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.
Author(s): Thomas A. Ivey, Joseph M. Landsberg
Series: Graduate Studies in Mathematics 175
Edition: 2
Publisher: American Mathematical Society
Year: 2016
Language: English
Commentary: decrypted from A6D7A8D1AD696321F859DFA4BD136BB3 source file
Pages: 455
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 ? and ? for some examples
2.3. Darboux frames and applications
2.4. What do ? and ? tell us?
2.5. Invariants for ?-dimensional submanifolds of \bee{?+?}
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. ?-structures
9.2. Connections on \cf_{?} and differential invariants of ?-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 ?-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 ?-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 ?₂
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
