Curved spaces: from classical geometries to elementary differential geometry
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2008
|
| Edition: | 1. publ. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | X, 186 S. Ill., graph. Darst. |
| ISBN: | 9780521886291 0521886295 9780521713900 0521713900 |
Staff View
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| 245 | 1 | 0 | |a Curved spaces |b from classical geometries to elementary differential geometry |c P. M. H. Wilson |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2008 | |
| 300 | |a X, 186 S. |b Ill., graph. Darst. | ||
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Record in the Search Index
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| adam_text | Contents
Preface
page
ix
Euclidean geometry
1
1.1
Euclidean space
1
4
9
11
15
17
22
25
25
26
29
31
34
39
42
45
47
Triangulations
and
Euler
numbers
51
3.1
Geometry of the
toras
51
3.2
Triangulations
55
3.3
Polygonal decompositions
59
3.4
Topology of the g-holed
toras
62
Exercises
67
Appendix on polygonal approximations
68
Riemannian metrics
75
4.1
Revision on derivatives and the Chain Rule
75
4.2
Riemannian metrics on open subsets of R2
79
1.2
Isometries
1.3
The group O(3,R)
1.4
Curves and their lengths
1.5
Completeness and compactness
1.6
Polygons in the Euclidean plane
Exercises
2
Spherical geometry
2.1
Introduction
2.2
Spherical triangles
2.3
Curves on the sphere
2.4
Finite groups of isometries
2.5
Gauss-Bonnet and spherical polygons
2.6
Möbius
geometry
2.7
The double cover of SO(3)
2.8
Circles on S2
Exercises
vin
CONTENTS
4.3
Lengths of curves
82
4.4
Isometries and areas
85
Exercises °
5
Hyperbolic geometry
89
5.1
Poincaré
models for the hyperbolic plane
89
5.2
Geometry of the upper half-plane model
H
92
5.3
Geometry of the disc model
D
96
5.4
Reflections in hyperbolic lines
98
5.5
Hyperbolic triangles
102
5.6
Parallel and
ultraparallel
lines
105
5.7 Hyperboloid
model of the hyperbolic plane
107
Exercises
112
6
Smooth embedded surfaces
115
6.1
Smooth parametrizations
115
6.2
Lengths and areas
118
6.3
Surfaces of revolution
121
6.4
Gaussian curvature of embedded surfaces
123
Exercises
130
7
Geodesies
133
7.1
Variations of smooth curves
133
7.2
Geodesies on embedded surfaces
138
7.3
Length and energy
140
7.4
Existence of geodesies
141
7.5
Geodesic polars and Gauss s lemma
144
Exercises
150
8
Abstract surfaces and Gauss-Bonnet
153
8.1
Gauss s
Theorema
Egregium
153
8.2
Abstract smooth surfaces and isometries
155
8.3
Gauss-Bonnet for geodesic triangles
159
8.4
Gauss-Bonnet for general closed surfaces
165
8.5
Plumbing joints and building blocks
170
Exercises
175
Postscript
177
References
179
Index
181
|
| adam_txt |
Contents
Preface
page
ix
Euclidean geometry
1
1.1
Euclidean space
1
4
9
11
15
17
22
25
25
26
29
31
34
39
42
45
47
Triangulations
and
Euler
numbers
51
3.1
Geometry of the
toras
51
3.2
Triangulations
55
3.3
Polygonal decompositions
59
3.4
Topology of the g-holed
toras
62
Exercises
67
Appendix on polygonal approximations
68
Riemannian metrics
75
4.1
Revision on derivatives and the Chain Rule
75
4.2
Riemannian metrics on open subsets of R2
79
1.2
Isometries
1.3
The group O(3,R)
1.4
Curves and their lengths
1.5
Completeness and compactness
1.6
Polygons in the Euclidean plane
Exercises
2
Spherical geometry
2.1
Introduction
2.2
Spherical triangles
2.3
Curves on the sphere
2.4
Finite groups of isometries
2.5
Gauss-Bonnet and spherical polygons
2.6
Möbius
geometry
2.7
The double cover of SO(3)
2.8
Circles on S2
Exercises
vin
CONTENTS
4.3
Lengths of curves
82
4.4
Isometries and areas
85
Exercises °
5
Hyperbolic geometry
89
5.1
Poincaré
models for the hyperbolic plane
89
5.2
Geometry of the upper half-plane model
H
92
5.3
Geometry of the disc model
D
96
5.4
Reflections in hyperbolic lines
98
5.5
Hyperbolic triangles
102
5.6
Parallel and
ultraparallel
lines
105
5.7 Hyperboloid
model of the hyperbolic plane
107
Exercises
112
6
Smooth embedded surfaces
115
6.1
Smooth parametrizations
115
6.2
Lengths and areas
118
6.3
Surfaces of revolution
121
6.4
Gaussian curvature of embedded surfaces
123
Exercises
130
7
Geodesies
133
7.1
Variations of smooth curves
133
7.2
Geodesies on embedded surfaces
138
7.3
Length and energy
140
7.4
Existence of geodesies
141
7.5
Geodesic polars and Gauss's lemma
144
Exercises
150
8
Abstract surfaces and Gauss-Bonnet
153
8.1
Gauss's
Theorema
Egregium
153
8.2
Abstract smooth surfaces and isometries
155
8.3
Gauss-Bonnet for geodesic triangles
159
8.4
Gauss-Bonnet for general closed surfaces
165
8.5
Plumbing joints and building blocks
170
Exercises
175
Postscript
177
References
179
Index
181 |
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| spelling | Wilson, Pelham M. H. Verfasser (DE-588)173983723 aut Curved spaces from classical geometries to elementary differential geometry P. M. H. Wilson 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2008 X, 186 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Curves, Algebraic Curves, Algebraic / Problems, exercises, etc Curves, Algebraic Problems, exercises, etc Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf (DE-588)4151278-9 Einführung gnd-content Differentialgeometrie (DE-588)4012248-7 s DE-604 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016411029&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Wilson, Pelham M. H. Curved spaces from classical geometries to elementary differential geometry Curves, Algebraic Curves, Algebraic / Problems, exercises, etc Curves, Algebraic Problems, exercises, etc Differentialgeometrie (DE-588)4012248-7 gnd |
| subject_GND | (DE-588)4012248-7 (DE-588)4151278-9 |
| title | Curved spaces from classical geometries to elementary differential geometry |
| title_auth | Curved spaces from classical geometries to elementary differential geometry |
| title_exact_search | Curved spaces from classical geometries to elementary differential geometry |
| title_exact_search_txtP | Curved spaces from classical geometries to elementary differential geometry |
| title_full | Curved spaces from classical geometries to elementary differential geometry P. M. H. Wilson |
| title_fullStr | Curved spaces from classical geometries to elementary differential geometry P. M. H. Wilson |
| title_full_unstemmed | Curved spaces from classical geometries to elementary differential geometry P. M. H. Wilson |
| title_short | Curved spaces |
| title_sort | curved spaces from classical geometries to elementary differential geometry |
| title_sub | from classical geometries to elementary differential geometry |
| topic | Curves, Algebraic Curves, Algebraic / Problems, exercises, etc Curves, Algebraic Problems, exercises, etc Differentialgeometrie (DE-588)4012248-7 gnd |
| topic_facet | Curves, Algebraic Curves, Algebraic / Problems, exercises, etc Curves, Algebraic Problems, exercises, etc Differentialgeometrie Einführung |
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