An Introduction to Differential Manifolds

Differential Calculus -- Manifolds: The Basics -- From Local to Global -- Lie Groups -- Differential Forms -- Integration and Applications -- Cohomology and Degree Theory -- Euler-Poincaré and Gauss-Bonnet.;This book is an introduction to differential manifolds. It gives solid preliminaries for more advanced topics: Riemannian manifolds, differential topology, Lie theory. It presupposes little background: the reader is only expected to master basic differential calculus, and a little point-set topology. The book covers the main topics of differential geometry: manifolds, tangent space, vector fields, differential forms, Lie groups, and a few more sophisticated topics such as de Rham cohomology, degree theory and the Gauss-Bonnet theorem for surfaces. Its ambition is to give solid foundations. In particular, the introduction of ĺlabstractĺl notions such as manifolds or differential forms is motivated via questions and examples from mathematics or theoretical physics. More than 150 exercises, some of them easy and classical, some others more sophisticated, will help the beginner as well as the more expert reader. Solutions are provided for most of them. The book should be of interest to various readers: undergraduate and graduate students for a first contact to differential manifolds, mathematicians from other fields and physicists who wish to acquire some feeling about this beautiful theory. The original French text Introduction aux vari©♭t©♭s diff©♭rentielles has been a best-seller in its category in France for many years. Jacques Lafontaine was successively assistant Professor at Paris Diderot University and Professor at the University of Montpellier, where he is presently emeritus. His main research interests are Riemannian and pseudo-Riemannian geometry, including some aspects of mathematical relativity. Besides his personal research articles, he was involved in several textbooks and research monographs. Preface......Page 6 Contents......Page 10 List of Figures......Page 16 Notations......Page 18 1.1.1. What Is Differential Calculus?......Page 21 1.1.2. In This Chapter......Page 23 1.2.1. Definition and Basic Properties......Page 24 Cauchy-Riemann equations......Page 27 Inverse of a matrix; determinant and trace......Page 28 1.2.3. Functions of Class Cp......Page 30 1.3. The Chain Rule......Page 31 1.4.1. Diffeomorphisms......Page 34 1.4.2. Local Diffeomorphisms......Page 36 1.4.3. Immersions, Submersions......Page 38 1.5.1. Basic Properties......Page 41 1.5.2. Examples: Spheres, Tori, and the Orthogonal Group......Page 43 1.5.3. Parametrizations......Page 45 1.5.4. Tangent Vectors, Tangent Space......Page 46 1.6. One-Parameter Subgroups of the Linear Group......Page 49 1.7. Critical Points......Page 53 1.8. Critical Values......Page 56 1.9. Differential Calculus in Infinite Dimensions......Page 58 Local models of maps......Page 60 Weakening of the regularity hypotheses......Page 61 1.11. Exercises......Page 62 2.1.1. A Typical Example: The Set of Lines in the Plane......Page 69 2.2.1. From Topological to Smooth Manifolds......Page 71 The sphere......Page 74 The manifold of affine straight lines of the plane......Page 75 2.3. Differentiable Functions; Diffeomorphisms......Page 76 2.4. Fundamental Theorem of Algebra......Page 80 2.5. Projective Spaces......Page 81 2.6.1. Tangent Space, Linear Tangent Map......Page 87 2.6.2. Local Diffeomorphisms, Immersions, Submersions, Submanifolds......Page 89 2.7. Covering Spaces......Page 93 2.7.1. Quotient of a Manifold by a Group......Page 94 2.7.2. Simply Connected Spaces......Page 101 2.8. Countability at Infinity......Page 103 Topological manifolds and smooth manifolds......Page 105 Flat structures......Page 106 2.10. Exercises......Page 107 3.1. Introduction......Page 117 3.2. Bump Functions; Embedding Manifolds......Page 118 3.3.1. Derivation at a Point......Page 123 3.3.2. Another Point of View on the Tangent Space......Page 126 3.3.3. Global Derivations......Page 128 3.4. Image of a Vector Field; Bracket......Page 130 3.5.1. The Manifold of Tangent Vectors......Page 133 3.5.2. Vector Bundles......Page 134 3.5.3. Vector Fields on Manifolds; The Hessian......Page 137 3.6. The Flow of a Vector Field......Page 139 3.7. Time-Dependent Vector Fields......Page 147 3.8. One-Dimensional Manifolds......Page 151 More about immersions and embeddings......Page 153 Morse theory......Page 154 Dynamical systems......Page 155 Connections......Page 156 3.10. Exercises......Page 157 4.1. Introduction......Page 166 4.2. Left Invariant Vector Fields......Page 167 4.3.1. Basic Properties; The Adjoint Representation......Page 173 4.3.2. From Lie Groups to Lie Algebras......Page 176 4.3.3. From Lie Algebras to Lie Groups......Page 177 4.4. A Digression on Topological Groups......Page 180 4.5.1. A Structure Theorem......Page 186 4.5.2. Towards Elliptic Curves......Page 189 4.6. Homogeneous Spaces......Page 190 Analytic structure......Page 195 “Infinite” Lie groups......Page 196 4.8. Exercises......Page 197 5.1.1. Why Differential Forms?......Page 203 5.1.2. Abstract......Page 204 5.2.1. Tensor Algebra......Page 205 5.2.2. Exterior Algebra......Page 207 5.2.3. Application: The Grassmannian of 2-Planes in 4 Dimensions......Page 211 5.3.1. Forms of Degree 1......Page 212 Passing from vector fields to differential forms in Euclidean space; gradient......Page 213 5.3.2. Forms of Arbitrary Degree......Page 214 5.4. Exterior Derivative......Page 217 5.5. Interior Product, Lie Derivative......Page 222 5.6.1. Star-Shaped Open Subsets......Page 227 b) integration with respect to the parameter.......Page 230 5.7. Differential Forms on a Manifold......Page 231 5.8.1. Minkowski Space......Page 236 5.8.2. The Electromagnetic Field as a Differential Form......Page 237 5.8.3. Electromagnetic Field and the Lorentz Group......Page 238 Differential forms and tensors......Page 240 Riemannian metrics......Page 241 Lorentzian manifolds......Page 242 5.10. Exercises......Page 243 6.1. Introduction......Page 252 6.2.1. Oriented Atlas......Page 254 6.2.2. Volume Forms......Page 256 6.2.3. Orientation Covering......Page 259 6.3.1. Integral of a Differential Form of Maximum Degree......Page 261 6.3.2. The Hairy Ball Theorem......Page 263 6.4.1. Integration on Compact Subsets......Page 265 6.4.2. Regular Domains and Their Boundary......Page 266 Orientation of the boundary......Page 268 6.4.3. Stokes’s Theorem in All of Its Forms......Page 270 b) Divergence theorem.......Page 272 6.5. Canonical Volume Form of a Submanifold of Euclidean Space......Page 274 6.6. Brouwer’s Theorem......Page 279 Integration on chains: towards algebraic topology......Page 282 6.8. Exercises......Page 283 7.1. Introduction......Page 289 7.2. De Rham Spaces......Page 291 7.3. Cohomology in Maximum Degree......Page 293 7.4.1. The Case of a Circle......Page 297 7.4.2. Definition and Basic Properties in the General Case......Page 299 7.4.3. Invariance of the Degree under Homotopy; Applications......Page 302 7.4.4. Index of a Vector Field......Page 305 7.5.1. Two Proofs of the Fundamental Theorem of Algebra Using Degree Theory......Page 308 7.5.2. Comparison of the Different Proofs of the Fundamental Theorem of Algebra......Page 309 7.6. Linking......Page 311 7.7. Invariance under Homotopy......Page 315 7.8.1. Exact Sequences......Page 319 7.8.2. The Mayer-Vietoris Sequence......Page 320 7.8.3. Application: A Few Cohomology Calculations......Page 323 7.8.4. The Noncompact Case......Page 325 7.9. Integral Methods......Page 326 7.10. Comments......Page 329 7.11. Exercises......Page 331 8.1.1. From Euclid to Carl-Friedrich Gauss and Pierre-Ossian Bonnet......Page 338 8.1.2. Sketch of a Proof of the Gauss-Bonnet Theorem......Page 340 8.2.1. Definition; Additivity......Page 341 8.2.2. Tilings......Page 343 8.3. Invitation to Riemannian Geometry......Page 346 Two curvature calculations......Page 350 8.4.2. A Residue Theorem......Page 351 8.5.1. Proof Using the Classification Theorem for Surfaces......Page 354 8.5.2. Proof Using Tilings: Sketch......Page 355 8.5.3. Putting the Preceding Arguments Together......Page 356 The case of embedded surfaces......Page 359 8.7. Exercises......Page 361 Appendix: The Fundamental Theorem of Differential Topology......Page 364 Chapter 1......Page 366 Chapter 2......Page 369 Chapter 3......Page 375 Chapter 4......Page 378 Chapter 5......Page 380 Chapter 6......Page 385 Chapter 7......Page 390 Chapter 8......Page 394 Other points of view......Page 397 Related subjects......Page 398 References......Page 399 Index......Page 406 This book is an introduction to differential manifolds. It gives solid preliminaries for more advanced topics: Riemannian manifolds, differential topology, Lie theory. It presupposes little background: the reader is only expected to master basic differential calculus, and a little point-set topology. The book covers the main topics of differential geometry: manifolds, tangent space, vector fields, differential forms, Lie groups, and a few more sophisticated topics such as de Rham cohomology, degree theory and the Gauss-Bonnet theorem for surfaces. Its ambition is to give solid foundations. In particular, the introduction of ́llabstract́ll notions such as manifolds or differential forms is motivated via questions and examples from mathematics or theoretical physics. More than 150 exercises, some of them easy and classical, some others more sophisticated, will help the beginner as well as the more expert reader. Solutions are provided for most of them. The book should be of interest to various readers: undergraduate and graduate students for a first contact to differential manifolds, mathematicians from other fields and physicists who wish to acquire some feeling about this beautiful theory. The original French text Introduction aux vari©♭t©♭s diff©♭rentielles has been a best-seller in its category in France for many years. Jacques Lafontaine was successively assistant Professor at Paris Diderot University and Professor at the University of Montpellier, where he is presently emeritus. His main research interests are Riemannian and pseudo-Riemannian geometry, including some aspects of mathematical relativity. Besides his personal research articles, he was involved in several textbooks and research monographs This book is an introduction to differential manifolds. It gives solid preliminaries for more advanced topics: Riemannian manifolds, differential topology, Lie theory. It presupposes little background: the reader is only expected to master basic differential calculus, and a little point-set topology. The book covers the main topics of differential geometry: manifolds, tangent space, vector fields, differential forms, Lie groups, and a few more sophisticated topics such as de Rham cohomology, degree theory and the Gauss-Bonnet theorem for surfaces. Its ambition is to give solid foundations. In particular, the introduction of "abstract" notions such as manifolds or differential forms is motivated via questions and examples from mathematics or theoretical physics. More than 150 exercises, some of them easy and classical, some others more sophisticated, will help the beginner as well as the more expert reader. Solutions are provided for most of them. The book should be of interest to various readers: undergraduate and graduate students for a first contact to differential manifolds, mathematicians from other fields and physicists who wish to acquire some feeling about this beautiful theory. The original French text Introduction aux variétés différentielles has been a best-seller in its category in France for many years. Jacques Lafontaine was successively assistant Professor at Paris Diderot University and Professor at the University of Montpellier, where he is presently emeritus. His main research interests are Riemannian and pseudo-Riemannian geometry, including some aspects of mathematical relativity. Besides his personal research articles, he was involved in several textbooks and research monographs This book is an introduction to differential manifolds. It gives solid preliminaries for more advanced topics: Riemannian manifolds, differential topology, Lie theory. It presupposes little background: the reader is only expected to master basic differential calculus, and a little point-set topology. The book covers the main topics of differential geometry: manifolds, tangent space, vector fields, differential forms, Lie groups, and a few more sophisticated topics such as de Rham cohomology, degree theory and the Gauss-Bonnet theorem for surfaces. Its ambition is to give solid foundations. In particular, the introduction of zabstracty notions such as manifolds or differential forms is motivated via questions and examples from mathematics or theoretical physics. More than 150 exercises, some of them easy and classical, some others more sophisticated, will help the beginner as well as the more expert reader. Solutions are provided for most of them. The book should be of interest to various readers: undergraduate and graduate students for a first contact to differential manifolds, mathematicians from other fields and physicists who wish to acquire some feeling about this beautiful theory. The original French text Introduction aux variétés différentielles has been a best-seller in its category in France for many years. Jacques Lafontaine was successively assistant Professor at Paris Diderot University and Professor at the University of Montpellier, where he is presently emeritus. His main research interests are Riemannian and pseudo-Riemannian geometry, including some aspects of mathematical relativity. Besides his personal research articles, he was involved in several textbooks and research monographs Ce livre scientifique est une initiation aux variétés différentielles, préalable à des enseignements plus spécialisés. Le lecteur devra posséder une compétence sur le calcul différentiel dans les espaces euclidiens. Sont abordées les principales notions de géométrie différentielle : variétés différentielles, espaces tangent et cotangent, champs de vecteurs, formes différentielles. De nombreux exemples sont traités en détail. Cet ensemble constitue une introduction aux groupes de Lie. Il est illustré par les éléments de théorie du degré et de cohomologie. Introduction aux variétés différentielles a pour objectif d'être un ouvrage de base. Il propose des exercices classiques pour l'étudiant et le débutant en la matière, d'autres plus délicats pour l'enseignant, le chercheur ou l'étudiant de niveau plus avancé. Les solutions d'un bon nombre d'entre eux sont données en fin de volume. Le succès de la première édition notamment auprès des étudiants a motivé les améliorations de cette édition. Un chapitre nouveau est proposé sur les caractéristiques d'Euler-Poincaré et le théorème de Gauss-Bonnet. Cet ouvrage est un pap-ebook : un site web corrélé propose des compléments et des annexes. Le lecteur peut ainsi s'appuyer sur des rappels, des exercices, des approfondissements sur le site compagnon présenté au début du livre. Destiné aux étudiants de master et des préparations à l'agrégation, aux universitaires, aux professeurs des lycées et des classes préparatoires. Les physiciens sont également concernés. Cet ouvrage est reconnu comme un des classiques (au programme de l'agrégation de mathématiques) et l'édition originale a été vendue à plus de 3000 exemplaires, ce qui est une indication pour un ouvrage de ce niveau. Il est dans toutes les listes des universités qui recommandent des ouvrages à ce niveau (M2, concours). La création du site web corrélé va accroître l'intérêt de l'ouvrage avec une possibilité de compléments qui peuvent évoluer avec le temps