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Intrinsic Geometry of Convex Surfaces A.D. ALEXANDROV SELECTED WORKS PART II Intrinsic Geometry of Convex Surfaces © 2006 by Taylor & Francis Group, LLC A.D. ALEXANDROV SELECTED WORKS PART II Intrinsic Geometry of Convex Surfaces Edited by S.S. Kutateladze Novosibirsk University, Russia Translated from Russian by S. Vakhrameyev Boca Raton London New York Singapore © 2006 by Taylor & Francis Group, LLC Published in 2006 by Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2006 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 987654321 International Standard Book Number-10: 0-415-29802-4 (Hardcover) International Standard Book Number-13: 978-0-415-29802-5 (Hardcover) Library of Congress Card Number 2004049387 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Alexsandrov, A.D. (Aleksandr Danilovich), 1912– [Vnutrenniaia geometriia vypuklykh poverkhnostei. English] Intrinsic geometry of convex surfaces / [edited by] S.S. Kutateladze ; translated by Sergei A. vakhrameyev. p. cm. – (A.D. Alexandrov, selected works ; pt. 2) Includes index. ISBN 0-415-29802-4 (alk. paper) 1. Convex surfaces. 2. Geometry, Differential. I. Kutateladze, S.S. (Semen Samsonovich) II. Title QA3.A358413 2004 [QA643] 510 s–dc22 [516.3¢6] 2004049387 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com Taylor & Francis Group and the CRC Press Web site at is the Academic Division of T&F Informa plc. http://www.crcpress.com © 2006 by Taylor & Francis Group, LLC CONTENTS Foreword ix Preface xi Chapter I Basic Concepts and Results 1 1. The General Concept and Problems of Intrinsic Geometry 1 2. Gaussian Intrinsic Geometry 8 3. A Polyhedral Metric 13 4. Development 17 5. Passage from Polyhedra to Arbitrary Surfaces 22 6. A Manifold with an Intrinsic Metric 23 7. Basic Concepts of Intrinsic Geometry 28 8. Curvature 34 9. Characteristic Properties of the Intrinsic Metric 38 10. Some Singularities of the Intrinsic Geometry of Convex Surfaces 46 11. Theorems of the Intrinsic Geometry of Convex Surfaces 52 Chapter II General Propositions About Intrinsic Metrics 57 1. General Theorems on Rectifiable Curves 57 2. General Theorems on Shortest Arcs 64 3. The Nonoverlapping Condition for Shortest Arcs 71 4. A Convex Neighborhood 73 5. General Properties of Convex Domains 80 6. Triangulation 83 Chapter III Characteristic Properties of the Intrinsic Metric 91 1. Convergence of the Metrics of Convergent Convex Surfaces 91 2. The Convexity Condition for a Polyhedral Metric 99 3. The Convexity Condition for the Metric of a Convex Surface 108 4. Consequences of the Convexity Condition 113 Chapter IV The Angle 121 1. General Theorems on Addition of Angles 121 2. Theorems on Addition of Angles on Convex Surfaces 128 3. The Angle of a Sector Bounded by Shortest Arcs 131 4. On Convergence of Angles 136 5. The Tangent Cone 141 6. The Spatial Meaning of the Angle between Shortest Arcs 147 © 2006 by Taylor & Francis Group, LLC vi Contents Chapter V Curvature 157 1. Intrinsic Curvature 157 2. The Area of a Spherical Image 163 3. Generalization of the Gauss Theorem 174 4. The Curvature of a Borel Set 181 5. Directions in Which It Is Impossible to Draw a Shortest Arc 186 6. Curvature as Measure of Non-Euclidicity of the Metric 188 Chapter VI Existence of a Convex Polyhedron with a Given Metric 197 1. On Determining a Metric from a Development 197 2. The Idea of the Proof of the Realization Theorem 204 3. Small Deformations of a Polyhedron 210 4. Deformation of a Convex Polyhedral Angle 213 5. The Rigidity Theorem 218 6. Realizability of the Metrics Close to Realized Metrics 222 7. Smooth Passage from a Given Metric to a Realizable Metric 225 8. Proof of the Realizability Theorem 233 Chapter VII Existence of a Closed Convex Surface 235 1. The Result and the Method of Proof 235 2. The Main Lemma on Convex Triangles 241 3. Corollaries of the Main Lemma on Convex Triangles 249 4. The Complete Angle at a Point 252 5. Curvature and Two Related Estimates 258 6. Approximation of a Metric of Positive Curvature 262 7. Realization of a Metric of Positive Curvature Given on a Sphere 269 Chapter VIII Other Existence Theorems 277 1. The Gluing Theorem 277 2. Application of the Gluing Theorem to the Realization Theorems 281 3. Realizability of a Complete Metric of Positive Curvature 284 4. Manifolds on Which a Metric of Positive Curvature Can Be Given 288 5. The Question of the Uniqueness of a Convex Surface with a Given Metric 295 6. Various Definitions of a Metric of Positive Curvature 298 Chapter IX Curves on Convex Surfaces 301 1. The Direction of a Curve 301 2. The Swerve of a Curve 308 3. The General Gluing Theorem 316 4. Convex Domains 320 5. Quasigeodesics 326 6. A Circle 332 Chapter X Area 341 1. The Intrinsic Definition of Area 341 2. The Extrinsic–Geometric Meaning of Area 350 3. Extremal Properties of Pyramids and Cones 356 Chapter XI The Role of Specific Curvature 365 1. Intrinsic Geometry of a Surface 365 2. Intrinsic Geometry of a Surface of Bounded Specific Curvature 376 3. The Shape of a Convex Surface Depending on Its Curvature 386 © 2006 by Taylor & Francis Group, LLC Contents vii Chapter XII Generalization 393 1. Convex Surfaces in Spaces of Constant Curvature 393 2. Realization Theorems in Spaces of Constant Curvature 398 3. Surfaces of Indefinite Curvature 402 Appendix Basics of Convex Bodies 409 1. Convex Domains and Curves 409 2. Convex Bodies. A Supporting Plane 411 3. A Convex Cone 414 4. Topological Types of Convex Bodies 415 5. A Convex Polyhedron and the Convex Hull 418 6. On Convergence of Convex Surfaces 421 © 2006 by Taylor & Francis Group, LLC FOREWORD Aleksandr Danilovich Alexandrov (1912–1999) is usually ranked as second to Gauss in surface theory. This appraisal rests primarily on his masterpiece whose English translation lies before the reader. There is no need in expatiating upon the content of this book since the main ideas of geometry “in the large” have been placed in the treasure trove of modern science half a century ago. The reader will be pleased to taste the original presentation of such fundamental concepts as shortest arc, intrinsic metric, angle, curvature, etc. For this reason my introductory words are kept at a minimum, confining merely to those of pride and apology. It was in the mid 1990s that Alexandrov asked me to help him in editing this translation he chose as the second volume of his Selected Works. We agreed with him from the very beginning that no modernizing comments will be added to the book since the area of geometry in the large is impossible to embrace these days; thus, the reader can feel the original flavor of invention which is undoubtedly spoiled in translation. I am proud of this opportunity to return a trifle to the brilliant person whose genius, charm, and sympathy are listed as decoration and consolation of about two decades of my life. I also offer my sincere words of apology to the reader for the crushing defeats I have suffered as editor in the severe battle against mistranslation and solecism. Unfortunately, my inspired teacher and long-time interlocutor had no chance to look through this translation but I am pretty sure that no blunders of mine nor the translator’s nor anybody else’s can ever conceal the keen mind and farsighted vision of Alexandrov, a geometrical giant of the 20th century. S.S. Kutateladze Novosibirsk, 2002 © 2006 by Taylor & Francis Group, LLC PREFACE Intrinsic geometry investigates the properties of a surface and of figures on it within the limits of the surface similar to the way plane geometry studies a plane without regard to the fact that it may be located in some space. The start of intrinsic geometry was made by Gauss’ paper “Disquisitiones generales circa superficies cur- vas,” which appeared in 1827. Since that time, intrinsic geometry has advanced so far that, at present, all of its major issues can be considered solved, at least those that deal with the geometry of small pieces of regular surfaces. However, in their conventional form the differential-geometry methods apply only to regular surfaces, i.e., the surfaces set up by equations with functions differentiable sufficiently many times, usually, at least three times.
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