Geometry from a Differentiable Viewpoint The development of geometry from Euclid to Euler to Lobachevski˘ı, Bolyai, Gauss, and Riemann is a story that is often broken into parts – axiomatic geome- try, non-Euclidean geometry, and differential geometry. This poses a problem for undergraduates: Which part is geometry? What is the big picture to which these parts belong? In this introduction to differential geometry, the parts are united with all of their interrelations, motivated by the history of the parallel postulate. Beginning with the ancient sources, the author first explores synthetic methods in Euclidean and non-Euclidean geometry and then introduces differential geometry in its classical formulation, leading to the modern formulation on manifolds such as space-time. The presentation is enlivened by historical diversions such as Hugyens’s clock and the mathematics of cartography. The intertwined approaches will help under- graduates understand the role of elementary ideas in the more general, differential setting. This thoroughly revised second edition includes numerous new exercises, together with newly prepared solutions to selected exercises. JOHN MCCLEARY is professor of mathematics at Vassar College on the Elizabeth Stillman Williams Chair. His research interests lie at the boundary between geom- etry and topology, especially where algebraic topology plays a role. His research papers have appeared in journals such as Inventiones Mathematicae and the American Journal of Mathematics, and he has also written expository papers in the American Mathematical Monthly. He is interested in the history of mathemat- ics, particularly the history of geometry in the nineteenth century and of topology in the twentieth century. He is the author of A User’s Guide to Spectral Sequences and A First Course in Topology: Continuity and Dimension, and he has edited pro- ceedings in topology and in history, as well as a volume of the collected works of John Milnor. Geometry from a Differentiable Viewpoint Second Edition JOHN MC CLEARY Vassar College CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Mexico City Cambridge University Press 32 Avenue of the Americas, New York, NY 10013-2473, USA www.cambridge.org Information on this title: www.cambridge.org/9780521133111 c Cambridge University Press 1995, 2013 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1995 Second edition published 2013 Printed in the United States of America A catalog record for this publication is available from the British Library. Library of Congress Cataloging in Publication Data McCleary, John, 1952– Geometry from a differentiable viewpoint / John McCleary. – 2nd ed. p. cm. Includes bibliographical references and indexes. ISBN 978-0-521-11607-7 (hardback) – ISBN 978-0-521-13311-1 (pbk.) 1. Geometry, Differential. I. Title. QA641.M38 2012 516.3 6–dc23 2012017159 ISBN 978-0-521-11607-7 Hardback ISBN 978-0-521-13311-1 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate. To my sisters Mary Ann, Denise, Rose • Contents Preface to the second edition page ix Introduction xi PART A Prelude and themes: Synthetic methods and results 1 Spherical geometry 3 2Euclid 12 Euclid’s theory of parallels 19 Appendix: The Elements: Book I 24 3 The theory of parallels 27 Uniqueness of parallels 28 Equidistance and boundedness of parallels 29 On the angle sum of a triangle 31 Similarity of triangles 34 The work of Saccheri 37 4 Non-Euclidean geometry 43 The work of Gauss 43 The hyperbolic plane 47 Digression: Neutral space 55 Hyperbolic space 62 Appendix: The Elements: Selections from Book XI 72 PART B Development: Differential geometry 5 Curves in the plane 77 Early work on plane curves (Huygens, Leibniz, Newton, and Euler) 81 The tractrix 84 Oriented curvature 86 Involutes and evolutes 89 6 Curves in space 99 Appendix: On Euclidean rigid motions 110 7 Surfaces 116 The tangent plane 124 The first fundamental form 128 Lengths, angles, and areas 130 7bis Map projections 138 Stereographic projection 143 vii viii Contents Central (gnomonic) projection 147 Cylindrical projections 148 Sinusoidal projection 152 Azimuthal projection 153 8 Curvature for surfaces 156 Euler’s work on surfaces 156 The Gauss map 159 9 Metric equivalence of surfaces 171 Special coordinates 179 10 Geodesics 185 Euclid revisited I: The Hopf–Rinow Theorem 195 11 The Gauss–Bonnet Theorem 201 Euclid revisited II: Uniqueness of lines 205 Compact surfaces 207 A digression on curves 211 12 Constant-curvature surfaces 218 Euclid revisited III: Congruences 223 The work of Minding 224 Hilbert’s Theorem 231 PART C Recapitulation and coda 13 Abstract surfaces 237 14 Modeling the non-Euclidean plane 251 The Beltrami disk 255 The Poincaré disk 262 The Poincaré half-plane 265 15 Epilogue: Where from here? 282 Manifolds (differential topology) 283 Vector and tensor fields 287 Metrical relations (Riemannian manifolds) 289 Curvature 294 Covariant differentiation 303 Riemann’s Habilitationsvortrag: On the hypotheses which lie at the foundations of geometry 313 Solutions to selected exercises 325 Bibliography 341 Symbol index 351 Name index 352 Subject index 354 • Preface to the second edition Giving an author a chance to rewrite a text is a mixed blessing. The temptation to rewrite every sentence is strong, as is the temptation to throw everything out and start over. When I asked colleagues who had taught from the book what I might change, I was surprised in one case to hear that I should change nothing. That presented a challenge to preparing a second edition. Several folks were kind enough to point out errors in the first edition that I have fixed. Many thanks to you. Among these errors was a mishandling of congruences that has led to additional material in this edition. The notion of congruence leads to the important theory of transformation groups and to Klein’s Erlangen Program. I have taken some of the opportunities to apply arguments using transformations, exposing another of the pillars of geometry (Stillwell 2005) to the reader. Succumbing to the second temptation, I have reordered the material signifi- cantly, resulting in one fewer chapter and a better story line. Chapter 4 is a more or less self-contained exposition of non-Euclidean geometry, and it now parallels Chapter 14 better. I have added material including Euclid’s geometry of space, fur- ther results on cycloids, another map projection, Clairaut’s relation, and reflections in the Beltrami disk. I have also added new exercises. Due to a loss of files for the first edition, all of the pictures have been redrawn and improved. In many places I have made small changes that I hope improve the clarity of my telling of this amazing and rich story. ix • Introduction AEMETPHTOMHEIEIIT ABOVETHEENTRANCETOPLATO’SACADEMY Oneofthemanyrolesofhistoryistotellastory.ThehistoryoftheParallelPos- tulateisagreatstory.Itspansmorethantwomillennia,starsanimpressivecastof characters,andcontainssomeofthemostbeautifulresultsinallofmathematics. Myimmodestgoalforthisbookistotellthisstory. Anotherroleofhistoryistofocusourattention.Wecanthenseeathreadof unitythroughaparadeofevents,people,andideas.Mymoremodestgoalisto provideafocuswithwhichtoviewthestandardtoolsofelementarydifferential geometry,anddiscoverhowtheirhistoryemergesoutofGeometrywritlarge,and howtheydevelopedintothemodern,globaledificeoftoday. Inrecentyears,toofferacourseindifferentialgeometrytoundergraduateshas becomealuxury.Whensuchacourseexists,itsstudentsoftenarrivewithamodern introductiontoanalysis,butwithouthavingseengeometrysincehighschool.Inthe UnitedStateshighschoolgeometryisgenerallyelementaryEuclideangeometry basedonHilbert’saxiomscheme.Suchanapproachisawelcomeintroduction totherigorsofaxiomaticthinking,butthebeautyofEuclideangeometrycanget lostinthecarefullywroughttwo-columnproof.Ifmentionedatall,themarvels ofnon-Euclideangeometryarerelegatedtoafootnote,enrichmentmaterial,ora “cultural”essay.Thissituationisalsothecaseinmostcurrentintroductionsto differentialgeometry.Themodernsubjectturnsonproblemsthathaveemerged fromthenewfoundationsthatarefarremovedfromtheancientrootsofgeometry. Whenweteachthenewandcutoffthepast,studentsarelefttofindtheirown waystoameaningofthewordgeometryindifferentialgeometry,orfailingthat,to identifytheiractivityassomethingdifferentandunconnected. ThisbookisanattempttocarrythereaderfromthefamiliarEuclidtothestate ofdevelopmentofdifferentialgeometryatthebeginningofthetwentiethcentury. Onenarrowthreadthatrunsthroughthislargehistoricalperiodisthesearchfora proofofEuclid’sPostulateV,theParallelPostulate,andtheeventualemergence of a new and non-Euclidean geometry. In the course of spinning this tale, another theme enters – the identification of properties of a surface that are intrinsic, that is, independent of the manner in which the surface is embedded in space. This idea, introduced by Gauss, provides the analytic key to properties that are really geometric, opening new realms to explore. Thebookiswritteninsonata-allegroform.PartAopenswithaprelude—asmall and orienting dose of spherical geometry, whose generalizations provide impor-