Raymond O. Wells, Jr. Differential and Complex Geometry: Origins, Abstractions and Embeddings Differential and Complex Geometry: Origins, Abstractions and Embeddings Raymond O. Wells, Jr. Differential and Complex Geometry: Origins, Abstractions and Embeddings 123 RaymondO.Wells, Jr. University of Colorado Boulder Boulder, CO USA and JacobsUniversity Bremen Bremen Germany ISBN978-3-319-58183-5 ISBN978-3-319-58184-2 (eBook) DOI 10.1007/978-3-319-58184-2 LibraryofCongressControlNumber:2017943176 MathematicsSubjectClassification(2010): 01-02,30-02,32-02,14-02,51-02,53-02,55-02 ©SpringerInternationalPublishingAG2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. Cover illustration reproduced from ‘Vorlesungen über Nicht-Euklidische Geometrie’ by Felix Klein (Volume 26 of the series Die Grundlehren der Mathematischen Wissenschaften), 1967, p. 300. With permissionofSpringerNature. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland To my friend, Howard Resnikoff Preface Abouttenyearsago,Ihadtheideaofwritingupasurveyofthemajorembedding theoremsofthetwentiethcentury.Thisbookrepresentstheculminationofthisidea, and I’m quite happy to be able to finally publish it after all this time. The embedding theorems represent important ideas in the modern fields of differential topology,differentialgeometry,complexmanifoldtheory,andthegeneraltheoryof functionsofseveralcomplexvariables,aswell astheoverallconceptofmanifolds in general. I thought it would be useful to review the origins of these various concepts as a way of hoping to give a deeper understanding of the theorems themselves. Consequently,Ispentafairamountoftimethesepastyearslookingatanumber of contributions by mathematicians during the seventeenth through the nineteenth centuries, where almost all of these concepts first appeared and then developed. In my book, I have tried to give the reader some sense of the language and under- standing of these earlier mathematicians as they gave voice to the many issues at hand. For instance, the developments of projective geometry and intrinsic differ- ential geometry both evolved at the same time in the first half of the nineteenth century, but in reading the literature of the time, it seems as if they were hardly aware of each other. Only in the last half of the nineteenth century did these seemingly disparate sets of ideas come to be part of a mathematical whole. I would not have been able to peruse these papers and books from these earlier timeshaditnotbeenfortheInternetandthefactthatthegreatlibrariesoftheworld put time and effort into digitizing their collections. I am very thankful that these ideas can be so readily shared today. I have had the support of three academic institutions over the past decades, whereithasbeenmyprivilegetoholdvariousacademicappointments,andIwant to thank them all for their continued support over the years: Rice University in Houston; Jacobs University in Bremen, Germany; and the University of Colorado in Boulder, Colorado, where I now live. Springer is the publisher of two of my earlier books, and I am very happy that theyarebringingthisnewworkofminetothepublic.Iwanttothank,inparticular, Rémi Lodh, who encouraged me and helped bring this book to fruition. vii viii Preface The comments of his reviewers were very helpful to me. Anne-Kathrin Birchley-Brun, also in the London Springer office, has been very helpful in the processofmanagingthedigitalfilesandusheringthemintotheproductionprocess. I want to thank Ina Mette, formerly of Springer and now an editor for the AmericanMathematicalSociety,forherencouragementforthisprojectovermany years now. I have dedicated this book to my very close friend, Howard Resnikoff. He has been an inspiration for me for over fifty years, and we have shared many things together.Hisreadingofvariousdraftsofthisbookandhisencouragingwordshave been very important to me. Finally, I want to thank my wife, Rena, for her continuous support in so many ways. Inparticular, sheread afinal draft andher comments and editorial pen were so very useful, as always. Boulder, CO, USA Raymond O. Wells, Jr. March 2017 Contents Introduction... .... .... .... ..... .... .... .... .... .... ..... .... xiii Part I Geometry in the Age of Enlightenment 1 Algebraic Geometry .... ..... .... .... .... .... .... ..... .... 5 1.1 Introduction .. .... ..... .... .... .... .... .... ..... .... 5 1.2 Descartes and Fermat .... .... .... .... .... .... ..... .... 7 1.3 Newton and Euler.. ..... .... .... .... .... .... ..... .... 11 2 Differential Geometry ... ..... .... .... .... .... .... ..... .... 17 2.1 Introduction .. .... ..... .... .... .... .... .... ..... .... 17 2.2 Huygens and Newton.... .... .... .... .... .... ..... .... 18 2.3 Curves in Space: Courbes à double courbure.. .... ..... .... 27 2.4 Curvature of a Surface: Euler in 1767 ... .... .... ..... .... 27 Part II Differential and Projective Geometry in the Nineteenth Century 3 Projective Geometry .... ..... .... .... .... .... .... ..... .... 33 3.1 Monge and Descriptive Geometry .. .... .... .... ..... .... 34 3.2 Poncelet’s “Propriétés Projectives”.. .... .... .... ..... .... 36 3.3 Analytic Projective Geometry.. .... .... .... .... ..... .... 45 4 Gauss and Intrinsic Differential Geometry ... .... .... ..... .... 49 4.1 Gaussian Curvature. ..... .... .... .... .... .... ..... .... 49 4.2 Gauss’s Theorema Egregrium.. .... .... .... .... ..... .... 54 5 Riemann’s Higher-Dimensional Geometry.... .... .... ..... .... 59 5.1 The Legacy of Riemann.. .... .... .... .... .... ..... .... 59 5.2 Higher-Dimensional Manifolds and a Quadratic Line Element . .... ..... .... .... .... .... .... ..... .... 62 5.3 Geodesic Normal Coordinates and a Definition of Curvature.. .... ..... .... .... .... .... .... ..... .... 65 ix x Contents Part III Origins of Complex Geometry 6 The Complex Plane. .... ..... .... .... .... .... .... ..... .... 75 6.1 Introduction .. .... ..... .... .... .... .... .... ..... .... 75 6.2 Caspar Wessel’s Cartography.. .... .... .... .... ..... .... 76 6.3 Argand and Gauss . ..... .... .... .... .... .... ..... .... 79 7 Elliptic and Abelian Integrals.. .... .... .... .... .... ..... .... 83 7.1 Introduction .. .... ..... .... .... .... .... .... ..... .... 83 7.2 Euler’s Addition Theorem .... .... .... .... .... ..... .... 85 7.3 Abel’s Addition Theorem. .... .... .... .... .... ..... .... 89 8 Elliptic Functions... .... ..... .... .... .... .... .... ..... .... 97 8.1 Introduction .. .... ..... .... .... .... .... .... ..... .... 97 8.2 Abel’s Recherches sur les fonctions elliptiques. .... ..... .... 99 8.3 Jacobi’s Fundamenta Nova.... .... .... .... .... ..... .... 108 8.4 Jacobi’s Theta Functions.. .... .... .... .... .... ..... .... 109 9 Complex Analysis .. .... ..... .... .... .... .... .... ..... .... 113 9.1 Introduction .. .... ..... .... .... .... .... .... ..... .... 113 9.2 Cauchy in 1814 ... ..... .... .... .... .... .... ..... .... 113 9.3 Cauchy’s 1825 Mémoire.. .... .... .... .... .... ..... .... 118 9.4 Riemann’s Dissertation from 1851.. .... .... .... ..... .... 120 9.5 The Lectures of Weierstrass ... .... .... .... .... ..... .... 128 9.6 The Mittag-Leffler Theorem ... .... .... .... .... ..... .... 133 10 Riemann Surfaces .. .... ..... .... .... .... .... .... ..... .... 137 10.1 Riemann’s Multilayered Surfaces... .... .... .... ..... .... 137 10.2 The Analysis Situs of Riemann .... .... .... .... ..... .... 139 10.3 Abelian Integrals and Abelian Functions . .... .... ..... .... 142 10.4 The Riemann–Roch Theorem.. .... .... .... .... ..... .... 153 11 Complex Geometry at the End of the Nineteenth Century.... .... 159 11.1 Klein and Lie. .... ..... .... .... .... .... .... ..... .... 159 11.2 The Uniformization Theorem for Riemann Surfaces. ..... .... 160 11.3 Point Set and Algebraic Topology .. .... .... .... ..... .... 162 11.4 Weyl’s Book, Die Idee der Riemannschen Fläche, in 1913.... 162 Part IV Twentieth-Century Embedding Theorems 12 Differentiable Manifolds . ..... .... .... .... .... .... ..... .... 175 12.1 Introduction .. .... ..... .... .... .... .... .... ..... .... 175 12.2 The Local Immersion Approximation.... .... .... ..... .... 179 12.3 Whitney’s Embedding Theorem.... .... .... .... ..... .... 181 12.4 Concluding Remarks..... .... .... .... .... .... ..... .... 184 Contents xi 13 Riemannian Manifolds .. ..... .... .... .... .... .... ..... .... 187 13.1 Introduction .. .... ..... .... .... .... .... .... ..... .... 187 13.2 Summary of the Proof of Nash’s Embedding Theorem ... .... 191 13.3 Nondegenerate Embeddings ... .... .... .... .... ..... .... 195 13.4 Nash’s Implicit Function Theorem .. .... .... .... ..... .... 197 13.5 Approximation of a Metric by an Induced Metric .. ..... .... 205 13.6 Closing Remarks .. ..... .... .... .... .... .... ..... .... 209 14 Compact Complex Manifolds.. .... .... .... .... .... ..... .... 211 14.1 Introduction .. .... ..... .... .... .... .... .... ..... .... 211 14.2 Holomorphic Line Bundles.... .... .... .... .... ..... .... 220 14.3 Sheaf Theory . .... ..... .... .... .... .... .... ..... .... 224 14.4 Hodge Theory. .... ..... .... .... .... .... .... ..... .... 233 14.5 Kodaira’s Vanishing Theorem . .... .... .... .... ..... .... 242 14.6 The Kodaira Embedding.. .... .... .... .... .... ..... .... 253 14.7 Riemann–Roch Theorems in Higher Dimensions ... ..... .... 262 15 Noncompact Complex Manifolds ... .... .... .... .... ..... .... 269 15.1 Introduction .. .... ..... .... .... .... .... .... ..... .... 269 15.2 Several Complex Variables.... .... .... .... .... ..... .... 270 15.3 Stein Manifolds ... ..... .... .... .... .... .... ..... .... 273 15.4 Generic Embeddings for a Class of Complex Manifolds .. .... 275 15.5 A Proper Embedding Theorem for Stein Manifolds . ..... .... 281 15.6 Grauert’s Solution to the Levi Problem .. .... .... ..... .... 286 15.7 The Grauert Real-Analytic Embedding Theorem ... ..... .... 297 Bibliography .. .... .... .... ..... .... .... .... .... .... ..... .... 303 Index .... .... .... .... .... ..... .... 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