Differential and Complex Geometry: Origins, Abstractions and Embeddings

Transcription

1 Differential and Complex Geometry: Origins, Abstractions and Embeddings

2 Raymond O. Wells, Jr. Differential and Complex Geometry: Origins, Abstractions and Embeddings 123

3 Raymond O. Wells, Jr. University of Colorado Boulder Boulder, CO USA and Jacobs University Bremen Bremen Germany ISBN ISBN (ebook) DOI / Library of Congress Control Number: Mathematics Subject Classification (2010): 01-02, 30-02, 32-02, 14-02, 51-02, 53-02, Springer International Publishing AG 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part 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 or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. 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 authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Cover illustration reproduced from Vorlesungen über Nicht-Euklidische Geometrie by Felix Klein (Volume 26 of the series Die Grundlehren der Mathematischen Wissenschaften), 1967, p With permission of Springer Nature. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

4 To my friend, Howard Resnikoff

5 Preface About ten years ago, I had the idea of writing up a survey of the major embedding theorems of the twentieth century. This book represents the culmination of this idea, 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, differential geometry, complex manifold theory, and the general theory of functions of several complex variables, as well as the overall concept of manifolds 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, I spent a fair amount of time these past years looking at a number 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 understanding of these earlier mathematicians as they gave voice to the many issues at hand. For instance, the developments of projective geometry and intrinsic differential 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 times had it not been for the Internet and the fact that the great libraries of the world 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, where it has been my privilege to hold various academic appointments, and I want 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 they are bringing this new work of mine to the public. I want to thank, in particular, Rémi Lodh, who encouraged me and helped bring this book to fruition. vii

6 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 process of managing the digital files and ushering them into the production process. I want to thank Ina Mette, formerly of Springer and now an editor for the American Mathematical Society, for her encouragement for this project over many 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. His reading of various drafts of this book and his encouraging words have been very important to me. Finally, I want to thank my wife, Rena, for her continuous support in so many ways. In particular, she read a final draft and her comments and editorial pen were so very useful, as always. Boulder, CO, USA March 2017 Raymond O. Wells, Jr.

7 Contents Introduction.... xiii Part I Geometry in the Age of Enlightenment 1 Algebraic Geometry Introduction Descartes and Fermat Newton and Euler Differential Geometry Introduction Huygens and Newton Curves in Space: Courbes à double courbure Curvature of a Surface: Euler in Part II Differential and Projective Geometry in the Nineteenth Century 3 Projective Geometry Monge and Descriptive Geometry Poncelet s Propriétés Projectives Analytic Projective Geometry Gauss and Intrinsic Differential Geometry Gaussian Curvature Gauss s Theorema Egregrium Riemann s Higher-Dimensional Geometry The Legacy of Riemann Higher-Dimensional Manifolds and a Quadratic Line Element Geodesic Normal Coordinates and a Definition of Curvature ix

8 x Contents Part III Origins of Complex Geometry 6 The Complex Plane Introduction Caspar Wessel s Cartography Argand and Gauss Elliptic and Abelian Integrals Introduction Euler s Addition Theorem Abel s Addition Theorem Elliptic Functions Introduction Abel s Recherches sur les fonctions elliptiques Jacobi s Fundamenta Nova Jacobi s Theta Functions Complex Analysis Introduction Cauchy in Cauchy s 1825 Mémoire Riemann s Dissertation from The Lectures of Weierstrass The Mittag-Leffler Theorem Riemann Surfaces Riemann s Multilayered Surfaces The Analysis Situs of Riemann Abelian Integrals and Abelian Functions The Riemann Roch Theorem Complex Geometry at the End of the Nineteenth Century Klein and Lie The Uniformization Theorem for Riemann Surfaces Point Set and Algebraic Topology Weyl s Book, Die Idee der Riemannschen Fläche, in Part IV Twentieth-Century Embedding Theorems 12 Differentiable Manifolds Introduction The Local Immersion Approximation Whitney s Embedding Theorem Concluding Remarks

9 Contents xi 13 Riemannian Manifolds Introduction Summary of the Proof of Nash s Embedding Theorem Nondegenerate Embeddings Nash s Implicit Function Theorem Approximation of a Metric by an Induced Metric Closing Remarks Compact Complex Manifolds Introduction Holomorphic Line Bundles Sheaf Theory Hodge Theory Kodaira s Vanishing Theorem The Kodaira Embedding Riemann Roch Theorems in Higher Dimensions Noncompact Complex Manifolds Introduction Several Complex Variables Stein Manifolds Generic Embeddings for a Class of Complex Manifolds A Proper Embedding Theorem for Stein Manifolds Grauert s Solution to the Levi Problem The Grauert Real-Analytic Embedding Theorem Bibliography Index

10 Introduction In 1913, Hermann Weyl wrote a very influential book that one can perceive of as a bridge between the geometry of earlier centuries and the geometry that evolved in the twentieth century. More specifically, using the newly discovered theories of set theory and point set topology, he created a theory of differentiable and complex manifolds, in particular, in the case of Riemann surfaces. In 1936, Hassler Whitney ushered in a new era of geometry when he formulated and proved the first embedding theorem for differentiable manifolds. Namely, he showed that any differentiable manifold can be embedded as a closed submanifold of a higher-dimensional Euclidean space. This was followed up over the next two decades by various mathematicians who provided similar characterizations: real-analytic submanifolds of Euclidean space (Grauert 1958), differentiable submanifolds with a Riemannian metric induced from the ambient Euclidean space (Nash 1956), complex submanifolds of complex Euclidean space (Remmert, Narasimhan, Bishop, ), and complex submanifolds of complex projective space (Kodaira 1954). All of these were embedding theorems of one sort or another. Their formulations and proofs depended on a variety of mathematical ideas, many of which had also evolved in the twentieth century. This book outlines a survey of roughly three centuries of mathematical work concerned with differential and complex geometry, culminating in the twentieth-century embedding theorems. The book is divided into four parts, which are described below in more detail. In Parts I III, we provide an overview of many of the geometric ideas that play an important role in the twentieth-century embedding theorems and which arose in various guises in the previous three centuries, and Part IV describes the embedding theorems in some detail. Our major source for this survey of mathematical ideas has been to look in some detail at the original papers and monographs of the principal authors whose works are the cornerstones of these developments. We have tried to look at the writings of these authors in the context of the mathematical knowledge known at the time. Part I looks at the way the geometry of curves and surfaces in two- and three-dimensional Euclidean space began to interact with the simultaneously xiii

11 xiv Introduction evolving theories of analysis that grew out of the late seventeenth century with the discoveries of differential and integral calculus. Here, the notion of tangent vectors and tangent spaces, first-order approximations to curves and surfaces, and curvature, measuring how far curves and surfaces deviated from straight lines and planes, all evolved in a systematic fashion. This became the essence of what we now call extrinsic differential geometry, where the notion of the distance between points is inherited from the ambient Euclidean space. Part II describes two parallel theories that evolved in the nineteenth century. The first was the discovery of intrinsic differential geometry that has become the foundation of contemporary differential geometry (Gauss, Riemann). The second was the creation of projective geometry, a generalization of classical Euclidean geometry that transcended the usual two- and three-dimensional space by asking new types of geometric questions and introducing points at infinity (Monge, Poncelet, and many others). This evolved into our contemporary notion of projective space and became the basis for much of algebraic geometry in the twentieth century. Part III is an outline of the origins of what became known in the twentieth century as complex geometry. It has its roots in the generalizations of trigonometric functions and their properties: Euler s addition theorems for elliptic integrals (Legendre, Abel), elliptic functions, and their generalizations, Abelian functions (Abel, Jacobi, Weierstrass, Riemann). Moreover, the development of function theory over many decades, starting with the pioneering work of Cauchy in the 1820s (Riemann, Weierstrass), and in the innovative work of Riemann in his creation of the theory of Riemann surfaces in the mid-nineteenth century, led to the developments of algebraic topology and the theory of manifolds in general at the end of the nineteenth century (Riemann, Betti, Poincaré, Weyl). The work of Klein and Lie on transformation groups in the latter half of the century was a very important contribution for modern geometry as well. Part IV of this book outlines in some detail the major twentieth-century embedding theorems. They are all philosophically related: A manifold of some sort can be embedded as a submanifold in some higher-dimensional Euclidean space or projective space, and the embedding characterizes all such submanifolds. Technically, they involve a broad range of mathematical tools and, for the most part, solved problems that had been formulated earlier and involved quite technical and often very difficult proofs. More specifically, they involve differentiable and real-analytic manifolds, arising from the work in Part II, and they involve complex manifolds, arising from the work in Part III. Each Part of this book has its own more detailed introduction to the material in that set of chapters. Here, we have given only a very brief overview of the whole book. This survey over several centuries tries to show how these various strands of mathematical thought have culminated in the powerful embedding theorems of the mid-twentieth century. Of course, there were many other areas of development of geometric ideas in the same time period which are not included in our survey, but we feel we have chosen a coherent family of ideas that have contributed greatly to our mathematical culture in the twentieth century.