The Laplacian on a Riemannian manifold: an introduction to analysis on manifolds
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge Univ. Press
1998
|
| Ausgabe: | Reprinted |
| Schriftenreihe: | London Mathematical Society: London Mathematical Society student texts
31 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | X, 174 S. |
| ISBN: | 0521463009 0521468310 |
Internformat
MARC
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| 245 | 1 | 0 | |a The Laplacian on a Riemannian manifold |b an introduction to analysis on manifolds |c Steven Rosenberg |
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| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 1998 | |
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Datensatz im Suchindex
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|---|---|
| adam_text | Contents
Introduction
vii
The Laplacian on a Riemannian Manifold
1
1.1
Basic Examples
............................ 2
1.1.1
The Laplacian on S1 and
R
................. 3
1.1.2
Heat Flow on S1 and
R
................... 5
1.2
The Laplacian on a Riemannian Manifold
............. 10
1.2.1
Riemannian Metrics
..................... 10
1.2.2
I? Spaces of Functions and Forms
............. 14
1.2.3
The Laplacian on Functions
................. 17
1.3
Hodge Theory for Functions and Forms
.............. 22
1.3.1
Analytic Preliminaries
.................... 22
1.3.2
The Heat Equation Proof of the Hodge Theorem for Func¬
tions
.............................. 27
1.3.3
The Hodge Theorem for Differential Forms
........ 33
1.3.4
Regularity Results
...................... 35
1.4 De Rham
Cohomology
........................ 39
1.5
The Kernel of the Laplacian on Forms
............... 46
Elements of Differential Geometry
52
2.1
Curvature
............................... 52
2.2
The Levi-Civita Connection and Bochner
Formula
................................ 63
2.2.1
The Levi-Civita Connection
................. 63
2.2.2 Weitzenböck
Formulas and Garding s Inequality
..... 67
2.3
Geodesies
............................... 79
2.4
The Laplacian in Exponential Coordinates
............. 85
The Construction of the Heat Kernel
90
3.1
Preliminary Results for the Heat Kernel
.............. 90
3.2
Construction of the Heat Kernel
.................. 92
3.2.1
Construction of the Parametrix
............... 92
3.2.2
The Heat Kernel for Functions
............... 96
3.3
The Asymptotics of the Heat Kernel
................ 101
3.4
Positivity
of the Heat Kernel
....................108
The Heat Equation Approach to the Atiyah-Singer Index The¬
orem 111
4.1
The Chem-Gauss-Bonnet Theorem
.................
Ill
4.1.1
The Heat Equation Approach
................112
4.1.2
Proof of the Chern-Gauss-Bonnet Theorem
........116
4.2
The Hirzebruch Signature Theorem and the Atiyah-Singer Index
Theorem
................................128
4.2.1
A Survey of Characteristic Forms
..............128
4.2.2
The Hirzebruch Signature Theorem
............134
4.2.3
The Atiyah-Singer Index Theorem
.............139
Zeta
Functions of Laplacians
144
5.1
The
Zeta
Function of a Laplacian
.................. 144
5.2
Isospectral
Manifolds
......................... 151
5.3
Reidemeister Torsion and Analytic Torsion
............ 153
5.3.1
Reidemeister Torsion
..................... 153
5.3.2
Analytic Torsion
....................... 154
5.3.3
The Families Index Theorem and Analytic Torsion
.... 163
Bibliography
166
Index
171
|
| any_adam_object | 1 |
| author | Rosenberg, Steven |
| author_facet | Rosenberg, Steven |
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| dewey-ones | 516 - Geometry |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| id | DE-604.BV012712949 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:32:22Z |
| institution | BVB |
| isbn | 0521463009 0521468310 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008641948 |
| oclc_num | 247917915 |
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| owner_facet | DE-355 DE-BY-UBR DE-29T DE-91G DE-BY-TUM DE-739 DE-188 |
| physical | X, 174 S. |
| publishDate | 1998 |
| publishDateSearch | 1998 |
| publishDateSort | 1998 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| series | London Mathematical Society: London Mathematical Society student texts |
| series2 | London Mathematical Society: London Mathematical Society student texts |
| spelling | Rosenberg, Steven Verfasser aut The Laplacian on a Riemannian manifold an introduction to analysis on manifolds Steven Rosenberg Reprinted Cambridge [u.a.] Cambridge Univ. Press 1998 X, 174 S. txt rdacontent n rdamedia nc rdacarrier London Mathematical Society: London Mathematical Society student texts 31 Laplace-Operator (DE-588)4166772-4 gnd rswk-swf Riemannscher Raum (DE-588)4128295-4 gnd rswk-swf Laplace-Operator (DE-588)4166772-4 s Riemannscher Raum (DE-588)4128295-4 s DE-604 London Mathematical Society: London Mathematical Society student texts 31 (DE-604)BV000841726 31 Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008641948&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Rosenberg, Steven The Laplacian on a Riemannian manifold an introduction to analysis on manifolds London Mathematical Society: London Mathematical Society student texts Laplace-Operator (DE-588)4166772-4 gnd Riemannscher Raum (DE-588)4128295-4 gnd |
| subject_GND | (DE-588)4166772-4 (DE-588)4128295-4 |
| title | The Laplacian on a Riemannian manifold an introduction to analysis on manifolds |
| title_auth | The Laplacian on a Riemannian manifold an introduction to analysis on manifolds |
| title_exact_search | The Laplacian on a Riemannian manifold an introduction to analysis on manifolds |
| title_full | The Laplacian on a Riemannian manifold an introduction to analysis on manifolds Steven Rosenberg |
| title_fullStr | The Laplacian on a Riemannian manifold an introduction to analysis on manifolds Steven Rosenberg |
| title_full_unstemmed | The Laplacian on a Riemannian manifold an introduction to analysis on manifolds Steven Rosenberg |
| title_short | The Laplacian on a Riemannian manifold |
| title_sort | the laplacian on a riemannian manifold an introduction to analysis on manifolds |
| title_sub | an introduction to analysis on manifolds |
| topic | Laplace-Operator (DE-588)4166772-4 gnd Riemannscher Raum (DE-588)4128295-4 gnd |
| topic_facet | Laplace-Operator Riemannscher Raum |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008641948&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000841726 |
| work_keys_str_mv | AT rosenbergsteven thelaplacianonariemannianmanifoldanintroductiontoanalysisonmanifolds |