Lectures on surfaces: (almost) everything you wanted to know about them
Summary: Surfaces are among the most common and easily visualized mathematical objects, and their study brings into focus fundamental ideas, concepts, and methods from geometry, topology, complex analysis, Morse theory, and group theory. This book introduces many of the principal actors - the round...
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, RI
American Mathematical Society
2008
|
| Series: | Student mathematical library
46 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Summary: | Summary: Surfaces are among the most common and easily visualized mathematical objects, and their study brings into focus fundamental ideas, concepts, and methods from geometry, topology, complex analysis, Morse theory, and group theory. This book introduces many of the principal actors - the round sphere, flat torus, Mobius strip, and Klein bottle. |
| Physical Description: | XV, 286 S. graph. Darst. |
| ISBN: | 9780821846797 |
Staff View
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| adam_text | Contents
Foreword: MASS and REU at Penn State University xi
Preface xiii
Chapter 1. Various Ways of Representing Surfaces and
Basic Examples 1
Lecture 1 1
a. First examples 1
b. Equations vs. other methods 4
c. Planar models 8
d. Projective plane and flat torus as factor spaces 9
Lecture 2 11
a. Equations for surfaces and local coordinates 11
b. Other ways of introducing local coordinates 14
c. Parametric representations 16
d. Metrics on surfaces 17
Lecture 3 18
a. More about the Mobius strip and projective plane 18
b. A first glance at geodesies 20
c. Parametric representations of curves 22
d. Difficulties with representation by embedding 24
e. Regularity conditions for parametrically defined
surfaces 27
v
Lecture 4 28
a. Remarks on metric spaces and topology 28
b. Homeomorphisms and isometries 31
c. Other notions of dimension 32
d. Geodesies 33
Lecture 5 34
a. Isometries of the Euclidean plane 34
b. Isometries of the sphere and the elliptic plane 38
Lecture 6 39
a. Classification of isometries of the sphere and the
elliptic plane 39
b. Area of a spherical triangle 41
Lecture 7 43
a. Spaces with lots of isometries 43
b. Symmetric spaces 45
c. Remarks concerning direct products 47
Chapter 2. Combinatorial Structure and Topological
Classification of Surfaces 49
Lecture 8 49
a. Topology and combinatorial structure on surfaces 49
b. Triangulation 52
c. Euler characteristic 56
Lecture 9 58
a. Continuation of the proof of Theorem 2.4 58
b. Calculation of Euler characteristic 65
Lecture 10 67
a. Erom triangulations to maps 67
b. Examples 70
Lecture 11 73
a. Euler characteristic of planar models 73
b. Attaching handles 74
c. Orientability 77
d. Inverted handles and Mobius caps 79
Lecture 12 80
a. Non-orientable surfaces and Mobius caps 80
b. Calculation of Euler characteristic 81
c. Covering non-orientable surfaces 83
d. Classification of orientable surfaces 85
Lecture 13 86
a. Proof of the classification theorem 86
b. Non-orientable surfaces: Classification and models 91
Lecture 14 92
a. Chain complexes and Betti numbers 92
b. Homology of surfaces 94
c. A second interpretation of Euler characteristic 96
Lecture 15 98
a. Interpretation of the Betti numbers 98
b. Torsion in the first homology and non-orientability 100
c. Another derivation of interpretation of Betti numbers 101
Chapter 3. Differentiable Structure on Surfaces: Real and
Complex 103
Lecture 16 103
a. Charts and atlases 103
b. First examples of atlases 106
Lecture 17 109
a. Differentiable manifolds 109
b. DifFeomorphisms 110
c. More examples of charts and atlases 113
Lecture 18 117
a. Embedded surfaces 117
b. Gluing surfaces 117
c. Quotient spaces 118
d. Removing singularities 120
Lecture 19 121
a. Riemann surfaces: Definition and first examples 121
b. Holomorphic equivalence of Riemann surfaces 125
c. Conformal property of holomorphic functions and
invariance of angles on Riemann surfaces 127
d. Complex tori and the modular surface 129
Lecture 20 130
a. Differentiable functions on real surfaces 130
b. Morse functions 135
c. The third incarnation of Euler characteristic 138
Lecture 21 141
a. Functions with degenerate critical points 141
b. Degree of a circle map 145
c. Brouwer s fixed point theorem 149
Lecture 22 150
a. Zeroes of a vector field and their indices 150
b. Calculation of index 153
c. Tangent vectors, tangent spaces, and the tangent
bundle 155
Chapter 4. Riemannian Metrics and Geometry of Surfaces 159
Lecture 23 159
a. Definition of a Riemannian metric 159
b. Partitions of unity 163
Lecture 24 165
a. Existence of partitions of unity 165
b. Global properties from local and infinitesimal 169
c. Lengths, angles, and areas 170
Lecture 25 172
a. Geometry via a Riemannian metric 172
b. Differential equations 174
c. Geodesies 175
Lecture 26 178
a. First glance at curvature 178
b. The hyperbolic plane: two conformal models 181
c. Geodesies and distances on H2 186
Lecture 27 189
a. Detailed discussion of geodesies and isometries in the
upper half-plane model 189
b. The cross-ratio 193
c. Circles in the hyperbolic plane 196
Lecture 28 198
a. Three approaches to hyperbolic geometry 198
b. Characterisation of isometries 199
Lecture 29 204
a. Classification of isometries 204
b. Geometric interpretation of isometries 213
Lecture 30 217
a. Area of triangles in different geometries 217
b. Area and angular defect in hyperbolic geometry 218
Lecture 31 224
a. Hyperbolic metrics on surfaces of higher genus 224
b. Curvature, area, and Euler characteristic 228
Lecture 32 231
a. Geodesic polar coordinates 231
b. Curvature as an error term in the circle length formula 233
c. The Gauss-Bonnet Theorem 235
d. Comparison with traditional approach 240
Chapter 5. Topology and Smooth Structure Revisited 243
Lecture 33 243
a. Back to degree and index 243
b. The Fundamental Theorem of Algebra 246
Lecture 34 249
a. Jordan Curve Theorem 249
b. Another interpretation of genus 253
Lecture 35 255
a. A remark on tubular neighbourhoods 255
b. Proving the Jordan Curve Theorem 256
c. Poincare-Hopf Index Formula 259
Lecture 36 260
a. Proving the Poincare-Hopf Index Formula 260
b. Gradients and index formula for general functions 265
c. Fixed points and index formula for maps 267
d. The ubiquitous Euler characteristic 269
Suggested Reading 271
Hints 275
Index 283
|
| adam_txt |
Contents
Foreword: MASS and REU at Penn State University xi
Preface xiii
Chapter 1. Various Ways of Representing Surfaces and
Basic Examples 1
Lecture 1 1
a. First examples 1
b. Equations vs. other methods 4
c. Planar models 8
d. Projective plane and flat torus as factor spaces 9
Lecture 2 11
a. Equations for surfaces and local coordinates 11
b. Other ways of introducing local coordinates 14
c. Parametric representations 16
d. Metrics on surfaces 17
Lecture 3 18
a. More about the Mobius strip and projective plane 18
b. A first glance at geodesies 20
c. Parametric representations of curves 22
d. Difficulties with representation by embedding 24
e. Regularity conditions for parametrically defined
surfaces 27
v
Lecture 4 28
a. Remarks on metric spaces and topology 28
b. Homeomorphisms and isometries 31
c. Other notions of dimension 32
d. Geodesies 33
Lecture 5 34
a. Isometries of the Euclidean plane 34
b. Isometries of the sphere and the elliptic plane 38
Lecture 6 39
a. Classification of isometries of the sphere and the
elliptic plane 39
b. Area of a spherical triangle 41
Lecture 7 43
a. Spaces with lots of isometries 43
b. Symmetric spaces 45
c. Remarks concerning direct products 47
Chapter 2. Combinatorial Structure and Topological
Classification of Surfaces 49
Lecture 8 49
a. Topology and combinatorial structure on surfaces 49
b. Triangulation 52
c. Euler characteristic 56
Lecture 9 58
a. Continuation of the proof of Theorem 2.4 58
b. Calculation of Euler characteristic 65
Lecture 10 67
a. Erom triangulations to maps 67
b. Examples 70
Lecture 11 73
a. Euler characteristic of planar models 73
b. Attaching handles 74
c. Orientability 77
d. Inverted handles and Mobius caps 79
Lecture 12 80
a. Non-orientable surfaces and Mobius caps 80
b. Calculation of Euler characteristic 81
c. Covering non-orientable surfaces 83
d. Classification of orientable surfaces 85
Lecture 13 86
a. Proof of the classification theorem 86
b. Non-orientable surfaces: Classification and models 91
Lecture 14 92
a. Chain complexes and Betti numbers 92
b. Homology of surfaces 94
c. A second interpretation of Euler characteristic 96
Lecture 15 98
a. Interpretation of the Betti numbers 98
b. Torsion in the first homology and non-orientability 100
c. Another derivation of interpretation of Betti numbers 101
Chapter 3. Differentiable Structure on Surfaces: Real and
Complex 103
Lecture 16 103
a. Charts and atlases 103
b. First examples of atlases 106
Lecture 17 109
a. Differentiable manifolds 109
b. DifFeomorphisms 110
c. More examples of charts and atlases 113
Lecture 18 117
a. Embedded surfaces 117
b. Gluing surfaces 117
c. Quotient spaces 118
d. Removing singularities 120
Lecture 19 121
a. Riemann surfaces: Definition and first examples 121
b. Holomorphic equivalence of Riemann surfaces 125
c. Conformal property of holomorphic functions and
invariance of angles on Riemann surfaces 127
d. Complex tori and the modular surface 129
Lecture 20 130
a. Differentiable functions on real surfaces 130
b. Morse functions 135
c. The third incarnation of Euler characteristic 138
Lecture 21 141
a. Functions with degenerate critical points 141
b. Degree of a circle map 145
c. Brouwer's fixed point theorem 149
Lecture 22 150
a. Zeroes of a vector field and their indices 150
b. Calculation of index 153
c. Tangent vectors, tangent spaces, and the tangent
bundle 155
Chapter 4. Riemannian Metrics and Geometry of Surfaces 159
Lecture 23 159
a. Definition of a Riemannian metric 159
b. Partitions of unity 163
Lecture 24 165
a. Existence of partitions of unity 165
b. Global properties from local and infinitesimal 169
c. Lengths, angles, and areas 170
Lecture 25 172
a. Geometry via a Riemannian metric 172
b. Differential equations 174
c. Geodesies 175
Lecture 26 178
a. First glance at curvature 178
b. The hyperbolic plane: two conformal models 181
c. Geodesies and distances on H2 186
Lecture 27 189
a. Detailed discussion of geodesies and isometries in the
upper half-plane model 189
b. The cross-ratio 193
c. Circles in the hyperbolic plane 196
Lecture 28 198
a. Three approaches to hyperbolic geometry 198
b. Characterisation of isometries 199
Lecture 29 204
a. Classification of isometries 204
b. Geometric interpretation of isometries 213
Lecture 30 217
a. Area of triangles in different geometries 217
b. Area and angular defect in hyperbolic geometry 218
Lecture 31 224
a. Hyperbolic metrics on surfaces of higher genus 224
b. Curvature, area, and Euler characteristic 228
Lecture 32 231
a. Geodesic polar coordinates 231
b. Curvature as an error term in the circle length formula 233
c. The Gauss-Bonnet Theorem 235
d. Comparison with traditional approach 240
Chapter 5. Topology and Smooth Structure Revisited 243
Lecture 33 243
a. Back to degree and index 243
b. The Fundamental Theorem of Algebra 246
Lecture 34 249
a. Jordan Curve Theorem 249
b. Another interpretation of genus 253
Lecture 35 255
a. A remark on tubular neighbourhoods 255
b. Proving the Jordan Curve Theorem 256
c. Poincare-Hopf Index Formula 259
Lecture 36 260
a. Proving the Poincare-Hopf Index Formula 260
b. Gradients and index formula for general functions 265
c. Fixed points and index formula for maps 267
d. The ubiquitous Euler characteristic 269
Suggested Reading 271
Hints 275
Index 283 |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516 |
| dewey-search | 516 |
| dewey-sort | 3516 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV035128515 |
| illustrated | Illustrated |
| index_date | 2024-07-02T22:23:49Z |
| indexdate | 2024-07-09T21:22:58Z |
| institution | BVB |
| isbn | 9780821846797 |
| language | English |
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| series | Student mathematical library |
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| spelling | Katok, Anatole 1944-2018 Verfasser (DE-588)111187125 aut Lectures on surfaces (almost) everything you wanted to know about them Anatole Katok ; Vaughn Climenhaga Providence, RI American Mathematical Society 2008 XV, 286 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Student mathematical library 46 Summary: Surfaces are among the most common and easily visualized mathematical objects, and their study brings into focus fundamental ideas, concepts, and methods from geometry, topology, complex analysis, Morse theory, and group theory. This book introduces many of the principal actors - the round sphere, flat torus, Mobius strip, and Klein bottle. Geometri Flächentheorie (DE-588)4475211-8 gnd rswk-swf Flächentheorie (DE-588)4475211-8 s DE-604 Climenhaga, Vaughn 1982- Verfasser (DE-588)139069801 aut Student mathematical library 46 (DE-604)BV013184751 46 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016796056&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Katok, Anatole 1944-2018 Climenhaga, Vaughn 1982- Lectures on surfaces (almost) everything you wanted to know about them Student mathematical library Geometri Flächentheorie (DE-588)4475211-8 gnd |
| subject_GND | (DE-588)4475211-8 |
| title | Lectures on surfaces (almost) everything you wanted to know about them |
| title_auth | Lectures on surfaces (almost) everything you wanted to know about them |
| title_exact_search | Lectures on surfaces (almost) everything you wanted to know about them |
| title_exact_search_txtP | Lectures on surfaces (almost) everything you wanted to know about them |
| title_full | Lectures on surfaces (almost) everything you wanted to know about them Anatole Katok ; Vaughn Climenhaga |
| title_fullStr | Lectures on surfaces (almost) everything you wanted to know about them Anatole Katok ; Vaughn Climenhaga |
| title_full_unstemmed | Lectures on surfaces (almost) everything you wanted to know about them Anatole Katok ; Vaughn Climenhaga |
| title_short | Lectures on surfaces |
| title_sort | lectures on surfaces almost everything you wanted to know about them |
| title_sub | (almost) everything you wanted to know about them |
| topic | Geometri Flächentheorie (DE-588)4475211-8 gnd |
| topic_facet | Geometri Flächentheorie |
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