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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Basel [u.a.]
Birkhäuser
2004
|
| Edition: | 2. ed. |
| Series: | Progress in mathematics
221 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references (p. [257]-271) and indexes |
| Physical Description: | XIII, 280 S. graph. Darst. |
| ISBN: | 3764369078 |
Staff View
MARC
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| 650 | 4 | |a Weyl, Problème de | |
| 650 | 4 | |a Geometry, Differential | |
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Record in the Search Index
| _version_ | 1804130444490309632 |
|---|---|
| adam_text | Contents
Preface to the Second Edition ix
Preface xi
1 An Introduction to Weyl s Tube Formula 1
1.1 The Formula and Its History 1
1.2 Weyl s Formula for Low Dimensions 2
1.3 Problems 11
2 Fermi Coordinates and Fermi Fields 13
2.1 Fermi Coordinates as Generalized Normal Coordinates 13
2.2 A Review of Curvature Fundamentals 19
2.3 Fermi Fields 20
2.4 The Generalized Gauss Lemma 25
2.5 Problems 29
3 The Riccati Equation for Second Fundamental Forms 31
3.1 The Second Fundamental Forms of the Tubular Hypersurfaees ... 32
3.2 The Infinitesimal Change of Volume Function 36
3.3 Volume of a Tube in Terms of Infinitesimal Change
of Volume Function 40
3.4 The Volume of a Tube in R in Terms of Its Second
Fundamental Forms 42
3.5 The Bishop Giinther Inequalities 44
3.6 Myers Theorem 48
3.7 Problems 51
4 The Proof of Weyl s Tube Formula 53
4.1 Double Forms 54
4.2 Invariants 57
4.3 Moments and Invariants 60
4.4 Averaging the Tube Integrand 62
4.5 Generalizations 67
4.6 Problems 68
v Contents
5 The Generalized Gauss Bonnet Theorem 71
5.1 Tubes around Tubular Hypersurfaces 72
5.2 The Euler Characteristic 73
5.3 The Pfaffian 75
5.4 The Gauss Bonnet Theorem for Hypersurfaces in K2 +1 78
5.5 The Tube Proof of the Generalized Gauss Bonnet Theorem .... 80
5.6 The History of the Gauss Bonnet Theorem 81
5.7 Problems 83
6 Chern Forms and Chern Numbers 85
6.1 The Chern Forms of a Kahler Manifold 86
6.2 Spaces of Constant Holomorphic Sectional Curvature 92
6.3 Locally Symmetric Spaces and Their Compatible
Submanifolds 94
6.4 Geodesic Balls in a Space K£ol(A) 98
6.5 Complex Projective Space CP (A) 100
6.6 The Chern Forms of K£ol(A) 101
6.7 Kahler Submanifolds and Wirtinger s Inequality 105
6.8 The Homology and Cohomology of Complex Projective
Space CPn(A) 108
6.9 Chern Numbers 109
6.10 Complex Hypersurfaces of Complex Projective
Space CPn+1(A) 110
6.11 Problems 112
7 The Tube Formula in the Complex Case 117
7.1 Higher Order Curvature Identities 118
7.2 Tubes about Complex Submanifolds of Cn 122
7.3 Tubes in a Space KJJol(A) of Constant Holomorphic
Sectional Curvature 123
7.4 Tubes and Chern Forms 126
7.5 The Projective Weyl Tube Formula 133
7.6 Tubes about Complex Hypersurfaces of Complex
Projective Space CPn+l X) 135
7.7 Kahler Deformations 138
7.8 Tubes about Totally Real Submanifolds of K£()1( A) 139
7.9 Problems 140
8 Comparison Theorems for Tube Volumes 143
8.1 Focal Points and Cut focal Points 144
8.2 Tubes about Submanifolds of a Space of Nonnegative or
Nonpositive Curvature 147
8.3 The Bishop Giinther Inequalities Generalized to Tubes 157
8.4 Tube Volume Estimates Involving Ricci Curvature 164
Contents vii
8.5 Comparison Theorems for the Volumes of Tubes about
Kahler Submanifolds 167
8.6 Some Inequalities of Heintze and Karcher 171
8.7 Gromov s Improvement of the Bishop Giinther Inequalities .... 174
8.8 Ball and Tube Comparison Theorems for Surfaces 177
8.9 Comparison Theorems for Riemannian Manifolds 180
8.10 Problems 182
9 Power Series Expansions for Tube Volumes 185
9.1 Power Series Expansions in Normal Coordinates 186
9.2 The Power Series Expansion for the Volume Vrf/ (r) of
a Small Geodesic Ball 194
9.3 Power Series Expansions in Fermi Coordinates 200
9.4 Problems 206
10 Steiner s Formula 209
10.1 Hypersurfaces 212
10.2 The Infinitesimal Change of Volume Function of
a Hypersurface 216
10.3 Hypersurfaces in Manifolds of Nonnegative or
Nonpositive Curvature 219
10.4 Steiner s Formula for a Space of Nonnegative or
Nonpositive Curvature 223
10.5 Inequalities that Generalize Steiner s Formula 225
10.6 Problems 227
11 Mean value Theorems 231
11.1 The Laplacian and the Euclidean Laplacian 232
11.2 Relations between the Two Laplacians and Curvature 236
11.3 The Power Expansion for the Mean Value Mm(r,f) 237
11.4 Problems 242
Appendix A 247
A.I The Volume of a Ball in En 247
A.2 Moments 249
A.3 Computation of the Volume of a Geodesic Ball 252
Appendix B 255
Index 273
Notation Index 277
Name Index 279
|
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| callnumber-subject | QC - Physics |
| classification_rvk | SK 350 SK 370 |
| ctrlnum | (OCoLC)53814572 (DE-599)BVBBV017718005 |
| dewey-full | 516.3/6 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/6 |
| dewey-search | 516.3/6 |
| dewey-sort | 3516.3 16 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV017718005 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T19:21:08Z |
| institution | BVB |
| isbn | 3764369078 |
| language | English |
| lccn | 2003063792 |
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| physical | XIII, 280 S. graph. Darst. |
| publishDate | 2004 |
| publishDateSearch | 2004 |
| publishDateSort | 2004 |
| publisher | Birkhäuser |
| record_format | marc |
| series | Progress in mathematics |
| series2 | Progress in mathematics |
| spelling | Gray, Alfred 1939-1998 Verfasser (DE-588)124637108 aut Tubes Alfred Gray 2. ed. Basel [u.a.] Birkhäuser 2004 XIII, 280 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Progress in mathematics 221 Includes bibliographical references (p. [257]-271) and indexes Géométrie différentielle Weyl, Problème de Geometry, Differential Weyl's problem Zylinderbereich (DE-588)4232981-4 gnd rswk-swf Röhrenfläche (DE-588)4637476-0 gnd rswk-swf Röhrenfläche (DE-588)4637476-0 s DE-604 Zylinderbereich (DE-588)4232981-4 s 1\p DE-604 Progress in mathematics 221 (DE-604)BV000004120 221 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010651209&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Gray, Alfred 1939-1998 Tubes Progress in mathematics Géométrie différentielle Weyl, Problème de Geometry, Differential Weyl's problem Zylinderbereich (DE-588)4232981-4 gnd Röhrenfläche (DE-588)4637476-0 gnd |
| subject_GND | (DE-588)4232981-4 (DE-588)4637476-0 |
| title | Tubes |
| title_auth | Tubes |
| title_exact_search | Tubes |
| title_full | Tubes Alfred Gray |
| title_fullStr | Tubes Alfred Gray |
| title_full_unstemmed | Tubes Alfred Gray |
| title_short | Tubes |
| title_sort | tubes |
| topic | Géométrie différentielle Weyl, Problème de Geometry, Differential Weyl's problem Zylinderbereich (DE-588)4232981-4 gnd Röhrenfläche (DE-588)4637476-0 gnd |
| topic_facet | Géométrie différentielle Weyl, Problème de Geometry, Differential Weyl's problem Zylinderbereich Röhrenfläche |
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