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E-raamat: Cartan for Beginners

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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.

Arvustused

[ 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

Introduction xi
Preface to the Second Edition xi
Preface to the First Edition xiv
Chapter 1 Moving Frames and Exterior Differential Systems
1(38)
§1.1 Geometry of surfaces in E3 in coordinates
2(3)
§1.2 Differential equations in coordinates
5(3)
§1.3 Introduction to differential equations without coordinates
8(5)
§1.4 Introduction to geometry without coordinates: curves in E2
13(3)
§1.5 Submanifolds of homogeneous spaces
16(2)
§1.6 The Maurer-Cartan form
18(4)
§1.7 Plane curves in other geometries
22(3)
§1.8 Curves in E3
25(3)
§1.9 Grassmannians
28(3)
§1.10 Exterior differential systems and jet spaces
31(8)
Chapter 2 Euclidean Geometry
39(26)
§2.1 Gauss and mean curvature via frames
40(3)
§2.2 Calculation of H and K for some examples
43(3)
§2.3 Darboux frames and applications
46(1)
§2.4 What do H and K tell us?
47(1)
§2.5 Invariants for n-dimensional submanifolds of En+S
48(4)
§2.6 Intrinsic and extrinsic geometry
52(3)
§2.7 Curves on surfaces
55(2)
§2.8 The Gauss-Bonnet and Poincare-Hopf theorems
57(8)
Chapter 3 Riemannian Geometry
65(30)
§3.1 Covariant derivatives and the fundamental lemma of Riemannian geometry
65(8)
§3.2 Nonorthonormal frames and a geometric interpretation of mean curvature
73(4)
§3.3 The Riemann curvature tensor
77(3)
§3.4 Space forms: the sphere and hyperbolic space
80(3)
§3.5 Representation theory for Riemannian geometry
83(2)
§3.6 Infinitesimal symmetries: Killing vector fields
85(4)
§3.7 Homogeneous Riemannian manifolds
89(3)
§3.8 The Laplacian
92(3)
Chapter 4 Projective Geometry I: Basic Definitions and Examples
95(20)
§4.1 Frames and the projective second fundamental form
96(5)
§4.2 Algebraic varieties
101(8)
§4.3 Varieties with degenerate Gauss mappings
109(6)
Chapter 5 Cartan-Kahler I: Linear Algebra and Constant-Coefficient Homogeneous Systems
115(20)
§5.1 Tableaux
116(4)
§5.2 First example
120(2)
§5.3 Second example
122(3)
§5.4 Third example
125(1)
§5.5 The general case
126(3)
§5.6 The characteristic variety of a tableau
129(6)
Chapter 6 Cartan-Kahler II: The Cartan Algorithm for Linear Pfaffian Systems
135(40)
§6.1 Linear Pfaffian systems
135(2)
§6.2 First example
137(1)
§6.3 Second example: constant coefficient homogeneous systems
138(3)
§6.4 The local isometric embedding problem
141(5)
§6.5 The Cartan algorithm formalized: tableau, torsion and prolongation
146(3)
§6.6 Summary of Cartan's algorithm for linear Pfaffian systems
149(2)
§6.7 Additional remarks on the theory
151(3)
§6.8 Examples
154(7)
§6.9 Functions whose Hessians commute, with remarks on singular solutions
161(3)
§6.10 The Cartan-Janet Isometric Embedding Theorem
164(2)
§6.11 Isometric embeddings of space forms (mostly flat ones)
166(3)
§6.12 Calibrated submanifolds
169(6)
Chapter 7 Applications to PDE
175(40)
§7.1 Symmetries and Cauchy characteristics
176(8)
§7.2 Second-order PDE and Monge characteristics
184(4)
§7.3 Derived systems and the method of Darboux
188(7)
§7.4 Monge-Ampere systems and Weingarten surfaces
195(9)
§7.5 Integrable extensions and Backlund transformations
204(11)
Chapter 8 Cartan-Kahler III: The General Case
215(26)
§8.1 Integral elements and polar spaces
216(7)
§8.2 Example: triply orthogonal systems
223(3)
§8.3 Statement and proof of Cartan-Kahler
226(3)
§8.4 Cartan's Test
229(3)
§8.5 More examples of Cartan's Test
232(9)
Chapter 9 Geometric Structures and Connections
241(44)
§9.1 G-structures
241(7)
§9.2 Connections on FG and differential invariants of G-structures
248(5)
§9.3 Overview of the Cartan algorithm
253(1)
§9.4 How to differentiate sections of vector bundles
254(3)
§9.5 Induced vector bundles and connections
257(3)
§9.6 Killing vector fields for G-structures
260(2)
§9.7 Holonomy
262(9)
§9.8 Extended example: path geometry
271(14)
Chapter 10 Superposition for Darboux-Integrable Systems
285(22)
§10.1 Decomposability
286(3)
§10.2 Integrability
289(5)
§10.3 Coframe adaptations
294(5)
§10.4 Some results on group actions
299(1)
§10.5 The superposition formula
300(7)
Chapter 11 Conformal Differential Geometry
307(24)
§11.1 Conformal geometry via Riemannian geometry
308(3)
§11.2 Conformal differential geometry as a G-structure
311(8)
§11.3 Conformal Killing vector fields
319(3)
§11.4 Conformal densities and the Laplacian
322(4)
§11.5 Einstein manifolds in a conformal class and the tractor bundle
326(5)
Chapter 12 Projective Geometry II: Moving Frames and Subvarieties of Projective Space
331(50)
§12.1 The Fubini cubic and higher order differential invariants
332(4)
§12.2 Fundamental forms of Veronese, Grassmann, and Segre varieties
336(3)
§12.3 Ruled and uniruled varieties
339(2)
§12.4 Dual varieties
341(5)
§12.5 Secant and tangential varieties
346(4)
§12.6 Cominuscule varieties and their differential invariants
350(6)
§12.7 Higher-order Fubini forms
356(7)
§12.8 Varieties with vanishing Fubini cubic
363(2)
§12.9 Associated varieties
365(2)
§12.10 More on varieties with degenerate Gauss maps
367(2)
§12.11 Rank restriction theorems
369(3)
§12.12 Local study of smooth varieties with degenerate tangential varieties
372(4)
§12.13 Generalized Monge systems
376(2)
§12.14 Complete intersections
378(3)
Appendix A Linear Algebra and Representation Theory
381(24)
§A.1 Dual spaces and tensor products
381(5)
§A.2 Matrix Lie groups
386(2)
§A.3 Complex vector spaces and complex structures
388(2)
§A.4 Lie algebras
390(4)
§A.5 Division algebras and the simple group G%
394(3)
§A.6 A smidgen of representation theory
397(3)
§A.7 Clifford algebras and spin groups
400(5)
Appendix B Differential Forms
405(6)
§B.1 Differential forms and vector fields
405(2)
§B.2 Three definitions of the exterior derivative
407(2)
§B.3 Basic and semi-basic forms
409(2)
Appendix C Complex Structures and Complex Manifolds
411(8)
§C.1 Complex manifolds
411(4)
§C.2 The Cauchy-Riemann equations
415(4)
Appendix D Initial Value Problems and the Cauchy-Kowalevski Theorem
419(6)
§D.1 Initial value problems
419(1)
§D.2 The Cauchy-Kowalevski Theorem
420(3)
§D.3 Generalizations
423(2)
Hints and Answers to Selected Exercises 425(12)
Bibliography 437(8)
Index 445
Thomas A. Ivey, College of Charleston, SC.

Joseph M. Landsberg, Texas A&M University, College Station, TX.
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