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Cartan for beginners: differential geometry via moving frames and exterior differential systems

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Bibliographische Detailangaben
Beteiligte Personen:	Ivey, Thomas A. 1963- (VerfasserIn), Landsberg, Joseph M. 1963- (VerfasserIn)
Format:	Buch
Sprache:	Englisch
Veröffentlicht:	Providence, Rhode Island American Mathematical Society [2016]
Ausgabe:	second edition
Schriftenreihe:	Graduate studies in mathematics 175
Schlagwörter:	Geometria diferencial Géométrie différentielle Sistemas diferenciais exteriores Systèmes différentiels extérieurs Exterior differential systems Geometry, Differential Äußeres Differentialsystem Differentialgeometrie
Links:	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029442848&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
Beschreibung:	Includes bibliographical references and index
Umfang:	xviii, 453 Seiten Illustrationen
ISBN:	9781470409869

Internformat

MARC


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Datensatz im Suchindex

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adam_text	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 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. 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 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 En+S §2.6. Intrinsic and extrinsic geometry xi xi xiv 1 2 5 8 13 16 18 22 25 28 31 39 40 43 46 47 48 52 v VI Contents §2.7. Curves on surfaces §2.8. The Gauss-Bonnet and Poincare-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 55 57 65 65 73 77 80 83 85 89 92 95 96 101 109 Chapter 5. Cartan-Kähler I: Linear Algebra and Constant-Coefficient Homogeneous Systems 115 §5.1. Tableaux 116 §5.2. First example 120 §5.3. Second example 122 §5.4. Third example 125 §5.5. The general case 126 §5.6. The characteristic variety of a tableau 129 Chapter 6. Cartan-Kähler II: The Cartan Algorithm for Linear Pfaffian Systems 135 §6.1. Linear Pfaffian systems 135 §6.2. First example 137 §6.3. Second example: constant coefficient homogeneous systems 138 §6.4. The local isometric embedding problem 141 §6.5. The Cartan algorithm formalized: tableau, torsion and prolongation 146 §6.6. Summary of Cartan’s algorithm for linear Pfaffian systems 149 §6.7. Additional remarks on the theory 151 Contents vu §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. Monge-Ampere systems and Weingarten surfaces §7.5. Integrable extensions and Bácklund transformations Chapter 8. Cartan-Káhler III: The General Case §8.1. Integral elements and polar spaces , §8.2. Example: triply orthogonal systems f liii §8.3. Statement and proof of Cartan-Káhler ; ... r,. f §8.4. Cartan’s Test - jy- §8.5. More examples of Cartan’s Test Ts { Chapter 9. Geometric Structures and Connections 7 §9.1. G-structures §9.2. Connections on Tq and differential invariants of G-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 G-structures §9.7. Holonomy §9.8. Extended example: path geometry Chapter §10.1. §10.2. §10.3. §10.4. §10.5. 10. Superposition for Darboux-Integrable Systems Decomposability Integrability Coframe adaptations Some results on group actions : The superposition formula 154 161 164 166 169 175 176 184 188 195 204 215 216 223 226 229 232 241 241 248 253 254 257 260 262 271 285 286 289 294 299 300 vin Contents Chapter 11. Conformal Differential Geometry 307 §11.1. Conformal geometry via Riemannian geometry 308 §11.2. Conformal differential geometry as a G-structure 311 §11.3. Conformal Killing vector fields 319 §11.4. Conformal densities and the Laplacian 322 §11.5. Einstein manifolds in a conformal class and the tractor bundle 326 Chapter 12. Projective Geometry II: Moving Frames and Subvarieties of Projective Space 331 §12.1. The Fubini cubic and higher order differential invariants 332 Fundamental forms of Veronese, Grassmann, and Segre varieties Ruled and uniruled varieties Dual varieties Secant and tangential varieties Cominuscule varieties and their differential invariants Higher-order Fubini forms Varieties with vanishing Fubini cubic 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.l. 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.l. Differential forms and vector fields §12.2. §12.3. §12.4. §12.5. §12.6. §12.7. §12.8. §12.9. 336 339 341 346 350 356 363 365 367 369 372 376 378 381 381 386 388 390 394 397 400 405 405 Contents IX §B.2. Three definitions of the exterior derivative §B.3. Basic and semi-basic forms Appendix C. Complex Structures and Complex Manifolds §C.l. Complex manifolds §C.2. The Cauchy-Riemann equations Appendix D. Initial Value Problems and the Cauchy-Kowalevski Theorem §D.l. Initial value problems §D.2. The Cauchy-Kowalevski Theorem §D.3. Generalizations Hints and Answers to Selected Exercises Bibliography Index 407 409 411 411 415 419 419 420 423 425 437 445
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spellingShingle	Ivey, Thomas A. 1963- Landsberg, Joseph M. 1963- Cartan for beginners differential geometry via moving frames and exterior differential systems Graduate studies in mathematics Geometria diferencial larpcal Géométrie différentielle Sistemas diferenciais exteriores larpcal Systèmes différentiels extérieurs Exterior differential systems Geometry, Differential Äußeres Differentialsystem (DE-588)4141545-0 gnd Differentialgeometrie (DE-588)4012248-7 gnd
subject_GND	(DE-588)4141545-0 (DE-588)4012248-7
title	Cartan for beginners differential geometry via moving frames and exterior differential systems
title_auth	Cartan for beginners differential geometry via moving frames and exterior differential systems
title_exact_search	Cartan for beginners differential geometry via moving frames and exterior differential systems
title_full	Cartan for beginners differential geometry via moving frames and exterior differential systems Thomas A. Ivey, Joseph M. Landsberg
title_fullStr	Cartan for beginners differential geometry via moving frames and exterior differential systems Thomas A. Ivey, Joseph M. Landsberg
title_full_unstemmed	Cartan for beginners differential geometry via moving frames and exterior differential systems Thomas A. Ivey, Joseph M. Landsberg
title_short	Cartan for beginners
title_sort	cartan for beginners differential geometry via moving frames and exterior differential systems
title_sub	differential geometry via moving frames and exterior differential systems
topic	Geometria diferencial larpcal Géométrie différentielle Sistemas diferenciais exteriores larpcal Systèmes différentiels extérieurs Exterior differential systems Geometry, Differential Äußeres Differentialsystem (DE-588)4141545-0 gnd Differentialgeometrie (DE-588)4012248-7 gnd
topic_facet	Geometria diferencial Géométrie différentielle Sistemas diferenciais exteriores Systèmes différentiels extérieurs Exterior differential systems Geometry, Differential Äußeres Differentialsystem Differentialgeometrie
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029442848&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
volume_link	(DE-604)BV009739289
work_keys_str_mv	AT iveythomasa cartanforbeginnersdifferentialgeometryviamovingframesandexteriordifferentialsystems AT landsbergjosephm cartanforbeginnersdifferentialgeometryviamovingframesandexteriordifferentialsystems

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