Isometric embedding of Riemannian manifolds in Euclidean spaces:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
[2006]
|
| Series: | Mathematical surveys and monographs
Volume 130 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xiv, 260 Seiten Diagramme |
| ISBN: | 9780821840719 0821840711 |
Staff View
MARC
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| 245 | 1 | 0 | |a Isometric embedding of Riemannian manifolds in Euclidean spaces |c Qing Han ; Jia-Xing Hong |
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| 490 | 1 | |a Mathematical surveys and monographs |v Volume 130 | |
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| 650 | 4 | |a Isométrie (Mathématiques) | |
| 650 | 4 | |a Riemann, Variétés de | |
| 650 | 4 | |a Algebraic spaces | |
| 650 | 4 | |a Isometrics (Mathematics) | |
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Record in the Search Index
| _version_ | 1804137337961054208 |
|---|---|
| adam_text | Contents
Preface
ix
A Brief History
xi
Part
1.
Isometric Embedding of Riemannian Manifolds
1
Chapter
1.
Fundamental Theorems
3
1.1.
Local Isometric Embedding of Analytic Metrics
4
1.2.
Local Isometric Embedding of Smooth Metrics
9
1.3.
Global Isometric Embedding of Smooth Metrics
16
Notes
31
Chapter
2.
Surfaces in Low Dimensional Euclidean Spaces
33
2.1.
Surfaces in M3
33
2.2.
Isometric Immersions of Surfaces of Constant Curvature
38
2.3.
Isometric Immersions of Surfaces in
Ш4
40
Notes
42
Part
2.
Local Isometric Embedding of Surfaces in R3
43
Chapter
3.
Basic Equations
45
3.1.
The Darboux Equation
45
3.2.
The
Gauss-Codazzi
System
47
3.3.
The
Rozhdestvenskiï-Poznyak
System
51
Notes
53
Chapter
4.
Nonzero Gauss Curvature
55
4.1.
Positive Gauss Curvature
56
4.2.
Negative Gauss Curvature
58
4.3.
Appendix: Sobolev Spaces
63
4.4.
Appendix: Some Lemmas
64
4.5.
Appendix: Symmetric Hyperbolic Linear Differential Systems
68
Notes
70
Chapter
5.
Gauss Curvature Changing Sign Cleanly
71
5.1.
The Setting
71
5.2.
Iterations
74
5.3.
Symmetric Positive Linear Differential Systems
79
Notes
86
vii
viii CONTENTS
Chapter
6.
Nonnegative
Gauss Curvature
87
6.1.
The Setting
87
6.2.
A Priori Estimates
90
6.3.
Iterations
96
Notes
107
Chapter
7.
Nonpositive
Gauss Curvature
109
7.1.
The Setting HO
7.2.
A Priori Estimates for the Linearized System
116
7.3.
A Priori Estimates for the Cauchy Problem
124
7.4.
Iterations
130
7.5.
Remarks on the Hypothesis
136
Notes
141
Part
3.
Global Isometric Embedding of Surfaces in M3
143
Chapter
8.
Deformation of Surfaces
145
8.1.
The Rigidity of Surfaces
145
8.2.
The Infinitesimal Rigidity of Surfaces
154
8.3.
A Positive Disk of Not Infinitesimal Rigidity
158
Notes
165
Chapter
9.
The Weyl Problem
167
9.1.
Geometric Preliminaries
168
9.2.
Openness
171
9.3.
Closedness
180
Notes
188
Chapter
10.
Complete Negatively Curved Surfaces
191
10.1.
Nonexistence of Isometric Immersions
192
10.2.
Existence of Isometric Immersions, Part I
201
10.3.
Existence of Isometric Immersions, Part II
214
Notes
224
Chapter
11.
Boundary Value Problems
225
11.1.
The Dirichlet Problem with Planar Boundaries
226
11.2.
The Darboux System and Interior C2-Estimates
229
11.3.
Global C2-Estimates
237
Notes
247
Bibliography
249
Index
259
|
| adam_txt |
Contents
Preface
ix
A Brief History
xi
Part
1.
Isometric Embedding of Riemannian Manifolds
1
Chapter
1.
Fundamental Theorems
3
1.1.
Local Isometric Embedding of Analytic Metrics
4
1.2.
Local Isometric Embedding of Smooth Metrics
9
1.3.
Global Isometric Embedding of Smooth Metrics
16
Notes
31
Chapter
2.
Surfaces in Low Dimensional Euclidean Spaces
33
2.1.
Surfaces in M3
33
2.2.
Isometric Immersions of Surfaces of Constant Curvature
38
2.3.
Isometric Immersions of Surfaces in
Ш4
40
Notes
42
Part
2.
Local Isometric Embedding of Surfaces in R3
43
Chapter
3.
Basic Equations
45
3.1.
The Darboux Equation
45
3.2.
The
Gauss-Codazzi
System
47
3.3.
The
Rozhdestvenskiï-Poznyak
System
51
Notes
53
Chapter
4.
Nonzero Gauss Curvature
55
4.1.
Positive Gauss Curvature
56
4.2.
Negative Gauss Curvature
58
4.3.
Appendix: Sobolev Spaces
63
4.4.
Appendix: Some Lemmas
64
4.5.
Appendix: Symmetric Hyperbolic Linear Differential Systems
68
Notes
70
Chapter
5.
Gauss Curvature Changing Sign Cleanly
71
5.1.
The Setting
71
5.2.
Iterations
74
5.3.
Symmetric Positive Linear Differential Systems
79
Notes
86
vii
viii CONTENTS
Chapter
6.
Nonnegative
Gauss Curvature
87
6.1.
The Setting
87
6.2.
A Priori Estimates
90
6.3.
Iterations
96
Notes
107
Chapter
7.
Nonpositive
Gauss Curvature
109
7.1.
The Setting HO
7.2.
A Priori Estimates for the Linearized System
116
7.3.
A Priori Estimates for the Cauchy Problem
124
7.4.
Iterations
130
7.5.
Remarks on the Hypothesis
136
Notes
141
Part
3.
Global Isometric Embedding of Surfaces in M3
143
Chapter
8.
Deformation of Surfaces
145
8.1.
The Rigidity of Surfaces
145
8.2.
The Infinitesimal Rigidity of Surfaces
154
8.3.
A Positive Disk of Not Infinitesimal Rigidity
158
Notes
165
Chapter
9.
The Weyl Problem
167
9.1.
Geometric Preliminaries
168
9.2.
Openness
171
9.3.
Closedness
180
Notes
188
Chapter
10.
Complete Negatively Curved Surfaces
191
10.1.
Nonexistence of Isometric Immersions
192
10.2.
Existence of Isometric Immersions, Part I
201
10.3.
Existence of Isometric Immersions, Part II
214
Notes
224
Chapter
11.
Boundary Value Problems
225
11.1.
The Dirichlet Problem with Planar Boundaries
226
11.2.
The Darboux System and Interior C2-Estimates
229
11.3.
Global C2-Estimates
237
Notes
247
Bibliography
249
Index
259 |
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| id | DE-604.BV023087195 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T19:39:39Z |
| indexdate | 2024-07-09T21:10:42Z |
| institution | BVB |
| isbn | 9780821840719 0821840711 |
| language | English |
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| spelling | Han, Qing 1964- Verfasser (DE-588)1102177458 aut Isometric embedding of Riemannian manifolds in Euclidean spaces Qing Han ; Jia-Xing Hong Providence, Rhode Island American Mathematical Society [2006] © 2006 xiv, 260 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Mathematical surveys and monographs Volume 130 Espaces algébriques Isométrie (Mathématiques) Riemann, Variétés de Algebraic spaces Isometrics (Mathematics) Riemannian manifolds Euklidischer Raum (DE-588)4309127-1 gnd rswk-swf Isometrische Einbettung (DE-588)4162532-8 gnd rswk-swf Isometrische Einbettung (DE-588)4162532-8 s Euklidischer Raum (DE-588)4309127-1 s DE-604 Hong, Jiaxing 1942- Verfasser (DE-588)140461035 aut Erscheint auch als Online-Ausgabe 978-1-4704-1357-6 Mathematical surveys and monographs Volume 130 (DE-604)BV000018014 130 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016290117&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Han, Qing 1964- Hong, Jiaxing 1942- Isometric embedding of Riemannian manifolds in Euclidean spaces Mathematical surveys and monographs Espaces algébriques Isométrie (Mathématiques) Riemann, Variétés de Algebraic spaces Isometrics (Mathematics) Riemannian manifolds Euklidischer Raum (DE-588)4309127-1 gnd Isometrische Einbettung (DE-588)4162532-8 gnd |
| subject_GND | (DE-588)4309127-1 (DE-588)4162532-8 |
| title | Isometric embedding of Riemannian manifolds in Euclidean spaces |
| title_auth | Isometric embedding of Riemannian manifolds in Euclidean spaces |
| title_exact_search | Isometric embedding of Riemannian manifolds in Euclidean spaces |
| title_exact_search_txtP | Isometric embedding of Riemannian manifolds in Euclidean spaces |
| title_full | Isometric embedding of Riemannian manifolds in Euclidean spaces Qing Han ; Jia-Xing Hong |
| title_fullStr | Isometric embedding of Riemannian manifolds in Euclidean spaces Qing Han ; Jia-Xing Hong |
| title_full_unstemmed | Isometric embedding of Riemannian manifolds in Euclidean spaces Qing Han ; Jia-Xing Hong |
| title_short | Isometric embedding of Riemannian manifolds in Euclidean spaces |
| title_sort | isometric embedding of riemannian manifolds in euclidean spaces |
| topic | Espaces algébriques Isométrie (Mathématiques) Riemann, Variétés de Algebraic spaces Isometrics (Mathematics) Riemannian manifolds Euklidischer Raum (DE-588)4309127-1 gnd Isometrische Einbettung (DE-588)4162532-8 gnd |
| topic_facet | Espaces algébriques Isométrie (Mathématiques) Riemann, Variétés de Algebraic spaces Isometrics (Mathematics) Riemannian manifolds Euklidischer Raum Isometrische Einbettung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016290117&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000018014 |
| work_keys_str_mv | AT hanqing isometricembeddingofriemannianmanifoldsineuclideanspaces AT hongjiaxing isometricembeddingofriemannianmanifoldsineuclideanspaces |