Differentiable manifolds:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boston [u.a.]
Birkhäuser
2008
|
| Edition: | 2. ed., reprint. |
| Series: | Modern Birkhäuser classics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 418 S. graph. Darst. |
| ISBN: | 9780817647667 |
Staff View
MARC
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| 245 | 1 | 0 | |a Differentiable manifolds |c Lawrence Conlon |
| 250 | |a 2. ed., reprint. | ||
| 264 | 1 | |a Boston [u.a.] |b Birkhäuser |c 2008 | |
| 300 | |a XII, 418 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-016540315 | ||
Record in the Search Index
| _version_ | 1804137717551857664 |
|---|---|
| adam_text | Contents
Preface
to the Second Edition
xi
Acknowledgments
xiii
Chapter
1.
Topologicei
Manifolds
1
1.1.
Locally Euclidean Spaces
1
1.2.
Topologica!
Manifolds
3
1.3.
Quotient Constructions and 2-Manifolds
6
1.4.
Partitions of Unity
17
1.5.
Imbeddings and Immersions
20
1.6.
Manifolds with Boundary
22
1.7.
Covering Spaces and the Fundamental Group
26
Chapter
2.
The Local Theory of Smooth Functions
41
2.1.
Differentiability Classes
41
2.2.
Tangent Vectors
42
2.3.
Smooth Maps and their Differentials
50
2.4.
Diffeomorphisms and Maps of Constant Rank
54
2.5.
Smooth Submanifolds of Euclidean Space
58
2.6.
Constructions of Smooth Functions
62
2.7.
Smooth Vector Fields
65
2.8.
Local Flows
71
2.9.
Critical Points and Critical Values
80
Chapter
3.
The Global Theory of Smooth Functions
87
3.1.
Smooth Manifolds and Mappings
87
3.2.
Diffeomorphic Structures
93
3.3.
The Tangent Bundle
94
3.4.
Cocycles and Geometric Structures
98
3.5.
Global Constructions of Smooth Functions
104
3.6.
Smooth Manifolds with Boundary
107
3.7.
Smooth Submanifolds
110
3.8.
Smooth Homotopy and Smooth Approximations
116
3.9.
Degree Theory Modulo
2* 119
3.10.Morse Functions*
124
Chapter
4.
Flows and Foliations
131
4.1.
Complete Vector Fields
131
4.2.
The Gradient Flow and Morse Functions*
136
4.3.
The Lie Bracket
142
4.4.
Commuting Flows
145
4.5.
Foliations
150
viii
CONTENTS
Chapter
5.
Lie Groups and Lie Algebras
161
5.1.
Basic Definitions and Facts
161
5.2.
Lie Subgroups and Subalgebras
170
5.3.
Closed Subgroups*
173
5.4.
Homogeneous Spaces*
178
Chapter
6.
Covectors and 1-Forms
183
6.1.
Dual Bundles
183
6.2.
The space of 1-forms
185
6.3.
Line Integrals
190
6.4.
The First Cohomology Space
195
6.5.
Degree Theory on S1*
202
Chapter
7.
Multilinear Algebra and Tensors
209
7.1.
Tensor Algebra
209
7.2.
Exterior Algebra
217
7.3.
Symmetric Algebra
226
7.4.
Multilinear Bundle Theory
227
7.5.
The Module of Sections
230
Chapter
8.
Integration of Forms and
de Rham
Cohomology
239
8.1.
The Exterior Derivative
239
8.2.
Stokes Theorem and Singular Homology
245
8.3.
The
Poincaré
Lemma
258
8.4.
Exact Sequences
264
8.5.
Mayer-Vietoris Sequences
267
8.6.
Computations of Cohomology
271
8.7.
Degree Theory*
274
8.8.
Poincaré
Duality*
276
8.9.
The
de Rham
Theorem*
281
Chapter
9.
Forms and Foliations
289
9.1.
The Frobenius Theorem Revisited
289
9.2.
The Normal Bundle and Transversality
293
9.3.
Closed, Nonsingular 1-forms*
296
Chapter
10.
Riemannian Geometry
303
10.1.
Connections
304
10.2.
Riemannian Manifolds
311
10.3.
Gauss Curvature
315
10.4.
Complete Riemannian Manifolds
322
10.5.
Geodesic Convexity
334
10.6.
The Cartan Structure Equations
337
10.7.
Riemannian Homogeneous Spaces*
342
Chapter
11.
Principal Bundles*
347
11.1.
The Frame Bundle
347
11.2.
Principal (7-Bundles
351
11.3.
Cocycles and Reductions
354
11.4.
Frame Bundles and the Equations of Structure
357
CONTENTS
ix
Appendix A. Construction of the Universal Covering
369
Appendix B. The Inverse Function Theorem
373
Appendix C. Ordinary Differential Equations
379
C.I. Existence and uniqueness of solutions
379
C.2. A digression concerning Banach spaces
382
C.3. Smooth dependence on initial conditions
383
C.4. The Linear Case
385
Appendix D. The
de Rhani
Cohomology Theorem
387
D.I.
Čech
cohomology
387
D.2. The
de Rham-Čech
complex
391
D.3. Singular Cohomology
397
Bibliography
403
Index
405
|
| adam_txt |
Contents
Preface
to the Second Edition
xi
Acknowledgments
xiii
Chapter
1.
Topologicei
Manifolds
1
1.1.
Locally Euclidean Spaces
1
1.2.
Topologica!
Manifolds
3
1.3.
Quotient Constructions and 2-Manifolds
6
1.4.
Partitions of Unity
17
1.5.
Imbeddings and Immersions
20
1.6.
Manifolds with Boundary
22
1.7.
Covering Spaces and the Fundamental Group
26
Chapter
2.
The Local Theory of Smooth Functions
41
2.1.
Differentiability Classes
41
2.2.
Tangent Vectors
42
2.3.
Smooth Maps and their Differentials
50
2.4.
Diffeomorphisms and Maps of Constant Rank
54
2.5.
Smooth Submanifolds of Euclidean Space
58
2.6.
Constructions of Smooth Functions
62
2.7.
Smooth Vector Fields
65
2.8.
Local Flows
71
2.9.
Critical Points and Critical Values
80
Chapter
3.
The Global Theory of Smooth Functions
87
3.1.
Smooth Manifolds and Mappings
87
3.2.
Diffeomorphic Structures
93
3.3.
The Tangent Bundle
94
3.4.
Cocycles and Geometric Structures
98
3.5.
Global Constructions of Smooth Functions
104
3.6.
Smooth Manifolds with Boundary
107
3.7.
Smooth Submanifolds
110
3.8.
Smooth Homotopy and Smooth Approximations
116
3.9.
Degree Theory Modulo
2* 119
3.10.Morse Functions*
124
Chapter
4.
Flows and Foliations
131
4.1.
Complete Vector Fields
131
4.2.
The Gradient Flow and Morse Functions*
136
4.3.
The Lie Bracket
142
4.4.
Commuting Flows
145
4.5.
Foliations
150
viii
CONTENTS
Chapter
5.
Lie Groups and Lie Algebras
161
5.1.
Basic Definitions and Facts
161
5.2.
Lie Subgroups and Subalgebras
170
5.3.
Closed Subgroups*
173
5.4.
Homogeneous Spaces*
178
Chapter
6.
Covectors and 1-Forms
183
6.1.
Dual Bundles
183
6.2.
The space of 1-forms
185
6.3.
Line Integrals
190
6.4.
The First Cohomology Space
195
6.5.
Degree Theory on S1*
202
Chapter
7.
Multilinear Algebra and Tensors
209
7.1.
Tensor Algebra
209
7.2.
Exterior Algebra
217
7.3.
Symmetric Algebra
226
7.4.
Multilinear Bundle Theory
227
7.5.
The Module of Sections
230
Chapter
8.
Integration of Forms and
de Rham
Cohomology
239
8.1.
The Exterior Derivative
239
8.2.
Stokes' Theorem and Singular Homology
245
8.3.
The
Poincaré
Lemma
258
8.4.
Exact Sequences
264
8.5.
Mayer-Vietoris Sequences
267
8.6.
Computations of Cohomology
271
8.7.
Degree Theory*
274
8.8.
Poincaré
Duality*
276
8.9.
The
de Rham
Theorem*
281
Chapter
9.
Forms and Foliations
289
9.1.
The Frobenius Theorem Revisited
289
9.2.
The Normal Bundle and Transversality
293
9.3.
Closed, Nonsingular 1-forms*
296
Chapter
10.
Riemannian Geometry
303
10.1.
Connections
304
10.2.
Riemannian Manifolds
311
10.3.
Gauss Curvature
315
10.4.
Complete Riemannian Manifolds
322
10.5.
Geodesic Convexity
334
10.6.
The Cartan Structure Equations
337
10.7.
Riemannian Homogeneous Spaces*
342
Chapter
11.
Principal Bundles*
347
11.1.
The Frame Bundle
347
11.2.
Principal (7-Bundles
351
11.3.
Cocycles and Reductions
354
11.4.
Frame Bundles and the Equations of Structure
357
CONTENTS
ix
Appendix A. Construction of the Universal Covering
369
Appendix B. The Inverse Function Theorem
373
Appendix C. Ordinary Differential Equations
379
C.I. Existence and uniqueness of solutions
379
C.2. A digression concerning Banach spaces
382
C.3. Smooth dependence on initial conditions
383
C.4. The Linear Case
385
Appendix D. The
de Rhani
Cohomology Theorem
387
D.I.
Čech
cohomology
387
D.2. The
de Rham-Čech
complex
391
D.3. Singular Cohomology
397
Bibliography
403
Index
405 |
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| id | DE-604.BV023356788 |
| illustrated | Illustrated |
| index_date | 2024-07-02T21:07:30Z |
| indexdate | 2024-07-09T21:16:44Z |
| institution | BVB |
| isbn | 9780817647667 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016540315 |
| oclc_num | 209332851 |
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| owner_facet | DE-355 DE-BY-UBR DE-20 DE-19 DE-BY-UBM |
| physical | XII, 418 S. graph. Darst. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Birkhäuser |
| record_format | marc |
| series2 | Modern Birkhäuser classics |
| spelling | Conlon, Lawrence 1933- Verfasser (DE-588)122970721 aut Differentiable manifolds Lawrence Conlon 2. ed., reprint. Boston [u.a.] Birkhäuser 2008 XII, 418 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Modern Birkhäuser classics Differentiable manifolds Differenzierbare Mannigfaltigkeit (DE-588)4012269-4 gnd rswk-swf Differenzierbare Mannigfaltigkeit (DE-588)4012269-4 s DE-604 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016540315&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Conlon, Lawrence 1933- Differentiable manifolds Differentiable manifolds Differenzierbare Mannigfaltigkeit (DE-588)4012269-4 gnd |
| subject_GND | (DE-588)4012269-4 |
| title | Differentiable manifolds |
| title_auth | Differentiable manifolds |
| title_exact_search | Differentiable manifolds |
| title_exact_search_txtP | Differentiable manifolds |
| title_full | Differentiable manifolds Lawrence Conlon |
| title_fullStr | Differentiable manifolds Lawrence Conlon |
| title_full_unstemmed | Differentiable manifolds Lawrence Conlon |
| title_short | Differentiable manifolds |
| title_sort | differentiable manifolds |
| topic | Differentiable manifolds Differenzierbare Mannigfaltigkeit (DE-588)4012269-4 gnd |
| topic_facet | Differentiable manifolds Differenzierbare Mannigfaltigkeit |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016540315&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT conlonlawrence differentiablemanifolds |