Cartan for Beginners: Differential Geometry Via Moving Frames and Exterior Differential Systems (Graduate Studies in Mathematics) (Graduate Studies in Mathematics, 175)

Two central aspects of Cartan's approach to differential geometry are the theory of exterior differential systems (EDS) and the method of moving frames. This book presents thorough and modern treatments of both subjects, including their applications to both classic and contemporary problems in geometry. It begins with the classical differential geometry of surfaces and basic Riemannian geometry in the language of moving frames, along with an elementary introduction to exterior differential systems. Key concepts are developed incrementally, with motivating examples leading to definitions, theorems, and proofs. Once the basics of the methods are established, the authors develop applications and advanced topics. One notable application is to complex algebraic geometry, where they expand and update important results from projective differential geometry. As well, the book features an introduction to $G$-structures and a treatment of the theory of connections. The techniques of EDS are also applied to obtain explicit solutions of PDEs via Darboux's method, the method of characteristics, and Cartan's method of equivalence. This text is suitable for a one-year graduate course in differential geometry, and parts of it can be used for a one-semester course. It has numerous exercises and examples throughout. It will also be useful to experts in areas such as geometry of PDE systems and complex algebraic geometry who want to learn how moving frames and exterior differential systems apply to their fields. The second edition features three new chapters: on Riemannian geometry, emphasizing the use of representation theory; on the latest developments in the study of Darboux-integrable systems; and on conformal geometry, written in a manner to introduce readers to the related parabolic geometry perspective. Cover Title page Contents Introduction Preface to the Second Edition Preface to the First Edition Chapter 1. Moving Frames and Exterior Differential Systems 1.1. Geometry of surfaces in \bee3 in coordinates 1.2. Differential equations in coordinates 1.3. Introduction to differential equations without coordinates 1.4. Introduction to geometry without coordinates: curves in \bee2 1.5. Submanifolds of homogeneous spaces 1.6. The Maurer-Cartan form 1.7. Plane curves in other geometries 1.8. Curves in \bee3 1.9. Grassmannians 1.10. Exterior differential systems and jet spaces Chapter 2. Euclidean Geometry 2.1. Gauss and mean curvature via frames 2.2. Calculation of H and K for some examples 2.3. Darboux frames and applications 2.4. What do H and K tell us? 2.5. Invariants for n-dimensional submanifolds of \bee{n+s} 2.6. Intrinsic and extrinsic geometry 2.7. Curves on surfaces 2.8. The Gauss-Bonnet and Poincaré-Hopf theorems Chapter 3. Riemannian Geometry 3.1. Covariant derivatives and the fundamental lemma of Riemannian geometry 3.2. Nonorthonormal frames and a geometric interpretation of mean curvature 3.3. The Riemann curvature tensor 3.4. Space forms: the sphere and hyperbolic space 3.5. Representation theory for Riemannian geometry 3.6. Infinitesimal symmetries: Killing vector fields 3.7. Homogeneous Riemannian manifolds 3.8. The Laplacian Chapter 4. Projective Geometry I: Basic Definitions and Examples 4.1. Frames and the projective second fundamental form 4.2. Algebraic varieties 4.3. Varieties with degenerate Gauss mappings Chapter 5. Cartan-Kähler I: Linear Algebra and Constant-Coefficient Homogeneous Systems 5.1. Tableaux 5.2. First example 5.3. Second example 5.4. Third example 5.5. The general case 5.6. The characteristic variety of a tableau Chapter 6. Cartan-Kähler II: The Cartan Algorithm for Linear Pfaffian Systems 6.1. Linear Pfaffian systems 6.2. First example 6.3. Second example: constant coefficient homogeneous systems 6.4. The local isometric embedding problem 6.5. The Cartan algorithm formalized: tableau, torsion and prolongation 6.6. Summary of Cartan’s algorithm for linear Pfaffian systems 6.7. Additional remarks on the theory 6.8. Examples 6.9. Functions whose Hessians commute, with remarks on singular solutions 6.10. The Cartan-Janet Isometric Embedding Theorem 6.11. Isometric embeddings of space forms (mostly flat ones) 6.12. Calibrated submanifolds Chapter 7. Applications to PDE 7.1. Symmetries and Cauchy characteristics 7.2. Second-order PDE and Monge characteristics 7.3. Derived systems and the method of Darboux 7.4. \MA/ systems and Weingarten surfaces 7.5. Integrable extensions and Bäcklund transformations Chapter 8. Cartan-Kähler III: The General Case 8.1. Integral elements and polar spaces 8.2. Example: triply orthogonal systems 8.3. Statement and proof of Cartan-Kähler 8.4. Cartan’s Test 8.5. More examples of Cartan’s Test Chapter 9. Geometric Structures and Connections 9.1. G-structures 9.2. Connections on \cf_{G} and differential invariants of G-structures 9.3. Overview of the Cartan algorithm 9.4. How to differentiate sections of vector bundles 9.5. Induced vector bundles and connections 9.6. Killing vector fields for G-structures 9.7. Holonomy 9.8. Extended example: path geometry Chapter 10. Superposition for Darboux-Integrable Systems 10.1. Decomposability 10.2. Integrability 10.3. Coframe adaptations 10.4. Some results on group actions 10.5. The superposition formula Chapter 11. Conformal Differential Geometry 11.1. Conformal geometry via Riemannian geometry 11.2. Conformal differential geometry as a G-structure 11.3. Conformal Killing vector fields 11.4. Conformal densities and the Laplacian 11.5. Einstein manifolds in a conformal class and the tractor bundle Chapter 12. Projective Geometry II: Moving Frames and Subvarieties of Projective Space 12.1. The Fubini cubic and higher order differential invariants 12.2. Fundamental forms of Veronese, Grassmann, and Segre varieties 12.3. Ruled and uniruled varieties 12.4. Dual varieties 12.5. Secant and tangential varieties 12.6. Cominuscule varieties and their differential invariants 12.7. Higher-order Fubini forms 12.8. Varieties with vanishing Fubini cubic 12.9. Associated varieties 12.10. More on varieties with degenerate Gauss maps 12.11. Rank restriction theorems 12.12. Local study of smooth varieties with degenerate tangential varieties 12.13. Generalized Monge systems 12.14. Complete intersections Appendix A. Linear Algebra and Representation Theory A.1. Dual spaces and tensor products A.2. Matrix Lie groups A.3. Complex vector spaces and complex structures A.4. Lie algebras A.5. Division algebras and the simple group G2 A.6. A smidgen of representation theory A.7. Clifford algebras and spin groups Appendix B. Differential Forms B.1. Differential forms and vector fields B.2. Three definitions of the exterior derivative B.3. Basic and semi-basic forms Appendix C. Complex Structures and Complex Manifolds C.1. Complex manifolds C.2. The Cauchy-Riemann equations Appendix D. Initial Value Problems and the Cauchy-Kowalevski Theorem D.1. Initial value problems D.2. The Cauchy-Kowalevski Theorem D.3. Generalizations Hints and Answers to Selected Exercises Bibliography Index Back Cover