Global affine differential geometry of hypersurfaces:
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
de Gruyter
1993
|
| Schriftenreihe: | De Gruyter expositions in mathematics
11 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIII, 328 S. |
| ISBN: | 3110127695 |
Internformat
MARC
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| 100 | 1 | |a Li, An-Min |d 1946- |e Verfasser |0 (DE-588)113507054 |4 aut | |
| 245 | 1 | 0 | |a Global affine differential geometry of hypersurfaces |c by An-Min Li ; Udo Simon ; Guosong Zhao |
| 264 | 1 | |a Berlin [u.a.] |b de Gruyter |c 1993 | |
| 300 | |a XIII, 328 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a De Gruyter expositions in mathematics |v 11 | |
| 650 | 7 | |a Géométrie différentielle globale |2 ram | |
| 650 | 7 | |a Hypersphère |2 Jussieu | |
| 650 | 7 | |a Hypersurfaces |2 ram | |
| 650 | 7 | |a Inégalité géométrique |2 Jussieu | |
| 650 | 7 | |a Problème variationnel |2 Jussieu | |
| 650 | 7 | |a Rigidité |2 Jussieu | |
| 650 | 4 | |a Global differential geometry | |
| 650 | 4 | |a Hypersurfaces | |
| 650 | 0 | 7 | |a Hyperfläche |0 (DE-588)4161054-4 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Globale Differentialgeometrie |0 (DE-588)4021286-5 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Affine Differentialgeometrie |0 (DE-588)4141563-2 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Globale Differentialgeometrie |0 (DE-588)4021286-5 |D s |
| 689 | 0 | 1 | |a Hyperfläche |0 (DE-588)4161054-4 |D s |
| 689 | 0 | |5 DE-604 | |
| 689 | 1 | 0 | |a Hyperfläche |0 (DE-588)4161054-4 |D s |
| 689 | 1 | 1 | |a Affine Differentialgeometrie |0 (DE-588)4141563-2 |D s |
| 689 | 1 | 2 | |a Globale Differentialgeometrie |0 (DE-588)4021286-5 |D s |
| 689 | 1 | |5 DE-604 | |
| 689 | 2 | 0 | |a Globale Differentialgeometrie |0 (DE-588)4021286-5 |D s |
| 689 | 2 | |5 DE-604 | |
| 700 | 1 | |a Simon, Udo |d 1938- |e Verfasser |0 (DE-588)172375525 |4 aut | |
| 700 | 1 | |a Zhao, Guosong |e Verfasser |4 aut | |
| 830 | 0 | |a De Gruyter expositions in mathematics |v 11 |w (DE-604)BV004069300 |9 11 | |
| 856 | 4 | 2 | |m HBZ Datenaustausch |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005393842&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-005393842 | ||
Datensatz im Suchindex
| _version_ | 1804122558117707776 |
|---|---|
| adam_text | Table of Contents
Introduction ix
Chapter 0
Preliminaries and Basic Structural Aspects
§0.1 Affine Spaces 1
§0.2 Euclidean Spaces 10
§ 0.3 Differential Geometric Structures of Affine and
Euclidean Spaces 12
§0.4 Klein s Erlanger Programme 14
§0.5 Motivation. A Short Sketch of the Euclidean Hypersurface
Theory 16
§ 0.6 Hypersurfaces in the Equiaffine Space 20
§0.7 Structural Motivation for Further Investigations 20
§0.8 Transversal Fields and Induced Structures 21
§ 0.9 Conormal Fields and Induced Structures 24
§0.10 Normalizations 25
§0.11 Non Degenerate Hypersurfaces 26
§0.12 Relative Normalizations 27
§0.13 Gauss Structure Equations for Conormal Fields 27
§0.14 Affine Invariance of the Induced Structures 29
§ 0.15 Comparison of Relative Normalizations 31
§0.16 Example. The Euclidean Normalization as
Relative Normalization 33
§0.17 The Equiaffine Normalization 33
§0.18 Equiaffine Structure Equations 36
§0.19 The Centroaffine Normalization 36
Chapter 1
Local Equiaffine Hypersurface Theory
§1.1 Berwald Blaschke Metric and Structure Equations 40
§ 1.2 The Affine Normal and the Fubini Pick Form 43
1.2.1 The Affine Normal 43
1.2.2 The Fubini Pick Form 48
1.2.3 Affine Curvatures 50
1.2.4 Geometric Meaning of the Affine Normal 52
vi Table of Contents
§ 1.3 The Equiaffine Conormal 56
1.3.1 Properties of the Equiaffine Conormal 56
1.3.2 The Affine Support Function 59
§1.4 Hyperquadrics 60
1.4.1 Hyperquadrics 60
1.4.2 Hypersurfaces with J = 0 66
§ 1.5 Integrability Conditions and
the Local Fundamental Theorem 71
1.5.1 Relations between the Coefficients 72
1.5.2 The Integrability Conditions 72
1.5.3 The Fundamental Theorem 76
Chapter 2
Affine Hyperspheres
§2.1 Definitions and Basic Results for Affine Hyperspheres 85
2.1.1 Definition of Affine Hyperspheres 85
2.1.2 Differential Equations for Affine Hyperspheres 87
2.1.3 A Composition Formula 93
§ 2.2 Affine Hyperspheres with Constant Sectional Curvature 95
2.2.1 Examples 95
2.2.2 Local Classification of Two dimensional Affine Spheres
with Constant Scalar Curvature 99
2.2.3 Generalization to Higher Dimensions 102
§2.3 Affine Completeness and Euclidean Completeness 110
§2.4 Affine Complete Elliptic Affine Hyperspheres 118
§ 2.5 A Differential Inequality on
a Complete Riemannian Manifold 121
§ 2.6 Estimates of the Ricci Curvatures of Affine Complete Affine
Hyperspheres of Parabolic or Hyperbolic Type 126
§ 2.7 Classification of Complete Hyperbolic Affine Hyperspheres 130
2.7.1 Euclidean Complete Affine Hyperspheres of
Hyperbolic Type 130
2.7.2 Affine Complete Affine Hyperspheres of
Hyperbolic Type 136
2.7.3 Proof of the Second Part of the Calabi Conjecture 144
§2.8 Complete Hyperbolic Affine 2 Spheres 151
§2.9 Appendix: Recent Results about Affine Spheres 161
Chapter 3
Rigidity and Uniqueness Theorems
§ 3.1 Integral Formulas for Affine Hypersurfaces and
Their Applications 163
Table of Contents vii
3.1.1 Minkowski s Integral Formulas for
Affine Hypersurfaces 164
3.1.2 Characterization of Ellipsoids 165
3.1.3 Some Further Characterizations of Ellipsoids 169
3.1.4 Global Solutions of a Differential Equation of
Schrodinger Type 174
3.1.5 Rigidity Theorems for Ovaloids 176
3.1.6 Some Results for Hypersurfaces with Boundary 179
§3.2 The Index Method 188
3.2.1 Fields of Line Elements and Nets 188
3.2.2 Vekua s System of Partial Differential Equations 193
3.2.3 Affine Weingarten Surfaces 195
3.2.4 An Affine Analogue of the Cohn Vossen Theorem 204
Chapter 4
Variational Problems and Affine Maximal Surfaces
§4.1 Variational Formulas for Higher Affine Mean Curvatures 208
§4.2 Affine Maximal Surfaces 215
4.2.1 Definitions and Fundamental Results 215
4.2.2 An Affine Analogue of the Weierstrass Representation 220
4.2.3 Computation of AJ 226
4.2.4 The Gauss Map 232
Chapter 5
Geometric Inequalities
§ 5.1 The Affine Isoperimetric Inequality 237
5.1.1 Steiner Symmetrization 238
5.1.2 A Characterization of Ellipsoids 241
5.1.3 The Affine Isoperimetric Inequality 243
§ 5.2 Inequalities for Higher Affine Mean Curvatures 245
5.2.1 Mixed Volumes 245
5.2.2 Integral Inequalities for Curvature Functions 247
5.2.3 Total Centroaffme Area 251
Appendix 1
Basic Concepts from Differential Geometry
§ Al.l Tensors and Exterior Algebra 253
A 1.1.1 Tensors 253
A 1.1.2 Exterior Algebra 256
viii Table of Contents
§A1.2 Differentiable Manifolds 260
Al.2.1 Differentiable Manifolds and Submanifolds 260
A 1.2.2 Tensor Fields on Manifolds 264
Al.2.3 Integration on Manifolds 267
§A1.3 Affine Connections and Riemannian Geometry.
Basic Facts 269
A 1.3.1 Affine Connections 269
A 1.3.2 Riemannian Manifolds 274
A 1.3.3 Manifolds of Constant Curvature.
Einstein Manifolds 278
A 1.3.4 Exponential Mapping and Completeness 280
§A1.4 Green s Formula 283
Appendix 2
Laplacian Comparison Theorem 285
Bibliography 291
List of Symbols 325
Index 327
|
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| author | Li, An-Min 1946- Simon, Udo 1938- Zhao, Guosong |
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| indexdate | 2024-07-09T17:15:47Z |
| institution | BVB |
| isbn | 3110127695 |
| language | English |
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| physical | XIII, 328 S. |
| publishDate | 1993 |
| publishDateSearch | 1993 |
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| publisher | de Gruyter |
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| series | De Gruyter expositions in mathematics |
| series2 | De Gruyter expositions in mathematics |
| spelling | Li, An-Min 1946- Verfasser (DE-588)113507054 aut Global affine differential geometry of hypersurfaces by An-Min Li ; Udo Simon ; Guosong Zhao Berlin [u.a.] de Gruyter 1993 XIII, 328 S. txt rdacontent n rdamedia nc rdacarrier De Gruyter expositions in mathematics 11 Géométrie différentielle globale ram Hypersphère Jussieu Hypersurfaces ram Inégalité géométrique Jussieu Problème variationnel Jussieu Rigidité Jussieu Global differential geometry Hypersurfaces Hyperfläche (DE-588)4161054-4 gnd rswk-swf Globale Differentialgeometrie (DE-588)4021286-5 gnd rswk-swf Affine Differentialgeometrie (DE-588)4141563-2 gnd rswk-swf Globale Differentialgeometrie (DE-588)4021286-5 s Hyperfläche (DE-588)4161054-4 s DE-604 Affine Differentialgeometrie (DE-588)4141563-2 s Simon, Udo 1938- Verfasser (DE-588)172375525 aut Zhao, Guosong Verfasser aut De Gruyter expositions in mathematics 11 (DE-604)BV004069300 11 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005393842&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Li, An-Min 1946- Simon, Udo 1938- Zhao, Guosong Global affine differential geometry of hypersurfaces De Gruyter expositions in mathematics Géométrie différentielle globale ram Hypersphère Jussieu Hypersurfaces ram Inégalité géométrique Jussieu Problème variationnel Jussieu Rigidité Jussieu Global differential geometry Hypersurfaces Hyperfläche (DE-588)4161054-4 gnd Globale Differentialgeometrie (DE-588)4021286-5 gnd Affine Differentialgeometrie (DE-588)4141563-2 gnd |
| subject_GND | (DE-588)4161054-4 (DE-588)4021286-5 (DE-588)4141563-2 |
| title | Global affine differential geometry of hypersurfaces |
| title_auth | Global affine differential geometry of hypersurfaces |
| title_exact_search | Global affine differential geometry of hypersurfaces |
| title_full | Global affine differential geometry of hypersurfaces by An-Min Li ; Udo Simon ; Guosong Zhao |
| title_fullStr | Global affine differential geometry of hypersurfaces by An-Min Li ; Udo Simon ; Guosong Zhao |
| title_full_unstemmed | Global affine differential geometry of hypersurfaces by An-Min Li ; Udo Simon ; Guosong Zhao |
| title_short | Global affine differential geometry of hypersurfaces |
| title_sort | global affine differential geometry of hypersurfaces |
| topic | Géométrie différentielle globale ram Hypersphère Jussieu Hypersurfaces ram Inégalité géométrique Jussieu Problème variationnel Jussieu Rigidité Jussieu Global differential geometry Hypersurfaces Hyperfläche (DE-588)4161054-4 gnd Globale Differentialgeometrie (DE-588)4021286-5 gnd Affine Differentialgeometrie (DE-588)4141563-2 gnd |
| topic_facet | Géométrie différentielle globale Hypersphère Hypersurfaces Inégalité géométrique Problème variationnel Rigidité Global differential geometry Hyperfläche Globale Differentialgeometrie Affine Differentialgeometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005393842&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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