Lectures on differential geometry:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English Russian |
| Published: |
Zürich
European Mathematical Society
[2008]
|
| Series: | EMS series of lectures in mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | viii, 211 Seite Illustrationen |
| ISBN: | 9783037190500 |
Staff View
MARC
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| 240 | 1 | 0 | |a Lekcii po differencial'noj geometrii |
| 245 | 1 | 0 | |a Lectures on differential geometry |c Iskander A. Taimanov |
| 264 | 1 | |a Zürich |b European Mathematical Society |c [2008] | |
| 264 | 4 | |c © 2008 | |
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Record in the Search Index
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|---|---|
| adam_text | Contents
Preface
Part I Curves and surfaces
1
1
Theory of curves
3
1.1
Basic notions of the theory of curves
.................. 3
1.2
Plane curves
............................... 5
1.3
Curves in three-dimensional space
................... 8
1.4
The orthogonal group
.......................... 10
2
Theory of surfaces
15
2.1
Metrics on regular surfaces
....................... 15
2.2
Curvature of a curve on a surface
.................... 17
2.3
Gaussian curvature
........................... 20
2.4
Derivational equations and Bonnet s theorem
............. 22
2.5
The Gauss theorem
........................... 27
2.6
Covariant derivative and geodesies
................... 28
2.7
The Euler-Lagrange equations
..................... 32
2.8
The Gauss-Bonnet fonnula
....................... 38
2.9
Minimal surfaces
............................ 44
Part II Riemannian geometry
47
3
Smooth manifolds
49
3.1
Topological spaces
........................... 49
3.2
Smooth manifolds and maps
...................... 51
3.3
Tensors
................................. 58
3.4
Action of maps on tensors
........................ 63
3.5
Embedding of smooth manifolds into the Euclidean space
....... 67
4
Riemannian manifolds
69
4.1
Metric tensor
.............................. 69
4.2 Affine
connection and covariant derivative
............... 70
4.3
Riemannian connections
........................ 74
4.4
Curvature
................................ 77
4.5
Geodesies
................................ 83
viii Contents
5
The Lobachevskii plane and the Minkowski space
89
5.1
The Lobachevskii plane
......................... 89
5.2
Pseudo-Euclidean spaces and their applications in physics
...... 95
Partili
Supplement chapters
101
6
Minimal surfaces and complex analysis
103
6.1
Conformai
parameterization of surfaces
................ 103
6.2
The theory of surfaces in terms of the
conformai
parameter
...... 107
6.3
The
Weierstrass
representation
.....................
Ill
7
Elements of Lie group theory
117
7.1
Linear Lie groups
............................ 117
7.2
Lie algebras
............................... 124
7.3
Geometry of the simplest linear groups
................. 129
8
Elements of representation theory
135
8.1
The basic notions of representation theory
............... 135
8.2
Representations of finite groups
.................... 140
8.3
On representations of Lie groups
.................... 147
9
Elements of
Poisson
and symplectic geometry
154
9.1
The
Poisson
bracket and Hamilton s equations
............. 154
9.2
Lagrangian formalism
.......................... 163
9.3
Examples of
Poisson
manifolds
..................... 166
9.4
Darboux s theorem and Liouville s theorem
.............. 170
9.5
Hamilton s variational principle
.................... 177
9.6
Reduction of the order of the system
.................. 180
9.7
Euler s equations
............................ 190
9.8
Integrable Hamiltonian
systems
.................... 194
Bibliography
205
Index
207
|
| adam_txt |
Contents
Preface
Part I Curves and surfaces
1
1
Theory of curves
3
1.1
Basic notions of the theory of curves
. 3
1.2
Plane curves
. 5
1.3
Curves in three-dimensional space
. 8
1.4
The orthogonal group
. 10
2
Theory of surfaces
15
2.1
Metrics on regular surfaces
. 15
2.2
Curvature of a curve on a surface
. 17
2.3
Gaussian curvature
. 20
2.4
Derivational equations and Bonnet's theorem
. 22
2.5
The Gauss theorem
. 27
2.6
Covariant derivative and geodesies
. 28
2.7
The Euler-Lagrange equations
. 32
2.8
The Gauss-Bonnet fonnula
. 38
2.9
Minimal surfaces
. 44
Part II Riemannian geometry
47
3
Smooth manifolds
49
3.1
Topological spaces
. 49
3.2
Smooth manifolds and maps
. 51
3.3
Tensors
. 58
3.4
Action of maps on tensors
. 63
3.5
Embedding of smooth manifolds into the Euclidean space
. 67
4
Riemannian manifolds
69
4.1
Metric tensor
. 69
4.2 Affine
connection and covariant derivative
. 70
4.3
Riemannian connections
. 74
4.4
Curvature
. 77
4.5
Geodesies
. 83
viii Contents
5
The Lobachevskii plane and the Minkowski space
89
5.1
The Lobachevskii plane
. 89
5.2
Pseudo-Euclidean spaces and their applications in physics
. 95
Partili
Supplement chapters
101
6
Minimal surfaces and complex analysis
103
6.1
Conformai
parameterization of surfaces
. 103
6.2
The theory of surfaces in terms of the
conformai
parameter
. 107
6.3
The
Weierstrass
representation
.
Ill
7
Elements of Lie group theory
117
7.1
Linear Lie groups
. 117
7.2
Lie algebras
. 124
7.3
Geometry of the simplest linear groups
. 129
8
Elements of representation theory
135
8.1
The basic notions of representation theory
. 135
8.2
Representations of finite groups
. 140
8.3
On representations of Lie groups
. 147
9
Elements of
Poisson
and symplectic geometry
154
9.1
The
Poisson
bracket and Hamilton's equations
. 154
9.2
Lagrangian formalism
. 163
9.3
Examples of
Poisson
manifolds
. 166
9.4
Darboux's theorem and Liouville's theorem
. 170
9.5
Hamilton's variational principle
. 177
9.6
Reduction of the order of the system
. 180
9.7
Euler's equations
. 190
9.8
Integrable Hamiltonian
systems
. 194
Bibliography
205
Index
207 |
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| illustrated | Illustrated |
| index_date | 2024-07-02T21:49:59Z |
| indexdate | 2024-07-09T21:20:37Z |
| institution | BVB |
| isbn | 9783037190500 |
| language | English Russian |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016700498 |
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| physical | viii, 211 Seite Illustrationen |
| publishDate | 2008 |
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| series2 | EMS series of lectures in mathematics |
| spelling | Tajmanov, Iskander Asanovič 1961- Verfasser (DE-588)1051919274 aut Lekcii po differencial'noj geometrii Lectures on differential geometry Iskander A. Taimanov Zürich European Mathematical Society [2008] © 2008 viii, 211 Seite Illustrationen txt rdacontent n rdamedia nc rdacarrier EMS series of lectures in mathematics Geometry, Differential Textbooks Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Differentialgeometrie (DE-588)4012248-7 s DE-604 Erscheint auch als Online-Ausgabe Taimanov, Iskander Asanovich Lectures on differential geometry 978-3-03719-550-5 (DE-604)BV036706065 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016700498&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Tajmanov, Iskander Asanovič 1961- Lectures on differential geometry Geometry, Differential Textbooks Differentialgeometrie (DE-588)4012248-7 gnd |
| subject_GND | (DE-588)4012248-7 (DE-588)4123623-3 |
| title | Lectures on differential geometry |
| title_alt | Lekcii po differencial'noj geometrii |
| title_auth | Lectures on differential geometry |
| title_exact_search | Lectures on differential geometry |
| title_exact_search_txtP | Lectures on differential geometry |
| title_full | Lectures on differential geometry Iskander A. Taimanov |
| title_fullStr | Lectures on differential geometry Iskander A. Taimanov |
| title_full_unstemmed | Lectures on differential geometry Iskander A. Taimanov |
| title_short | Lectures on differential geometry |
| title_sort | lectures on differential geometry |
| topic | Geometry, Differential Textbooks Differentialgeometrie (DE-588)4012248-7 gnd |
| topic_facet | Geometry, Differential Textbooks Differentialgeometrie Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016700498&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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