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

This book is an introduction to Cartan's approach to differential geometry. Two central methods in Cartan's geometry are the theory of exterior differential systems 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. It begins with the classical 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. The book features an introduction to $G$-structures and a treatment of the theory of connections. The Cartan machinery is 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 PDEs and algebraic geometry who want to learn how moving frames and exterior differential systems apply to their fields. Title page Preface Chapter 1. Moving Frames and Exterior Differential Systems 1.1. Geometry of surfaces in E3 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 E2 1.5. Submanifolds of homogeneous spaces 1.6. The Maurer-Cartan form 1.7. Plane curves in other geometries 1.8. Curves in E3 1.9. Exterior differential systems and jet spaces Chapter 2. Euclidean Geometry and Riemannian 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 E^{n+s} 2.6. Intrinsic and extrinsic geometry 2.7. Space forms: the sphere and hyperbolic space 2.8. Curves on surfaces 2.9. The Gauss-Bonnet and Poincaré-Hopf theorems 2.10. Non-orthonormal £rames Chapter 3. Projective Geometry 3.1. Grassmannians 3.2. Frames and the projective second fundamental form 3.3. Algebraic varieties 3.4. Varieties with degenerate Gauss mappings 3.5. Higher-order differential invariants 3.6. Fundamental forms of some homogeneous varieties 3.7. Higher-order Fubini forms 3.8. Ruled and uniruled varieties 3.9. Varieties with vanishing Fubini cubic 3.10. Dual varieties 3.11. Associated varieties 3.12. More on varieties with degenerate Gauss maps 3.13. Secant and tangential varieties 3.14. Rank restriction theorems 3.15. Local study of smooth varieties with degenerate tangential varieties 3.16. Generalized Monge systems 3.17. Complete intersections Chapter 4. Cartan-Kähler I: Linear Algebra and Constant-Coefficient Homogeneous Systems 4.1. Tableaux 4.2. First example 4.3. Second example 4.4. Third example 4.5. The general case 4.6. The characteristic variety of a tableau Chapter 5. Cartan-Kähler II: The Cartan Algorithm for Linear Pfaffian Systems 5.1. Linear Pfaffian systems 5.2. First example 5.3. Second example: constant coefficient homogeneous systems 5.4. The local isometric embedding problem 5.5. 5.12. The Cartan algorithm formalized: tableau, torsion and prolongation 5.6. Summary of Cartan's algorithm for linear Pfaffian systems 5.7. Additional remarks on the theory 5.8. Examples 5.9. Functions whose Hessians commute, with remarks on singular solutions 5.10. The Cartan-Janet Isometric Embedding Theorem 5.11. Isometric embeddings of space forms (mostly flat ones) Calibrated submanifolds Chapter 6. Applications to PDE 6.1. Symmetries and Cauchy characteristics 6.2. Second-order PDE and Monge characteristics 6.3. Derived systems and the method of Darboux 6.4. Monge-Ampère systems and Weingarten surfaces 6.5. Integrable extensions and Backlund transformations Chapter 7. Cartan-Kähler III: The General Case 7.1. Integral elements and polar spaces 7.2. Example: Triply orthogonal systems 7.3. Statement and proof of Cartan-Kähler 7.4. Cartan's Test 7.5. More examples of Cartan's Test Chapter 8. Geometric Structures and Connections 8.1. G-structures 8.2. How to differentiate sections of vector bundles 8.3. Connections on F_G and differential invariants of G-structures 8.4. Induced vector bundles and connections on induced bundles 8.5. Holonomy 8.6. Extended example: Path geometry 8.7. Frobenius and generalized conformal structures 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 B.4. Differential ideals Appendix C. Complex Structures and Complex Manifolds C.1. Complex manifolds C.2. The Cauchy-Riemann equations Appendix D. Initial Value Problems Hints and Answers to Selected Exercises Bibliography Index "This book is an introduction to Cartan's approach to differential geometry. Two central methods in Cartan's geometry are the theory of exterior, differential systems and the method of moving frames. The book presents thorough and modern treatments of both subjects, including their applications to classic and contemporary problems." "The book begins with the classical 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, applications and advanced topics are developed. One particularly notable application is to complex algebraic geometry, where important results from projective differential geometry are expanded and updated. The book features an introduction to G-structures and a treatment of the theory of connections. The Cartan machinery is 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. It has numerous exercises and examples throughout. The book will also be of use to experts in such areas as PDEs and algebraic geometry who want to learn how moving frames and exterior differential systems apply to their fields."--Jacket

this Book Is An Introduction To Cartan's Approach To Differential Geometry. Two Central Methods In Cartan's Geometry Are The Theory Of Exterior, Differential Systems And The Method Of Moving Frames. The Book Presents Thorough And Modern Treatments Of Both Subjects, Including Their Applications To Classic And Contemporary Problems.

the Book Begins With The Classical 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, Applications And Advanced Topics Are Developed. One Particularly Notable Application Is To Complex Algebraic Geometry, Where Important Results From Projective Differential Geometry Are Expanded And Updated. The Book Features An Introduction To G-structures And A Treatment Of The Theory Of Connections. The Cartan Machinery Is 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. It Has Numerous Exercises And Examples Throughout. The Book Will Also Be Of Use To Experts In Such Areas As Pdes And Algebraic Geometry Who Want To Learn How Moving Frames And Exterior Differential Systems Apply To Their Fields.

This book is an introduction to Cartan's approach to differential geometry. Two central methods in Cartan's geometry are the theory of exterior differential systems 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. It begins with the classical 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. The book features an introduction to $G$-structures and a treatment of the theory of connections. The Cartan machinery is also applied to obtain explicit solutions of PDEs via Darboux's method, the method of characteristics, and Cartan's method of equivalence.