its totally for the 1st time for the reason that its 1948 book in Russian. Alexandrov's treatise starts with an overview of the fundamental options, definitions, and effects proper to intrinsic geometry. It stories the final concept, then offers the needful basic theorems on rectifiable curves and curves of minimal size. facts of a few of the final homes of the intrinsic metric of convex surfaces follows. The learn then splits into nearly autonomous traces: extra exploration of the intrinsic geometry of convex surfaces and evidence of the lifestyles of a floor with a given metric. the ultimate bankruptcy experiences the generalization of the entire idea to convex surfaces within the Lobachevskii house and within the round house, concluding with an overview of the speculation 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. growth in geometry in contemporary many years correlates with the resurrection of the factitious equipment of geometry and brings the tips 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
This second condition ensures the connectedness, since otherwise the development falls into parts from which no connected polygon could be glued. 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.
By the formulas similar to (1). , s0 (Li ) ≥ ρ(X(ti−1 )X(ti )). (3) Summing these inequalities and noting that the sum of the lengths of Li is equal to the length of the whole curve L, we obtain s0 (L) ≥ ρ(X(ti−1 )X(ti )). , s0 (L) ≥ s(L). (5) © 2006 by Taylor & Francis Group, LLC 6. A Manifold with an Intrinsic Metric 27 Therefore, the existence of the finite length s0 (L) implies the existence of the finite length s(L). , ρ0 (X(ti−1 )X(ti )) ≤ ρ(X(ti−1 )X(ti )). (6) This and formulae (1) and (2) make it clear that s0 (L) ≤ s(L).
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.