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

This is the second of two volumes on the qualitative theory of foliations. For this volume, the authors have selected three special topics: analysis on foliated spaces, characteristic classes of foliations, and foliated manifolds. Each of these is an example of deep interaction between foliation theory and some other highly-developed area of mathematics. In all cases, the authors present useful, in-depth introductions, which lead to further study using the extensive available literature. This comprehensive volume has something to offer a broad spectrum of readers: from beginners to advanced students to professional researchers. It contains exercises and many illustrations. The book would make an elegant supplementary text for a topics course at the advanced graduate level. ''''Foliations I'''' is Volume 23 in the AMS series, ''''Graduate Studies in Mathematics'''' Part 1. The Foundations -- Chapter 1. Foliated Manifolds 5 -- 1.1. What Is A Foliation? 5 -- 1.2. Foliated Atlases 23 -- 1.3. The Frobenius Theorem 34 -- Chapter 2. Holonomy 45 -- 2.1. Foliated Bundles 46 -- 2.2. The Holonomy Pseudogroup 55 -- 2.3. Germinal Holonomy 59 -- 2.4. Reeb Stability 67 -- Chapter 3. Basic Constructions 71 -- 3.1. Suspension 71 -- 3.2. Pullbacks 79 -- 3.3. Transverse Modifications 81 -- 3.4. Tangential Gluing 90 -- 3.5. Orientation Covers 94 -- 3.6. Deformations 95 -- Chapter 4. Asymptotic Properties 103 -- 4.1. Minimal Sets 103 -- 4.2. Ends Of Manifolds 110 -- 4.3. Ends Of Leaves And Limit Sets 115 -- Part 2. Codimension One -- Chapter 5. Basic Structures 123 -- 5.1. Biregular Covers 123 -- 5.2. Open, Saturated Sets 126 -- 5.3. Proper And Semiproper Leaves 133 -- Chapter 6. Compact Leaves 137 -- 6.1. The Set Of Compact Leaves 137 -- 6.2. The Thurston Stability Theorem 142 -- 6.3. Compact Leaves And Closed Transversals 145 -- Chapter 7. General Position 151 -- 7.1. The Morse Lemma And Homotopies 151 -- 7.2. Poincare -- Bendixson Theory 156 -- 7.3. Analytic Foliations 162 -- Chapter 8. Generalized Poincare -- Bendixson Theory 165 -- 8.1. Foliations And Freshman Calculus 166 -- 8.2. Sacksteder's Theorem 183 -- 8.3. The Theory Of Levels 187 -- 8.4. Leaves Of Finite Depth 197 -- Chapter 9. Foliations Without Holonomy 205 -- 9.1. Open, Saturated Sets Without Holonomy 205 -- 9.2. A Leaf-preserving Flow 208 -- 9.3. A Closed, Nonsingular 1 -- Form 217 -- 9.4. Tischler's Theorem 220 -- 9.5. Isotopy 222 -- Part 3. Arbitrary Codimension -- Chapter 10. Foliation Cycles 231 -- 10.1. Basic Notions 231 -- 10.2. Foliation Currents 239 -- 10.3. Foliations Of Dimension One 253 -- 10.4. Taut Foliations Of Codimension One 257 -- 10.5. Taut Foliations Of Arbitrary Codimension 261 -- Chapter 11. Foliated Spaces 273 -- 11.2. Foliated Atlases 275 -- 11.3. Some Constructions Of Foliated Spaces 281 -- 11.4. Geometry On Foliated Spaces 298 -- 11.5. Holonomy -- Invariant Measures 301 -- Chapter 12. Growth, Invariant Measures And Geometry Of Leaves 309 -- 12.1. Quasi-riemannian Geometry Of Leaves 309 -- 12.2. Growth 311 -- 12.3. Growth And Holonomy -- Invariant Measures 325 -- 12.4. Leaves Closed At Infinity 330 -- 12.5. Dimension Two 331 -- 12.6. Uniformization 341 -- Chapter 13. Entropy Of Foliations 347 -- 13.1. Entropy Of Maps 348 -- 13.2. Entropy Of Groups, Pseudogroups And Foliations 351 -- 13.3. Geometric Entropy 358 -- 13.4. Invariant Measures 373 -- 13.5. Resilient Leaves 379. Alberto Candel, Lawrence Conlon. Includes Bibliographical References And 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. This is the second of two volumes on foliations (the first is Volume 23 of this series). In this volume, three specialized topics are treated: analysis on foliated spaces, characteristic classes of foliations, and foliated three-manifolds. Each of these topics represents deep interaction between foliation theory and another highly developed area of mathematics. In each case, the goal is to provide students and other interested people with a substantial introduction to the topic leading to further study using the extensive available literature. A guide to the qualitative theory of foliations. It features topics including: analysis on foliated spaces, characteristic classes of foliations and foliated manifolds. It is suitable as a supplementary text for a topics course at the advanced graduate level.