Introduction to differential geometry:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin, Germany
Springer
[2022]
|
| Series: | Textbook
Springer Studium Mathematik (Master) |
| Subjects: | |
| Online Access: | Inhaltstext http://www.springer.com/ Inhaltsverzeichnis Inhaltsverzeichnis |
| Item Description: | Literaturverzeichnis Seite 409-412 |
| Physical Description: | xiii, 418 Seiten Illustrationen 24 cm, 658 g |
| ISBN: | 9783662643396 |
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MARC
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| 100 | 1 | |a Robbin, Joel W. |d 1941- |e Verfasser |0 (DE-588)105369041X |4 aut | |
| 245 | 1 | 0 | |a Introduction to differential geometry |c Joel W. Robbin, Dietmar A. Salamon |
| 264 | 1 | |a Berlin, Germany |b Springer |c [2022] | |
| 300 | |a xiii, 418 Seiten |b Illustrationen |c 24 cm, 658 g | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Textbook | |
| 490 | 0 | |a Springer Studium Mathematik (Master) | |
| 500 | |a Literaturverzeichnis Seite 409-412 | ||
| 650 | 0 | 7 | |a Differentialgeometrie |0 (DE-588)4012248-7 |2 gnd |9 rswk-swf |
| 653 | |a extrinsic differential geometry | ||
| 653 | |a extrinsic vs. intrinsic differential geometry | ||
| 653 | |a accessible differential geometry | ||
| 653 | |a Riemannian manifolds | ||
| 653 | |a smooth submanifolds of Euclidean space | ||
| 653 | |a isometries between manifolds | ||
| 653 | |a embeddings | ||
| 653 | |a geodesics | ||
| 653 | |a curvature | ||
| 689 | 0 | 0 | |a Differentialgeometrie |0 (DE-588)4012248-7 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Salamon, Dietmar |d 1953- |e Verfasser |0 (DE-588)1015895271 |4 aut | |
| 710 | 2 | |a Springer-Verlag GmbH |0 (DE-588)1065168780 |4 pbl | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |a Robbin, Joel W., 1941- |t Introduction to Differential Geometry |b 1st edition 2022 |d Berlin, Heidelberg : Springer Berlin Heidelberg, 2022 |h Online-Ressource |z 978-3-662-64340-2 |
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Record in the Search Index
| _version_ | 1817968342224338944 |
|---|---|
| adam_text |
CONTENTS
1
WHAT
IS
DIFFERENTIAL
GEOMETRY?
.
1
1.1
CARTOGRAPHY
AND
DIFFERENTIAL
GEOMETRY
.
1
1.2
COORDINATES
.
3
1.3
TOPOLOGICAL
MANIFOLDS*
.
7
1.4
SMOOTH
MANIFOLDS
DEFINED*
.
9
1.5
THE
MASTER
PLAN
.
12
2
FOUNDATIONS
.
13
2.1
SUBMANIFOLDS
OF
EUCLIDEAN
SPACE
.
13
2.2
TANGENT
SPACES
AND
DERIVATIVES
.
21
2.2.1
TANGENT
SPACE
.
22
2.2.2
DERIVATIVE
.
26
2.2.3
THE
INVERSE
FUNCTION
THEOREM
.
28
2.2.4
REGULAR
VALUES
.
29
2.3
SUBMANIFOLDS
AND
EMBEDDINGS
.
31
2.4
VECTOR
FIELDS
AND
FLOWS
.
35
2.4.1
VECTOR
FIELDS
.
35
2.4.2
THE
FLOW
OF
A
VECTOR
FIELD
.
37
2.4.3
THE
GROUP
OF
DIFFEOMORPHISMS
.
42
2.4.4
THE
LIE
BRACKET
.
42
2.5
LIE
GROUPS
.
48
2.5.1
DEFINITION
AND
EXAMPLES
.
49
2.5.2
THE
LIE
ALGEBRA
OF
A
LIE
GROUP
.
52
2.5.3
LIE
GROUP
HOMOMORPHISMS
.
55
2.5.4
CLOSED
SUBGROUPS
.
59
2.5.5
LIE
GROUPS
AND
DIFFEOMORPHISMS
.
64
2.5.6
SMOOTH
MAPS
AND
ALGEBRA
HOMOMORPHISMS
.
66
2.5.7
VECTOR
FIELDS
AND
DERIVATIONS
.
68
IX
X
CONTENTS
2.6
VECTOR
BUNDLES
AND
SUBMERSIONS
.
69
2.6.1
SUBMERSIONS
.
69
2.6.2
VECTOR
BUNDLES
.
71
2.6.3
THE
IMPLICIT
FUNCTION
THEOREM
.
76
2.7
THE
THEOREM
OF
FROBENIUS
.
77
2.8
THE
INTRINSIC
DEFINITION
OF
A
MANIFOLD*
.
83
2.8.1
DEFINITION
AND
EXAMPLES
.
83
2.8.2
SMOOTH
MAPS
AND
DIFFEOMORPHISMS
.
88
2.8.3
TANGENT
SPACES
AND
DERIVATIVES
.
89
2.8.4
SUBMANIFOLDS
AND
EMBEDDINGS
.
91
2.8.5
TANGENT
BUNDLE
AND
VECTOR
FIELDS
.
93
2.8.6
COORDINATE
NOTATION
.
95
2.9
CONSEQUENCES
OF
PARACOMPACTNESS*
.
97
2.9.1
PARACOMPACTNESS
.
97
2.9.2
PARTITIONS
OF
UNITY
.
99
2.9.3
EMBEDDING
IN
EUCLIDEAN
SPACE
.
102
2.9.4
LEAVES
OF
A
FOLIATION
.
108
2.9.5
PRINCIPAL
BUNDLES
.
109
2.9.6
STANDING
ASSUMPTION
.
115
3
THE
LEVI-CIVITA
CONNECTION
.
117
3.1
SECOND
FUNDAMENTAL
FORM
.
117
3.2
COVARIANT
DERIVATIVE
.
123
3.3
PARALLEL
TRANSPORT
.
125
3.4
THE
FRAME
BUNDLE
.
132
3.4.1
FRAMES
OF
A
VECTOR
SPACE
.
132
3.4.2
THE
FRAME
BUNDLE
.
133
3.4.3
HORIZONTAL
LIFTS
.
136
3.5
MOTIONS
AND
DEVELOPMENTS
.
141
3.5.1
MOTION
.
141
3.5.2
SLIDING
.
143
3.5.3
TWISTING
AND
WOBBLING
.
145
3.5.4
DEVELOPMENT
.
148
3.6
CHRISTOFFEL
SYMBOLS
.
154
3.7
RIEMANNIAN
METRICS*
.
160
3.7.1
EXISTENCE
OF
RIEMANNIAN
METRICS
.
160
3.7.2
TWO
EXAMPLES
.
162
3.7.3
THE
LEVI-CIVITA
CONNECTION
.
163
3.7.4
BASIC
VECTOR
FIELDS
IN
THE
INTRINSIC
SETTING
.
166
4
GEODESICS
.
169
4.1
LENGTH
AND
ENERGY
.
169
4.1.1
THE
LENGTH
AND
ENERGY
FUNCTIONALS
.
169
4.1.2
THE
SPACE
OF
PATHS
.
172
4.1.3
CHARACTERIZATION
OF
GEODESICS
.
173
CONTENTS
XI
4.2
DISTANCE
.
177
4.3
THE
EXPONENTIAL
MAP
.
184
4.3.1
GEODESIC
SPRAY
.
184
4.3.2
THE
EXPONENTIAL
MAP
.
185
4.3.3
EXAMPLES
AND
EXERCISES
.
188
4.3.4
GEODESICS
IN
LOCAL
COORDINATES
.
190
4.4
MINIMAL
GEODESICS
.
191
4.4.1
CHARACTERIZATION
OF
MINIMAL
GEODESICS
.
191
4.4.2
LOCAL
EXISTENCE
OF
MINIMAL
GEODESICS
.
192
4.4.3
EXAMPLES
AND
EXERCISES
.
196
4.5
CONVEX
NEIGHBORHOODS
.
199
4.6
COMPLETENESS
AND
HOPF-RINOW
.
203
4.6.1
GEODESIC
COMPLETENESS
.
203
4.6.2
GLOBAL
EXISTENCE
OF
MINIMAL
GEODESICS
.
204
4.6.3
PROOF
OF
THE
HOPF-RINOW
THEOREM
.
206
4.7
GEODESICS
IN
THE
INTRINSIC
SETTING*
.
211
4.7.1
INTRINSIC
DISTANCE
.
211
4.7.2
GEODESICS
AND
THE
LEVI-CIVITA
CONNECTION
.
213
4.7.3
EXAMPLES
AND
EXERCISES
.
214
5
CURVATURE
.
217
5.1
ISOMETRIES
.
217
5.2
THE
RIEMANN
CURVATURE
TENSOR
.
226
5.2.1
DEFINITION
AND
GAUB-CODAZZI
.
226
5.2.2
COVARIANT
DERIVATIVE
OF
A
GLOBAL
VECTOR
FIELD
.
228
5.2.3
A
GLOBAL
FORMULA
.
231
5.2.4
SYMMETRIES
.
233
5.2.5
RIEMANNIAN
METRICS
ON
LIE
GROUPS
.
235
5.3
GENERALIZED
THEOREMA
EGREGIUM
.
238
5.3.1
PUSHFORWARD
.
238
5.3.2
THEOREMA
EGREGIUM
.
239
5.3.3
GAUBIAN
CURVATURE
.
243
5.4
CURVATURE
IN
LOCAL
COORDINATES*
.247
5.4.1
RIEMANN
.247
5.4.2
GAUB
.248
6
GEOMETRY
AND
TOPOLOGY
.
251
6.1
THE
CARTAN-AMBROSE-HICKS
THEOREM
.
251
6.1.1
HOMOTOPY
.
251
6.1.2
THE
GLOBAL
C-A-H
THEOREM
.
253
6.1.3
THE
LOCAL
C-A-H
THEOREM
.
259
6.2
FLAT
SPACES
.
261
XII
CONTENTS
6.3
SYMMETRIC
SPACES
.266
6.3.1
SYMMETRIC
SPACES
.
266
6.3.2
COVARIANT
DERIVATIVE
OF
THE
CURVATURE
.268
6.3.3
COVARIANT
DERIVATIVE
OF
THE
CURVATURE
IN
LOCAL
COORDINATES
271
6.3.4
EXAMPLES
AND
EXERCISES
.
272
6.4
CONSTANT
CURVATURE
.
273
6.4.1
SECTIONAL
CURVATURE
.
273
6.4.2
CONSTANT
SECTIONAL
CURVATURE
.
275
6.4.3
EXAMPLES
AND
EXERCISES
.
278
6.4.4
HYPERBOLIC
SPACE
.
279
6.5
NONPOSITIVE
SECTIONAL
CURVATURE
.
286
6.5.1
THE
CARTAN-HADAMARD
THEOREM
.
286
6.5.2
CARTAN
'
S
FIXED
POINT
THEOREM
.
292
6.5.3
POSITIVE
DEFINITE
SYMMETRIC
MATRICES
.
295
6.6
POSITIVE
RICCI
CURVATURE*
.
302
6.6.1
THE
RICCI
TENSOR
IN
LOCAL
COORDINATES
.
303
6.6.2
THE
BONNET-MYERS
THEOREM
.
303
6.6.3
POSITIVE
SECTIONAL
CURVATURE
.
305
6.7
SCALAR
CURVATURE*
.
306
6.7.1
DEFINITION
AND
BASIC
PROPERTIES
.
307
6.7.2
SCALAR
CURVATURE
IN
LOCAL
COORDINATES
.
309
6.7.3
POSITIVE
SCALAR
CURVATURE
.
309
6.7.4
CONSTANT
SCALAR
CURVATURE
.
310
6.8
THE
WEYL
TENSOR*
.
312
6.8.1
DEFINITION
AND
BASIC
PROPERTIES
.
313
6.8.2
THE
WEYL
TENSOR
IN
LOCAL
COORDINATES
.
314
6.8.3
CONFORMAL
INVARIANCE
.
315
6.8.4
SELF-DUAL
FOUR-MANIFOLDS
.
315
7
TOPICS
IN
GEOMETRY
.
321
7.1
CONJUGATE
POINTS
AND
THE
MORSE
INDEX*
.
321
7.1.1
CONJUGATE
POINTS
.
322
7.1.2
THE
MORSE
INDEX
THEOREM
.
323
7.1.3
LOCALLY
MINIMAL
GEODESICS
.
328
7.2
THE
INJECTIVITY
RADIUS*
.
332
7.3
THE
GROUP
OF
ISOMETRIES*
.
336
7.3.1
THE
MYERS-STEENROD
THEOREM
.
336
7.3.2
THE
TOPOLOGY
ON
THE
SPACE
OF
ISOMETRIES
.
338
7.3.3
KILLING
VECTOR
FIELDS
.
341
7.3.4
PROOF
OF
THE
MYERS-STEENROD
THEOREM
.
345
7.3.5
EXAMPLES
AND
EXERCISES
.
354
7.4
ISOMETRIES
OF
COMPACT
LIE
GROUPS*
.
356
CONTENTS
XIII
7.5
CONVEX
FUNCTIONS
ON
HADAMARD
MANIFOLDS*
.
362
7.5.1
CONVEX
FUNCTIONS
AND
THE
SPHERE
AT
INFINITY
.
363
7.5.2
INNER
PRODUCTS
AND
WEIGHTED
FLAGS
.
371
7.5.3
LENGTHS
OF
VECTORS
.
375
7.6
SEMISIMPLE
LIE
ALGEBRAS*
.
385
7.6.1
SYMMETRIC
INNER
PRODUCTS
.
385
7.6.2
SIMPLE
LIE
ALGEBRAS
.
388
7.6.3
SEMISIMPLE
LIE
ALGEBRAS
.
392
7.6.4
COMPLEX
LIE
ALGEBRAS
.
397
APPENDIX
A:
NOTES
.
403
A.
1
MAPS
AND
FUNCTIONS
.403
A.2
NORMAL
FORMS
.
404
A.3
EUCLIDEAN
SPACES
.
406
REFERENCES
.
409
INDEX
.
;
.413 |
| any_adam_object | 1 |
| author | Robbin, Joel W. 1941- Salamon, Dietmar 1953- |
| author_GND | (DE-588)105369041X (DE-588)1015895271 |
| author_facet | Robbin, Joel W. 1941- Salamon, Dietmar 1953- |
| author_role | aut aut |
| author_sort | Robbin, Joel W. 1941- |
| author_variant | j w r jw jwr d s ds |
| building | Verbundindex |
| bvnumber | BV049750642 |
| classification_rvk | SK 370 |
| ctrlnum | (OCoLC)1312645101 (DE-599)DNB1241129533 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV049750642 |
| illustrated | Illustrated |
| indexdate | 2024-12-09T13:08:35Z |
| institution | BVB |
| institution_GND | (DE-588)1065168780 |
| isbn | 9783662643396 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-035092322 |
| oclc_num | 1312645101 |
| open_access_boolean | |
| owner | DE-573 DE-739 |
| owner_facet | DE-573 DE-739 |
| physical | xiii, 418 Seiten Illustrationen 24 cm, 658 g |
| publishDate | 2022 |
| publishDateSearch | 2022 |
| publishDateSort | 2022 |
| publisher | Springer |
| record_format | marc |
| series2 | Textbook Springer Studium Mathematik (Master) |
| spelling | Robbin, Joel W. 1941- Verfasser (DE-588)105369041X aut Introduction to differential geometry Joel W. Robbin, Dietmar A. Salamon Berlin, Germany Springer [2022] xiii, 418 Seiten Illustrationen 24 cm, 658 g txt rdacontent n rdamedia nc rdacarrier Textbook Springer Studium Mathematik (Master) Literaturverzeichnis Seite 409-412 Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf extrinsic differential geometry extrinsic vs. intrinsic differential geometry accessible differential geometry Riemannian manifolds smooth submanifolds of Euclidean space isometries between manifolds embeddings geodesics curvature Differentialgeometrie (DE-588)4012248-7 s DE-604 Salamon, Dietmar 1953- Verfasser (DE-588)1015895271 aut Springer-Verlag GmbH (DE-588)1065168780 pbl Erscheint auch als Online-Ausgabe Robbin, Joel W., 1941- Introduction to Differential Geometry 1st edition 2022 Berlin, Heidelberg : Springer Berlin Heidelberg, 2022 Online-Ressource 978-3-662-64340-2 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=c418fcac4e9442be9b678c3641fcf8ef&prov=M&dok_var=1&dok_ext=htm Inhaltstext X:MVB http://www.springer.com/ B:DE-101 application/pdf https://d-nb.info/1241129533/04 Inhaltsverzeichnis DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=035092322&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p dnb 20220804 DE-101 https://d-nb.info/provenance/plan#dnb |
| spellingShingle | Robbin, Joel W. 1941- Salamon, Dietmar 1953- Introduction to differential geometry Differentialgeometrie (DE-588)4012248-7 gnd |
| subject_GND | (DE-588)4012248-7 |
| title | Introduction to differential geometry |
| title_auth | Introduction to differential geometry |
| title_exact_search | Introduction to differential geometry |
| title_full | Introduction to differential geometry Joel W. Robbin, Dietmar A. Salamon |
| title_fullStr | Introduction to differential geometry Joel W. Robbin, Dietmar A. Salamon |
| title_full_unstemmed | Introduction to differential geometry Joel W. Robbin, Dietmar A. Salamon |
| title_short | Introduction to differential geometry |
| title_sort | introduction to differential geometry |
| topic | Differentialgeometrie (DE-588)4012248-7 gnd |
| topic_facet | Differentialgeometrie |
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