Differential and complex geometry : origins, abstractions and embeddings / Raymond O. Wells, Jr.

Preface; Contents; Introduction; Part I Geometry in the Age of Enlightenment; Introduction; 1 Algebraic Geometry; 1.1 Introduction; 1.2 Descartes and Fermat; 1.3 Newton and Euler; 2 Differential Geometry; 2.1 Introduction; 2.2 Huygens and Newton; 2.3 Curves in Space: Courbes à double courbure; 2.4 Curvature of a Surface: Euler in 1767; Part II Differential and Projective Geometry in the Nineteenth Century; Introduction; 3 Projective Geometry; 3.1 Monge and Descriptive Geometry; 3.2 Poncelet's ``Propriétés Projectives''; 3.3 Analytic Projective Geometry

4 Gauss and Intrinsic Differential Geometry4.1 Gaussian Curvature; 4.2 Gauss's Theorema Egregrium; 5 Riemann's Higher-Dimensional Geometry; 5.1 The Legacy of Riemann; 5.2 Higher-Dimensional Manifolds and a Quadratic Line Element; 5.3 Geodesic Normal Coordinates and a Definition of Curvature; Part III Origins of Complex Geometry; Introduction; 6 The Complex Plane; 6.1 Introduction; 6.2 Caspar Wessel's Cartography; 6.3 Argand and Gauss; 7 Elliptic and Abelian Integrals; 7.1 Introduction; 7.2 Euler's Addition Theorem; 7.3 Abel's Addition Theorem; 8 Elliptic Functions; 8.1 Introduction

8.2 Abel's Recherches sur les fonctions elliptiques8.3 Jacobi's Fundamenta Nova; 8.4 Jacobi's Theta Functions; 9 Complex Analysis; 9.1 Introduction; 9.2 Cauchy in 1814; 9.3 Cauchy's 1825 Mémoire; 9.4 Riemann's Dissertation from 1851; 9.5 The Lectures of Weierstrass; 9.6 The Mittag-Leffler Theorem; 10 Riemann Surfaces; 10.1 Riemann's Multilayered Surfaces; 10.2 The Analysis Situs of Riemann; 10.3 Abelian Integrals and Abelian Functions; 10.4 The Riemann
Roch Theorem; 11 Complex Geometry at the End of the Nineteenth Century; 11.1 Klein and Lie

11.2 The Uniformization Theorem for Riemann Surfaces11.3 Point Set and Algebraic Topology; 11.4 Weyl's Book, Die Idee der Riemannschen Fläche, in 1913; Part IV Twentieth-Century Embedding Theorems; Introduction; 12 Differentiable Manifolds; 12.1 Introduction; 12.2 The Local Immersion Approximation; 12.3 Whitney's Embedding Theorem; 12.4 Concluding Remarks; 13 Riemannian Manifolds ; 13.1 Introduction; 13.2 Summary of the Proof of Nash's Embedding Theorem; 13.3 Nondegenerate Embeddings; 13.4 Nash's Implicit Function Theorem; 13.5 Approximation of a Metric by an Induced Metric

13.6 Closing Remarks14 Compact Complex Manifolds; 14.1 Introduction; 14.2 Holomorphic Line Bundles; 14.3 Sheaf Theory; 14.4 Hodge Theory; 14.5 Kodaira's Vanishing Theorem; 14.6 The Kodaira Embedding; 14.7 Riemann
Roch Theorems in Higher Dimensions; 15 Noncompact Complex Manifolds; 15.1 Introduction; 15.2 Several Complex Variables; 15.3 Stein Manifolds; 15.4 Generic Embeddings for a Class of Complex Manifolds; 15.5 A Proper Embedding Theorem for Stein Manifolds; 15.6 Grauert's Solution to the Levi Problem; 15.7 The Grauert Real-Analytic Embedding Theorem; Bibliography; Index