Cartan for beginners: differential geometry via moving frames and exterior differential systems
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, RI
American Mathematical Society
2003
|
| Series: | Graduate studies in mathematics
61 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 378 S. Ill. |
| ISBN: | 9780821833759 0821833758 |
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MARC
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| 100 | 1 | |a Ivey, Thomas A. |d 1963- |e Verfasser |0 (DE-588)141806532 |4 aut | |
| 245 | 1 | 0 | |a Cartan for beginners |b differential geometry via moving frames and exterior differential systems |c Thomas A. Ivey ; J. M. Landsberg |
| 264 | 1 | |a Providence, RI |b American Mathematical Society |c 2003 | |
| 300 | |a XIV, 378 S. |b Ill. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Graduate studies in mathematics |v 61 | |
| 650 | 7 | |a Geometria diferencial |2 larpcal | |
| 650 | 4 | |a Géométrie différentielle | |
| 650 | 7 | |a Sistemas diferenciais exteriores |2 larpcal | |
| 650 | 4 | |a Systèmes différentiels extérieurs | |
| 650 | 4 | |a Exterior differential systems | |
| 650 | 4 | |a Geometry, Differential | |
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Record in the Search Index
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|---|---|
| adam_text | Contents
Preface ix
Chapter 1. Moving Frames and Exterior Differential Systems 1
§1.1. Geometry of surfaces in E3 in coordinates 2
§1.2. Differential equations in coordinates 5
§1.3. Introduction to differential equations without coordinates 8
§1.4. Introduction to geometry without coordinates:
curves in E2 12
§1.5. Submanifolds of homogeneous spaces 15
§1.6. The Maurer Cartan form 16
§1.7. Plane curves in other geometries 20
§1.8. Curves in E3 23
§1.9. Exterior differential systems and jet spaces 26
Chapter 2. Euclidean Geometry and Riemannian Geometry 35
§2.1. Gauss and mean curvature via frames 36
§2.2. Calculation of H and K for some examples 39
§2.3. Darboux frames and applications 42
§2.4. What do H and K tell us? 43
§2.5. Invariants for n dimensional submanifolds of En+S 45
§2.6. Intrinsic and extrinsic geometry 47
§2.7. Space forms: the sphere and hyperbolic space 57
§2.8. Curves on surfaces 58
§2.9. The Gauss Bonnet and Poincare Hopf theorems 61
v
vi Contents
§2.10. Non orthonormal frames 66
Chapter 3. Projective Geometry 71
§3.1. Grassmannians 72
§3.2. Frames and the projective second fundamental form 76
§3.3. Algebraic varieties 81
§3.4. Varieties with degenerate Gauss mappings 89
§3.5. Higher order differential invariants 94
§3.6. Fundamental forms of some homogeneous varieties 98
§3.7. Higher order Fubini forms 107
§3.8. Ruled and uniruled varieties 113
§3.9. Varieties with vanishing Fubini cubic 115
§3.10. Dual varieties 118
§3.11. Associated varieties 123
§3.12. More on varieties with degenerate Gauss maps 125
§3.13. Secant and tangential varieties 128
§3.14. Rank restriction theorems 132
§3.15. Local study of smooth varieties with degenerate tangential
varieties 134
§3.16. Generalized Monge systems 137
§3.17. Complete intersections 139
Chapter 4. Cartan Kahler I: Linear Algebra and Constant Coefficient
Homogeneous Systems 143
§4.1. Tableaux 144
§4.2. First example 148
§4.3. Second example 150
§4.4. Third example 153
§4.5. The general case 154
§4.6. The characteristic variety of a tableau 157
Chapter 5. Cartan Kahler II: The Cartan Algorithm for Linear
Pfaffian Systems 163
§5.1. Linear Pfaffian systems 163
§5.2. First example 165
§5.3. Second example: constant coefficient homogeneous systems 166
§5.4. The local isometric embedding problem 169
Contents vii
§5.5. The Cartan algorithm formalized:
tableau, torsion and prolongation 173
§5.6. Summary of Cartan s algorithm for linear PfafRan systems 177
§5.7. Additional remarks on the theory 179
§5.8. Examples 182
§5.9. Functions whose Hessians commute, with remarks on singular
solutions 189
§5.10. The Cartan Janet Isometric Embedding Theorem 191
§5.11. Isometric embeddings of space forms (mostly flat ones) 194
§5.12. Calibrated submanifolds 197
Chapter 6. Applications to PDE 203
§6.1. Symmetries and Cauchy characteristics 204
§6.2. Second order PDE and Monge characteristics 212
§6.3. Derived systems and the method of Darboux 215
§6.4. Monge Ampere systems and Weingarten surfaces 222
§6.5. Integrable extensions and Backlund transformations 231
Chapter 7. Cartan Kahler III: The General Case 243
§7.1. Integral elements and polar spaces 244
§7.2. Example: Triply orthogonal systems 251
§7.3. Statement and proof of Cartan Kahler 254
§7.4. Cartan s Test 256
§7.5. More examples of Cartan s Test 259
Chapter 8. Geometric Structures and Connections 267
§8.1. G structures 267
§8.2. How to differentiate sections of vector bundles 275
§8.3. Connections on Tq and differential invariants of G structures 278
§8.4. Induced vector bundles and connections on induced bundles 283
§8.5. Holonomy 286
§8.6. Extended example: Path geometry 295
§8.7. Frobenius and generalized conformal structures 308
Appendix A. Linear Algebra and Representation Theory 311
§A.l. Dual spaces and tensor products 311
§A.2. Matrix Lie groups 316
§A.3. Complex vector spaces and complex structures 318
viii Contents
§A.4. Lie algebras 320
§A.5. Division algebras and the simple group Gi 323
§A.6. A smidgen of representation theory 326
§A.7. Clifford algebras and spin groups 330
Appendix B. Differential Forms 335
§B.l. Differential forms and vector fields 335
§B.2. Three definitions of the exterior derivative 337
§B.3. Basic and semi basic forms 339
§B.4. Differential ideals 340
Appendix C. Complex Structures and Complex Manifolds 343
§C.l. Complex manifolds 343
§C2. The Cauchy Riemann equations 347
Appendix D. Initial Value Problems 349
Hints and Answers to Selected Exercises 355
Bibliography 363
Index 371
|
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| dewey-raw | 516.3/6 |
| dewey-search | 516.3/6 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| spelling | Ivey, Thomas A. 1963- Verfasser (DE-588)141806532 aut Cartan for beginners differential geometry via moving frames and exterior differential systems Thomas A. Ivey ; J. M. Landsberg Providence, RI American Mathematical Society 2003 XIV, 378 S. Ill. txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics 61 Geometria diferencial larpcal Géométrie différentielle Sistemas diferenciais exteriores larpcal Systèmes différentiels extérieurs Exterior differential systems Geometry, Differential Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf Äußeres Differentialsystem (DE-588)4141545-0 gnd rswk-swf Differentialgeometrie (DE-588)4012248-7 s Äußeres Differentialsystem (DE-588)4141545-0 s DE-604 Landsberg, Joseph M. 1963- Verfasser (DE-588)14180677X aut Erscheint auch als Online-Ausgabe 978-1-4704-1801-4 Graduate studies in mathematics 61 (DE-604)BV009739289 61 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012817608&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| 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 Differentialgeometrie (DE-588)4012248-7 gnd Äußeres Differentialsystem (DE-588)4141545-0 gnd |
| subject_GND | (DE-588)4012248-7 (DE-588)4141545-0 |
| 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 ; J. M. Landsberg |
| title_fullStr | Cartan for beginners differential geometry via moving frames and exterior differential systems Thomas A. Ivey ; J. M. Landsberg |
| title_full_unstemmed | Cartan for beginners differential geometry via moving frames and exterior differential systems Thomas A. Ivey ; J. 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 Differentialgeometrie (DE-588)4012248-7 gnd Äußeres Differentialsystem (DE-588)4141545-0 gnd |
| topic_facet | Geometria diferencial Géométrie différentielle Sistemas diferenciais exteriores Systèmes différentiels extérieurs Exterior differential systems Geometry, Differential Differentialgeometrie Äußeres Differentialsystem |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012817608&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV009739289 |
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