Introduction to Möbius differential geometry:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York
Cambridge University Press
2003
|
| Edition: | 1. publ. |
| Series: | London Mathematical Society lecture note series
300 |
| Subjects: | |
| Online Access: | Publisher description Table of contents Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XI, 413 S. graph. Darst. |
| ISBN: | 0521535697 |
Staff View
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Record in the Search Index
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| adam_text | Contents
Introduction 1
1.1 Mobius geometry: models and applications 2
1.2 Philosophy and style 8
1.3 Acknowledgments 10
Preliminaries. The Riemannian point of view 12
P. 1 Conformal maps 14
P.2 Transformation formulas 16
P.3 The Weyl and Schouten tensors 17
P.4 Conformal flatness 18
P.5 The Weyl-Schouten theorem 20
P.6 Submanifolds 23
P.7 Spheres 28
P.8 Mobius transformations and Liouville s theorem 31
1 The projective model 33
1.1 Penta- and polyspherical coordinates 35
1.2 Sphere pencils and sphere complexes 40
1.3 The Mobius group 43
1.4 The metric subgeometries 47
1.5 Liouville s theorem revisited 52
1.6 Sphere congruences and their envelopes 53
1.7 Frame formulas and compatibility conditions 56
1.8 Channel hypersurfaces 62
2 Application: Conformally flat hypersurfaces 68
2.1 Cartan s theorem 70
2.2 Curved flats 73
2.3 The conformal fundamental forms 78
2.4 Guichard nets 85
2.5 Cyclic systems 93
2.6 Linear Weingarten surfaces in space forms 97
2.7 Bonnet s theorem 99
3 Application: Isothermic and Willmore surfaces 102
3.1 Blaschke s problem 106
3.2 Darboux pairs of isothermic surfaces 109
3.3 Aside: Transformations of isothermic surfaces Ill
3.4 Dual pairs of conformally minimal surfaces 120
3.5 Aside: Willmore surfaces 125
3.6 Thomsen s theorem 130
3.7 Isothermic and Willmore channel surfaces 132
4 A quaternionic model 146
vi Contents
4.1 Quaternionic linear algebra 148
4.2 The Study determinant 151
4.3 Quaternionic Hermitian forms 153
4.4 The Mobius group 157
4.5 The space of point pairs 160
4.6 The space of single points 164
4.7 Envelopes 165
4.8 Involutions and 2-spheres 169
4.9 The cross-ratio 175
5 Application: Smooth and discrete isothermic surfaces 184
5.1 Isothermic surfaces revisited 189
5.2 Christoffel s transformation 193
5.3 Goursat s transformation 205
5.4 Darboux s transformation 213
5.5 Calapso s transformation 227
5.6 Bianchi s permutability theorems 246
5.7 Discrete isothermic nets and their transformations 259
6 A Clifford algebra model 277
6.1 The Grassmann algebra 279
6.2 The Clifford algebra 280
6.3 The spin group 283
6.4 Spheres and the Clifford dual 288
6.5 The cross-ratio 292
6.6 Sphere congruences and their envelopes 294
6.7 Frames 296
7 A Clifford algebra model: Vahlen matrices 301
7.1 The quaternionic model revisited 303
7.2 Vahlen matrices and the spin group 308
7.3 Mobius transformations and spheres 315
7.4 Frames and structure equations 321
7.5 Miscellaneous 326
8 Applications: Orthogonal systems, isothermic surfaces 334
8.1 Dupin s theorem and Lame s equations 337
8.2 Ribaucour pairs of orthogonal systems 341
8.3 Ribaucour pairs of discrete orthogonal nets 345
8.4 Sphere constructions 351
8.5 Demoulin s family and Bianchi permutability 355
8.6 Isothermic surfaces revisited 361
8.7 Curved flats and the Darboux transformation 372
Further Reading 380
References 384
Index 408
|
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| isbn | 0521535697 |
| language | English |
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| spelling | Hertrich-Jeromin, Udo Verfasser aut Introduction to Möbius differential geometry Udo Hertrich-Jeromin 1. publ. New York Cambridge University Press 2003 XI, 413 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier London Mathematical Society lecture note series 300 Includes bibliographical references and index Differential geometry Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf Möbius-Geometrie (DE-588)4750877-2 gnd rswk-swf Differentialgeometrie (DE-588)4012248-7 s DE-604 Möbius-Geometrie (DE-588)4750877-2 s London Mathematical Society lecture note series 300 (DE-604)BV000000130 300 http://www.loc.gov/catdir/description/cam032/2002041686.html Publisher description http://www.loc.gov/catdir/toc/cam031/2002041686.html Table of contents HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010154485&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hertrich-Jeromin, Udo Introduction to Möbius differential geometry London Mathematical Society lecture note series Differential geometry Differentialgeometrie (DE-588)4012248-7 gnd Möbius-Geometrie (DE-588)4750877-2 gnd |
| subject_GND | (DE-588)4012248-7 (DE-588)4750877-2 |
| title | Introduction to Möbius differential geometry |
| title_auth | Introduction to Möbius differential geometry |
| title_exact_search | Introduction to Möbius differential geometry |
| title_full | Introduction to Möbius differential geometry Udo Hertrich-Jeromin |
| title_fullStr | Introduction to Möbius differential geometry Udo Hertrich-Jeromin |
| title_full_unstemmed | Introduction to Möbius differential geometry Udo Hertrich-Jeromin |
| title_short | Introduction to Möbius differential geometry |
| title_sort | introduction to mobius differential geometry |
| topic | Differential geometry Differentialgeometrie (DE-588)4012248-7 gnd Möbius-Geometrie (DE-588)4750877-2 gnd |
| topic_facet | Differential geometry Differentialgeometrie Möbius-Geometrie |
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