Download A.D. Alexandrov: Selected Works Part II: Intrinsic Geometry by S.S. Kutateladze PDF

By S.S. Kutateladze

A.D. Alexandrov is taken into account by means of many to be the daddy of intrinsic geometry, moment simply to Gauss in floor conception. That appraisal stems basically from this masterpiece--now to be had in its totally for the 1st time when you consider that its 1948 e-book in Russian. Alexandrov's treatise  starts off with an overview of the elemental techniques, definitions, and effects correct to intrinsic geometry. It experiences the overall thought, then offers the considered necessary basic theorems on rectifiable curves and curves of minimal size. evidence of a few of the overall houses of the intrinsic metric of convex surfaces follows. The examine then splits into nearly autonomous 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 experiences the generalization of the total conception to convex surfaces within the Lobachevskii house and within the round area, concluding with an overview of the idea of nonconvex surfaces. Alexandrov's paintings was once either unique and intensely influential. This publication 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 substitute 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.

Show description

Read or Download A.D. Alexandrov: Selected Works Part II: Intrinsic Geometry of Convex Surfaces PDF

Best differential geometry books

Foundations of mechanics

Within the Spring of 1966, I gave a sequence of lectures within the Princeton collage division of Physics, geared toward contemporary mathematical leads to mechanics, specially the paintings of Kolmogorov, Arnold, and Moser and its software to Laplace's query of balance of the sun process. Mr. Marsden's notes of the lectures, with a few revision and enlargement through either one of us, turned this publication.

Lectures on classical differential geometry

First-class short advent provides primary thought of curves and surfaces and applies them to a couple of examples. issues contain curves, conception of surfaces, basic equations, geometry on a floor, envelopes, conformal mapping, minimum surfaces, extra. Well-illustrated, with considerable difficulties and ideas.

New Developments in Differential Geometry (Mathematics and Its Applications)

This quantity comprises thirty-six examine articles offered at the Colloquium on Differential Geometry, which used to be held in Debrecen, Hungary, July 26-30, 1994. The convention used to be a continuation in the sequence of the Colloquia of the J? nos Bolyai Society. the diversity coated displays present job in differential geometry.

Riemannian geometry during the second half of the twentieth century

In the course of its first hundred years, Riemannian geometry loved regular, yet undistinguished progress as a box of arithmetic. within the final fifty years of the 20 th century, besides the fact that, it has exploded with task. Berger marks the beginning of this era with Rauch's pioneering paper of 1951, which incorporates the 1st genuine pinching theorem and an awesome bounce within the intensity of the relationship among geometry and topology.

Extra resources for A.D. Alexandrov: Selected Works Part II: Intrinsic Geometry of Convex Surfaces

Example text

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.

As is known, this property © 2006 by Taylor & Francis Group, LLC 36 Ch. I. Basic Concepts and Results distinguishes regular convex surfaces from all regular surfaces in general; we shall prove that this property is also characteristic of all convex surfaces. Roughly speaking, we shall prove that each point of a manifold with intrinsic metric in which the curvature of every set is nonnegative has a neighborhood isometric to a convex surface. Here, we are rushing ahead; a precise formulation of this result will be given in the next section.

E = m + n; therefore, f − k + m + n = 1. (3) Finally, each triangle has three sides, but there are only n sides lying on the boundary, each of which belongs to one triangle, and each of the sides belongs to two triangles. Therefore, 3f = 2k − n. (4) Multiplying equation (3) by equation (2) and putting 2k = 3f + n in (3), we obtain f = 2m + n − 2. (5) Substituting this expression for f into formula (2), we obtain n m 2πm + πn − 2π = αi + i=1 or n m αi − (n − 2)π = i=1 θj j=1 (2π − θj ); j=1 as required.

Download PDF sample

Rated 4.10 of 5 – based on 26 votes