Differential and Complex Geometry: Origins, Abstractions and Embeddings

320 Pages·2017·11.92 MB·English

by  Wells

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
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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 .... .... .... .... .... ..... .... .... .... .... .... ..... .... 313