**Graduate Studies in Mathematics**

Volume: 175;
2016;
455 pp;
Hardcover

MSC: Primary 35; 37; 53; 58;

Print ISBN: 978-1-4704-0986-9

Product Code: GSM/175

List Price: $89.00

AMS Member Price: $71.20

MAA Member Price: $80.10

**Electronic ISBN: 978-1-4704-3591-2
Product Code: GSM/175.E**

List Price: $89.00

AMS Member Price: $71.20

MAA Member Price: $80.10

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#### Supplemental Materials

# Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems, Second Edition

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*Thomas A. Ivey; Joseph M. Landsberg*

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.

#### Readership

Graduate students and researchers interested in differential geometry, in particular, in exterior systems and the moving frames method and in its applications in algebraic geometry, PDE, and other areas of mathematics

#### Reviews & Endorsements

[T]his book, like the first edition, is an excellent source for graduate students and professional mathematicians who want to learn about moving frames and G-structures in trying to understand differential geometry.

-- Thomas Garrity, Mathematical Reviews

All the material is carefully developed, many examples supporting the understanding. The reviewer warmly recommends this volume to mathematical university libraries.

-- Gabriel Eduard Vilcu, Zentralblatt MATH

#### Table of Contents

# Table of Contents

## Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems, Second Edition

- Cover Cover11
- Title page iii4
- Contents v6
- Introduction xi12
- Chapter 1. Moving Frames and Exterior Differential Systems 120
- 1.1. Geometry of surfaces in \bee3 in coordinates 221
- 1.2. Differential equations in coordinates 524
- 1.3. Introduction to differential equations without coordinates 827
- 1.4. Introduction to geometry without coordinates: curves in \bee2 1332
- 1.5. Submanifolds of homogeneous spaces 1635
- 1.6. The Maurer-Cartan form 1837
- 1.7. Plane curves in other geometries 2241
- 1.8. Curves in \bee3 2544
- 1.9. Grassmannians 2847
- 1.10. Exterior differential systems and jet spaces 3150

- Chapter 2. Euclidean Geometry 3958
- 2.1. Gauss and mean curvature via frames 4059
- 2.2. Calculation of 𝐻 and 𝐾 for some examples 4362
- 2.3. Darboux frames and applications 4665
- 2.4. What do 𝐻 and 𝐾 tell us? 4766
- 2.5. Invariants for 𝑛-dimensional submanifolds of \bee{𝑛+𝑠} 4867
- 2.6. Intrinsic and extrinsic geometry 5271
- 2.7. Curves on surfaces 5574
- 2.8. The Gauss-Bonnet and Poincaré-Hopf theorems 5776

- Chapter 3. Riemannian Geometry 6584
- 3.1. Covariant derivatives and the fundamental lemma of Riemannian geometry 6584
- 3.2. Nonorthonormal frames and a geometric interpretation of mean curvature 7392
- 3.3. The Riemann curvature tensor 7796
- 3.4. Space forms: the sphere and hyperbolic space 8099
- 3.5. Representation theory for Riemannian geometry 83102
- 3.6. Infinitesimal symmetries: Killing vector fields 85104
- 3.7. Homogeneous Riemannian manifolds 89108
- 3.8. The Laplacian 92111

- Chapter 4. Projective Geometry I: Basic Definitions and Examples 95114
- Chapter 5. Cartan-Kähler I: Linear Algebra and Constant-Coefficient Homogeneous Systems 115134
- Chapter 6. Cartan-Kähler II: The Cartan Algorithm for Linear Pfaffian Systems 135154
- 6.1. Linear Pfaffian systems 135154
- 6.2. First example 137156
- 6.3. Second example: constant coefficient homogeneous systems 138157
- 6.4. The local isometric embedding problem 141160
- 6.5. The Cartan algorithm formalized: tableau, torsion and prolongation 146165
- 6.6. Summary of Cartan’s algorithm for linear Pfaffian systems 149168
- 6.7. Additional remarks on the theory 151170
- 6.8. Examples 154173
- 6.9. Functions whose Hessians commute, with remarks on singular solutions 161180
- 6.10. The Cartan-Janet Isometric Embedding Theorem 164183
- 6.11. Isometric embeddings of space forms (mostly flat ones) 166185
- 6.12. Calibrated submanifolds 169188

- Chapter 7. Applications to PDE 175194
- Chapter 8. Cartan-Kähler III: The General Case 215234
- Chapter 9. Geometric Structures and Connections 241260
- 9.1. 𝐺-structures 241260
- 9.2. Connections on \cf_{𝐺} and differential invariants of 𝐺-structures 248267
- 9.3. Overview of the Cartan algorithm 253272
- 9.4. How to differentiate sections of vector bundles 254273
- 9.5. Induced vector bundles and connections 257276
- 9.6. Killing vector fields for 𝐺-structures 260279
- 9.7. Holonomy 262281
- 9.8. Extended example: path geometry 271290

- Chapter 10. Superposition for Darboux-Integrable Systems 285304
- Chapter 11. Conformal Differential Geometry 307326
- Chapter 12. Projective Geometry II: Moving Frames and Subvarieties of Projective Space 331350
- 12.1. The Fubini cubic and higher order differential invariants 332351
- 12.2. Fundamental forms of Veronese, Grassmann, and Segre varieties 336355
- 12.3. Ruled and uniruled varieties 339358
- 12.4. Dual varieties 341360
- 12.5. Secant and tangential varieties 346365
- 12.6. Cominuscule varieties and their differential invariants 350369
- 12.7. Higher-order Fubini forms 356375
- 12.8. Varieties with vanishing Fubini cubic 363382
- 12.9. Associated varieties 365384
- 12.10. More on varieties with degenerate Gauss maps 367386
- 12.11. Rank restriction theorems 369388
- 12.12. Local study of smooth varieties with degenerate tangential varieties 372391
- 12.13. Generalized Monge systems 376395
- 12.14. Complete intersections 378397

- Appendix A. Linear Algebra and Representation Theory 381400
- A.1. Dual spaces and tensor products 381400
- A.2. Matrix Lie groups 386405
- A.3. Complex vector spaces and complex structures 388407
- A.4. Lie algebras 390409
- A.5. Division algebras and the simple group 𝐺₂ 394413
- A.6. A smidgen of representation theory 397416
- A.7. Clifford algebras and spin groups 400419

- Appendix B. Differential Forms 405424
- Appendix C. Complex Structures and Complex Manifolds 411430
- Appendix D. Initial Value Problems and the Cauchy-Kowalevski Theorem 419438
- Hints and Answers to Selected Exercises 425444
- Bibliography 437456
- Index 445464
- Back Cover Back Cover1474