From Frenet to Cartan: the method of moving frames
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, Rhode Island
American Mathematical Society
[2017]
|
| Schriftenreihe: | Graduate studies in mathematics
178 |
| Schlagworte: |
Differential geometry
> Classical differential geometry
> Differential invariants (local theory), geometric objects
Global analysis, analysis on manifolds
> General theory of differentiable manifolds
> Differential forms
|
| Online-Zugang: | Inhaltsverzeichnis Inhaltsverzeichnis |
| Beschreibung: | xvi, 414 Seiten Diagramme |
| ISBN: | 9781470429522 |
Internformat
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| 245 | 1 | 0 | |a From Frenet to Cartan |b the method of moving frames |c Jeanne N. Clelland |
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| 650 | 4 | |a Geometry, Differential | |
| 650 | 4 | |a Mathematical physics | |
| 650 | 4 | |a Topological groups, Lie groups -- Noncompact transformation groups -- Homogeneous spaces | |
| 650 | 4 | |a Differential geometry -- Classical differential geometry -- Curves in Euclidean space | |
| 650 | 4 | |a Differential geometry -- Classical differential geometry -- Surfaces in Euclidean space | |
| 650 | 4 | |a Differential geometry -- Classical differential geometry -- Affine differential geometry | |
| 650 | 4 | |a Differential geometry -- Classical differential geometry -- Projective differential geometry | |
| 650 | 4 | |a Differential geometry -- Classical differential geometry -- Differential invariants (local theory), geometric objects | |
| 650 | 4 | |a Differential geometry -- Local differential geometry -- Local submanifolds | |
| 650 | 4 | |a Differential geometry -- Local differential geometry -- Lorentz metrics, indefinite metrics | |
| 650 | 4 | |a Global analysis, analysis on manifolds -- General theory of differentiable manifolds -- Differential forms | |
| 650 | 4 | |a Global analysis, analysis on manifolds -- General theory of differentiable manifolds -- Exterior differential systems (Cartan theory) | |
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Datensatz im Suchindex
| _version_ | 1804177541312806912 |
|---|---|
| adam_text | Titel: From Frenet to Cartan
Autor: Clelland, Jeanne N
Jahr: 2017
Contents Preface xi Acknowledgments xv Part 1. Background material Chapter 1. Assorted notions from differential geometry 3 §1.1. Manifolds 3 §1.2. Tensors, indices, and the Einstein summation convention 9 §1.3. Differentiable maps, tangent spaces, and vector fields 15 §1.4. Lie groups and matrix groups 26 §1.5. Vector bundles and principal bundles 32 Chapter 2. Differential forms 35 §2.1. Introduction 35 §2.2. Dual spaces, the cotangent bundle, and tensor products 35 §2.3. 1-forms on M n 40 §2.4. p-forms on M n 41 §2.5. The exterior derivative 43 §2.6. Closed and exact forms and the Poincare lemma 46 §2.7. Differential forms on manifolds 47 §2.8. Pullbacks 49 §2.9. Integration and Stokes’s theorem 53 §2.10. Cartan’s lemma 55 vii
Contents viii §2.11. The Lie derivative 56 §2.12. Introduction to the Cartan package for Maple 59 Part 2. Curves and surfaces in homogeneous spaces via the method of moving frames Chapter 3. Homogeneous spaces 69 §3.1. Introduction 69 §3.2. Euclidean space 70 §3.3. Orthonormal frames on Euclidean space 75 §3.4. Homogeneous spaces 84 §3.5. Minkowski space 85 §3.6. Equi-affine space 92 §3.7. Projective space 96 §3.8. Maple computations 103 Chapter 4. Curves and surfaces in Euclidean space 107 §4.1. Introduction 107 §4.2. Equivalence of submanifolds of a homogeneous space 108 §4.3. Moving frames for curves in E 3 111 §4.4. Compatibility conditions and existence of submanifolds with prescribed invariants §4.5. Moving frames for surfaces in E 3 §4.6. Maple computations Chapter 5. Curves and surfaces in Minkowski space §5.1. Introduction §5.2. Moving frames for timelike curves in M 1,2 §5.3. Moving frames for timelike surfaces in M 1 ’ 2 §5.4. An alternate construction for timelike surfaces §5.5. Maple computations Chapter 6. Curves and surfaces in equi-affine space §6.1. Introduction §6.2. Moving frames for curves in A 3 §6.3. Moving frames for surfaces in A 3 §6.4. Maple computations 115 117 134 143 143 144 149 161 166 171 171 172 178 191
Contents IX Chapter 7. Curves and surfaces in projective space 203 §7.1. Introduction 203 §7.2. Moving frames for curves in P 2 204 §7.3. Moving frames for curves in E 3 214 §7.4. Moving frames for surfaces in E 3 220 §7.5. Maple computations 235 Part 3. Applications of moving frames Chapter 8. Minimal surfaces in E 3 and A 3 251 §8.1. Introduction 251 §8.2. Minimal surfaces in E 3 251 §8.3. Minimal surfaces in A 3 268 §8.4. Maple computations 280 Chapter 9. Pseudospherical surfaces and Bâcklund’s theorem 287 §9.1. Introduction 287 §9.2. Line congruences 288 §9.3. Backlund’s theorem 289 §9.4. Pseudospherical surfaces and the sine-Gordon equation 293 §9.5. The Bàcklund transformation for the sine-Gordon equation 297 §9.6. Maple computations 303 Chapter 10. Two classical theorems 311 §10.1. Doubly ruled surfaces in 1R 3 311 §10.2. The Cauchy-Crofton formula 324 §10.3. Maple computations 329 Part 4. Beyond the flat case: Moving frames on Riemannian manifolds Chapter 11. Curves and surfaces in elliptic and hyperbolic spaces 339 §11.1. Introduction 339 §11.2. The homogeneous spaces § and HP 340 §11.3. A more intrinsic view of §” and HP 345 §11.4. Moving frames for curves in § 3 and HI 3 348 §11.5. Moving frames for surfaces in § 3 and H 3 351 §11.6. Maple computations 357
X Contents Chapter 12. The nonhomogeneous case: Moving frames on Riemannian manifolds 361 §12.1, Introduction 361 §12.2. Orthonormal frames and connections on Riemannian manifolds 362 §12.3. The Levi-Civita connection 370 §12.4. The structure equations 373 §12.5. Moving frames for curves in 3-dimensional Riemannian manifolds 379 §12.6. Moving frames for surfaces in 3-dimensional Riemannian manifolds 381 §12.7. Maple computations 388 Bibliography 397 Index 403
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| id | DE-604.BV044322149 |
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| indexdate | 2024-07-10T07:49:43Z |
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| isbn | 9781470429522 |
| language | English |
| lccn | 2016041073 |
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| physical | xvi, 414 Seiten Diagramme |
| publishDate | 2017 |
| publishDateSearch | 2017 |
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| series | Graduate studies in mathematics |
| series2 | Graduate studies in mathematics |
| spelling | Clelland, Jeanne N. 1970- Verfasser (DE-588)1131589696 aut From Frenet to Cartan the method of moving frames Jeanne N. Clelland Providence, Rhode Island American Mathematical Society [2017] © 2017 xvi, 414 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics 178 Mathematische Physik Frames (Vector analysis) Vector analysis Exterior differential systems Geometry, Differential Mathematical physics Topological groups, Lie groups -- Noncompact transformation groups -- Homogeneous spaces Differential geometry -- Classical differential geometry -- Curves in Euclidean space Differential geometry -- Classical differential geometry -- Surfaces in Euclidean space Differential geometry -- Classical differential geometry -- Affine differential geometry Differential geometry -- Classical differential geometry -- Projective differential geometry Differential geometry -- Classical differential geometry -- Differential invariants (local theory), geometric objects Differential geometry -- Local differential geometry -- Local submanifolds Differential geometry -- Local differential geometry -- Lorentz metrics, indefinite metrics Global analysis, analysis on manifolds -- General theory of differentiable manifolds -- Differential forms Global analysis, analysis on manifolds -- General theory of differentiable manifolds -- Exterior differential systems (Cartan theory) Frame Mathematik (DE-588)4528312-6 gnd rswk-swf Frame Mathematik (DE-588)4528312-6 s DE-604 Graduate studies in mathematics 178 (DE-604)BV009739289 178 http://digitool.hbz-nrw.de:1801/webclient/DeliveryManager?pid=7213145&custom_att_2=simple_viewer From Frenet to Cartan Inhaltsverzeichnis HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029725592&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Clelland, Jeanne N. 1970- From Frenet to Cartan the method of moving frames Graduate studies in mathematics Mathematische Physik Frames (Vector analysis) Vector analysis Exterior differential systems Geometry, Differential Mathematical physics Topological groups, Lie groups -- Noncompact transformation groups -- Homogeneous spaces Differential geometry -- Classical differential geometry -- Curves in Euclidean space Differential geometry -- Classical differential geometry -- Surfaces in Euclidean space Differential geometry -- Classical differential geometry -- Affine differential geometry Differential geometry -- Classical differential geometry -- Projective differential geometry Differential geometry -- Classical differential geometry -- Differential invariants (local theory), geometric objects Differential geometry -- Local differential geometry -- Local submanifolds Differential geometry -- Local differential geometry -- Lorentz metrics, indefinite metrics Global analysis, analysis on manifolds -- General theory of differentiable manifolds -- Differential forms Global analysis, analysis on manifolds -- General theory of differentiable manifolds -- Exterior differential systems (Cartan theory) Frame Mathematik (DE-588)4528312-6 gnd |
| subject_GND | (DE-588)4528312-6 |
| title | From Frenet to Cartan the method of moving frames |
| title_auth | From Frenet to Cartan the method of moving frames |
| title_exact_search | From Frenet to Cartan the method of moving frames |
| title_full | From Frenet to Cartan the method of moving frames Jeanne N. Clelland |
| title_fullStr | From Frenet to Cartan the method of moving frames Jeanne N. Clelland |
| title_full_unstemmed | From Frenet to Cartan the method of moving frames Jeanne N. Clelland |
| title_short | From Frenet to Cartan |
| title_sort | from frenet to cartan the method of moving frames |
| title_sub | the method of moving frames |
| topic | Mathematische Physik Frames (Vector analysis) Vector analysis Exterior differential systems Geometry, Differential Mathematical physics Topological groups, Lie groups -- Noncompact transformation groups -- Homogeneous spaces Differential geometry -- Classical differential geometry -- Curves in Euclidean space Differential geometry -- Classical differential geometry -- Surfaces in Euclidean space Differential geometry -- Classical differential geometry -- Affine differential geometry Differential geometry -- Classical differential geometry -- Projective differential geometry Differential geometry -- Classical differential geometry -- Differential invariants (local theory), geometric objects Differential geometry -- Local differential geometry -- Local submanifolds Differential geometry -- Local differential geometry -- Lorentz metrics, indefinite metrics Global analysis, analysis on manifolds -- General theory of differentiable manifolds -- Differential forms Global analysis, analysis on manifolds -- General theory of differentiable manifolds -- Exterior differential systems (Cartan theory) Frame Mathematik (DE-588)4528312-6 gnd |
| topic_facet | Mathematische Physik Frames (Vector analysis) Vector analysis Exterior differential systems Geometry, Differential Mathematical physics Topological groups, Lie groups -- Noncompact transformation groups -- Homogeneous spaces Differential geometry -- Classical differential geometry -- Curves in Euclidean space Differential geometry -- Classical differential geometry -- Surfaces in Euclidean space Differential geometry -- Classical differential geometry -- Affine differential geometry Differential geometry -- Classical differential geometry -- Projective differential geometry Differential geometry -- Classical differential geometry -- Differential invariants (local theory), geometric objects Differential geometry -- Local differential geometry -- Local submanifolds Differential geometry -- Local differential geometry -- Lorentz metrics, indefinite metrics Global analysis, analysis on manifolds -- General theory of differentiable manifolds -- Differential forms Global analysis, analysis on manifolds -- General theory of differentiable manifolds -- Exterior differential systems (Cartan theory) Frame Mathematik |
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