Topics in differential geometry
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, RI
American Mathematical Society
2008
|
| Schriftenreihe: | Graduate studies in mathematics
93 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
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| 100 | 1 | |a Michor, Peter W. |d 1949- |e Verfasser |0 (DE-588)172276144 |4 aut | |
| 245 | 1 | 0 | |a Topics in differential geometry |c Peter W. Michor |
| 264 | 1 | |a Providence, RI |b American Mathematical Society |c 2008 | |
| 300 | |a XI, 494 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Graduate studies in mathematics |v 93 | |
| 650 | 0 | 7 | |a Differentialgeometrie |0 (DE-588)4012248-7 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Differentialgeometrie |0 (DE-588)4012248-7 |D s |
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| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-1-4704-1161-9 |
| 830 | 0 | |a Graduate studies in mathematics |v 93 |w (DE-604)BV009739289 |9 93 | |
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| 856 | 4 | 2 | |m Digitalisierung UB Bayreuth |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016736220&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |3 Klappentext |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-016736220 | ||
Datensatz im Suchindex
| _version_ | 1804138014788550656 |
|---|---|
| adam_text | Contents
Preface
ix
CHAPTER I. Manifolds and Vector Fields
1
1.
Differentiable Manifolds
1
2.
Submersions and Immersions
16
3.
Vector Fields and Flows
21
CHAPTER II. Lie Groups and Group Actions
41
4.
Lie Groups I
41
5.
Lie Groups II. Lie Subgroups and Homogeneous Spaces
60
6.
Transformation Groups and G-Manifolds
66
7.
Polynomial and Smooth Invariant Theory
85
CHAPTER HI. Differential Forms and
de Rham
Cohomology
99
8.
Vector Bundles
99
9.
Differential Forms
113
10.
Integration on Manifolds
122
11. De Rham
Cohomology
129
12.
Cohomology with Compact Supports and
Poincaré
Duality
139
13. De Rham
Cohomology of Compact Manifolds
151
14.
Lie Groups III. Analysis on Lie Groups
158
15.
Extensions of Lie Algebras and Lie Groups
169
vu
viii Contents
CHAPTER IV. Bundles and Connections
191
16.
Derivations on the Algebra of Differential Forms
191
17.
Fiber Bundles and Connections
200
18.
Principal Fiber Bundles and G-Bundles
210
19.
Principal and Induced Connections
229
20.
Characteristic Classes
251
21.
Jets
266
CHAPTER V. Riemann Manifolds
273
22.
Pseudo-Riemann Metrics and Covariant Derivatives
273
23.
Geometry of Geodesies
289
24.
Parallel Transport and Curvature
298
25.
Computing with Adapted Frames and Examples
310
26.
Riemann Immersions and Submersions
327
27.
Jacobi Fields
345
CHAPTER VI. Isometric Group Actions or Riemann G-Manifolds
363
28.
Isometries, Homogeneous Manifolds, and Symmetric Spaces
363
29.
Riemann G-Manifolds
371
30.
Polar Actions
385
CHAPTER
VII. Symplectic
and
Poisson
Geometry
411
31.
Symplectic Geometry and Classical Mechanics
411
32.
Completely
Integrable Hamiltonian
Systems
433
33.
Poisson
Manifolds
439
34.
Hamiltonian Group Actions and Momentum Mappings
451
List of Symbols
477
Bibliography
479
Index
489
This book treats the fundamentals of differential geometry: manifolds, flows,
Lie groups and their actions, invariant theory, differential forms and
de Rham
cohomology, bundles and connections, Riemann manifolds, isometric actions,
and symplectic and
Poisson
geometry.
The layout of the material stresses
natu
ral i
ty and functoriality from the beginning and is as
coordinate-free as possible. Coordinate formulas are always derived as extra information. Some
attractive unusual aspects of this book are as follows:
•
Initial submanifolds and the Frobenius theorem for distributions of
nonconstant
rank (the
Stefan-Sussman theory) are discussed.
•
Lie groups and their actions are treated early on, including the slice theorem and invariant
theory.
•
De Rham
cohomology includes that of compact Lie groups, leading to the study of (nonabelian)
extensions of Lie algebras and Lie groups.
•
The
Frölicher-Nijenhuis
bracket for tangent bundle valued differential forms is used to express
any kind of curvature and second
Bianchi
identity, even for fiber bundles (without structure
groups). Riemann geometry starts with a careful treatment of connections to geodesic struc¬
tures to sprays to connectors and back to connections, going via the second and third tangent
bundles. The Jacobi flow on the second tangent bundle is a new aspect coming from this point
of view.
•
Symplectic and
Poisson
geometry emphasizes group actions, momentum mappings, and reduc¬
tions.
This book gives the careful reader working knowledge in a wide range of topics of modern
coordinate-free differential geometry in not too many pages. A prerequisite for using this book
is a good knowledge of undergraduate analysis and linear algebra.
|
| adam_txt |
Contents
Preface
ix
CHAPTER I. Manifolds and Vector Fields
1
1.
Differentiable Manifolds
1
2.
Submersions and Immersions
16
3.
Vector Fields and Flows
21
CHAPTER II. Lie Groups and Group Actions
41
4.
Lie Groups I
41
5.
Lie Groups II. Lie Subgroups and Homogeneous Spaces
60
6.
Transformation Groups and G-Manifolds
66
7.
Polynomial and Smooth Invariant Theory
85
CHAPTER HI. Differential Forms and
de Rham
Cohomology
99
8.
Vector Bundles
99
9.
Differential Forms
113
10.
Integration on Manifolds
122
11. De Rham
Cohomology
129
12.
Cohomology with Compact Supports and
Poincaré
Duality
139
13. De Rham
Cohomology of Compact Manifolds
151
14.
Lie Groups III. Analysis on Lie Groups
158
15.
Extensions of Lie Algebras and Lie Groups
169
vu
viii Contents
CHAPTER IV. Bundles and Connections
191
16.
Derivations on the Algebra of Differential Forms
191
17.
Fiber Bundles and Connections
200
18.
Principal Fiber Bundles and G-Bundles
210
19.
Principal and Induced Connections
229
20.
Characteristic Classes
251
21.
Jets
266
CHAPTER V. Riemann Manifolds
273
22.
Pseudo-Riemann Metrics and Covariant Derivatives
273
23.
Geometry of Geodesies
289
24.
Parallel Transport and Curvature
298
25.
Computing with Adapted Frames and Examples
310
26.
Riemann Immersions and Submersions
327
27.
Jacobi Fields
345
CHAPTER VI. Isometric Group Actions or Riemann G-Manifolds
363
28.
Isometries, Homogeneous Manifolds, and Symmetric Spaces
363
29.
Riemann G-Manifolds
371
30.
Polar Actions
385
CHAPTER
VII. Symplectic
and
Poisson
Geometry
411
31.
Symplectic Geometry and Classical Mechanics
411
32.
Completely
Integrable Hamiltonian
Systems
433
33.
Poisson
Manifolds
439
34.
Hamiltonian Group Actions and Momentum Mappings
451
List of Symbols
477
Bibliography
479
Index
489
This book treats the fundamentals of differential geometry: manifolds, flows,
Lie groups and their actions, invariant theory, differential forms and
de Rham
cohomology, bundles and connections, Riemann manifolds, isometric actions,
and symplectic and
Poisson
geometry.
The layout of the material stresses
natu
ral i
ty and functoriality from the beginning and is as
coordinate-free as possible. Coordinate formulas are always derived as extra information. Some
attractive unusual aspects of this book are as follows:
•
Initial submanifolds and the Frobenius theorem for distributions of
nonconstant
rank (the
Stefan-Sussman theory) are discussed.
•
Lie groups and their actions are treated early on, including the slice theorem and invariant
theory.
•
De Rham
cohomology includes that of compact Lie groups, leading to the study of (nonabelian)
extensions of Lie algebras and Lie groups.
•
The
Frölicher-Nijenhuis
bracket for tangent bundle valued differential forms is used to express
any kind of curvature and second
Bianchi
identity, even for fiber bundles (without structure
groups). Riemann geometry starts with a careful treatment of connections to geodesic struc¬
tures to sprays to connectors and back to connections, going via the second and third tangent
bundles. The Jacobi flow on the second tangent bundle is a new aspect coming from this point
of view.
•
Symplectic and
Poisson
geometry emphasizes group actions, momentum mappings, and reduc¬
tions.
This book gives the careful reader working knowledge in a wide range of topics of modern
coordinate-free differential geometry in not too many pages. A prerequisite for using this book
is a good knowledge of undergraduate analysis and linear algebra. |
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| illustrated | Illustrated |
| index_date | 2024-07-02T22:03:01Z |
| indexdate | 2024-07-09T21:21:27Z |
| institution | BVB |
| isbn | 9780821820032 |
| language | English |
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| physical | XI, 494 S. graph. Darst. |
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| publisher | American Mathematical Society |
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| series | Graduate studies in mathematics |
| series2 | Graduate studies in mathematics |
| spelling | Michor, Peter W. 1949- Verfasser (DE-588)172276144 aut Topics in differential geometry Peter W. Michor Providence, RI American Mathematical Society 2008 XI, 494 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics 93 Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf Differentialgeometrie (DE-588)4012248-7 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4704-1161-9 Graduate studies in mathematics 93 (DE-604)BV009739289 93 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016736220&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016736220&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Michor, Peter W. 1949- Topics in differential geometry Graduate studies in mathematics Differentialgeometrie (DE-588)4012248-7 gnd |
| subject_GND | (DE-588)4012248-7 |
| title | Topics in differential geometry |
| title_auth | Topics in differential geometry |
| title_exact_search | Topics in differential geometry |
| title_exact_search_txtP | Topics in differential geometry |
| title_full | Topics in differential geometry Peter W. Michor |
| title_fullStr | Topics in differential geometry Peter W. Michor |
| title_full_unstemmed | Topics in differential geometry Peter W. Michor |
| title_short | Topics in differential geometry |
| title_sort | topics in differential geometry |
| topic | Differentialgeometrie (DE-588)4012248-7 gnd |
| topic_facet | Differentialgeometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016736220&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016736220&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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