Visual geometry and topology:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English Russian |
| Published: |
Berlin u.a.
Springer
1994
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Aus dem Russ. übers. |
| Physical Description: | XVI, 324 S. Ill., graph. Darst. |
| ISBN: | 3540533613 0387533613 |
Staff View
MARC
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| 100 | 1 | |a Fomenko, Anatolij Timofeevič |d 1945- |e Verfasser |0 (DE-588)119092689 |4 aut | |
| 240 | 1 | 0 | |a Nagljadnaja geometrija i topologija |
| 245 | 1 | 0 | |a Visual geometry and topology |c Anatolij Fomenko |
| 264 | 1 | |a Berlin u.a. |b Springer |c 1994 | |
| 300 | |a XVI, 324 S. |b Ill., graph. Darst. | ||
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Record in the Search Index
| _version_ | 1804122121100591104 |
|---|---|
| adam_text | Table of Contents
1 Polyhedra. Simplicial Complexes. Homologies
1.1 Polyhedra 1
1.1.1 Introductory Remarks 1
1.1.2 The Concept of an n Dimensional Simplex
Barycentric Coordinates 5
1.1.3 Polyhedra. Simplicial Subdivisions of Polyhedra.
Simplicial Complexes 8
1.1.4 Examples of Polyhedra 10
1.1.5 Barycentric Subdivision 14
1.1.6 Visual Material 16
1.2 Simplicial Homology Groups of Simplicial Complexes
(Polyhedra) 18
1.2.1 Simplicial Chains 18
1.2.2 Chain Boundary 23
1.2.3 The Simplest Properties of the Boundary
Operator Cycles. Boundaries 26
1.2.4 Examples of Calculations of the Boundary Operator .... 27
1.2.5 Simplicial Homology Groups 29
1.2.6 Examples of Calculations of Homology Groups.
Homologies of Two dimensional Surfaces 32
1.2.7 Visual Material 46
1.3 General Properties of Simplicial Homology Groups . . 49
1.3.1 Incidence Matrices 49
1.3.2 The Method of Calculation of Homology Groups
Using Incidence Matrices 50
XII Table of Contents
1.3.3 Traces of Cell Homologies Inside Simplicial Ones ... 56
1.3.4 Chain Homotopy. Independence of Simplicial Homologies
of a Polyhedron of the Choice of Triangulation 59
1.3.5 Visual Material 65
2 Low Dimensional Manifolds
2.1 Basic Concepts of Differential Geometry 75
2.1.1 Coordinates in a Region.
Transformations of Curvilinear Coordinates 75
2.1.2 The Concept of a Manifold. Smooth Manifolds.
Submanifolds and Ways of Defining Them. Manifolds
with Boundary. Tangent Space and Tangent Bundle .... 79
2.1.3 Orientability and Non Orientability. The Differential
of a Mapping. Regular Values and Regular Points.
Embeddings and Immersions of Manifolds.
Critical Points of Smooth Functions on Manifolds.
Index of Nondegenerate Critical Points
and Morse Functions 85
2.1.4 Vector and Covector Fields. Integral Trajectories.
Vector Field Commutators. The Lie Algebra
of Vector Fields on a Manifold 91
2.1.5 Visual Material 95
2.2 Visual Properties of One Dimensional Manifolds .... 98
2.2.1 Isotopies, Frames 98
2.2.2 Visual Material 102
2.3 Visual Properties of Two Dimensional Manifolds . . . Ill
2.3.1 Two Dimensional Manifolds with Boundary Ill
2.3.2 Examples of Two Dimensional Manifolds 113
2.3.3 Modelling of a Projective Plane
in a Three Dimensional Space 115
2.3.4 Two Series of Two Dimensional Closed Manifolds . . . .120
Table of Contents XIII
2.3.5 Classification of Closed 2 Manifolds 125
2.3.6 Inversion of a Two Dimensional Sphere 128
2.3.7 Visual Material 130
2.4 Cohomology Groups and Differential Forms 135
2.4.1 Differential 1 Forms on a Smooth Manifold 135
2.4.2 Closed and Exact Forms on a Two Dimensional Manifold 136
2.4.3 An Important Property of Cohomology Groups 138
2.4.4 Direct Calculation of One Dimensional Cohomology
Groups of One Dimensional Manifolds 139
2.4.5 Direct Calculation of One Dimensional Cohomology
Groups of a Plane, a Two Dimensional Sphere
and a Torus 141
2.4.6 Direct Calculation of One Dimensional Cohomology
Groups of Oriented Surfaces, i.e. Spheres with Handles . . 148
2.4.7 An Algorithm for Recognition of Two Dimensional
Manifolds. Elements of Two Dimensional
Computer Geometry 152
2.4.8 Calculation of One Dimensional Cohomologies
of a Surface Using Triangulation 154
2.4.9 Visual Material 154
2.5 Visual Properties of Three Dimensional Manifolds . . .159
2.5.1 Heegaard Splittings (or Diagrams) 159
2.5.2 Examples of Three Dimensional Manifolds 162
2.5.3 Equivalence of Heegaard Splittings 164
2.5.4 Spines 165
2.5.5 Special Spines 168
2.5.6 Filtration of 3 Manifolds with Respect to Matveev s
Complexity 170
2.5.7 Simplification of Special Spines 173
2.5.8 The Use of Computers in Three Dimensional Topology.
Enumeration of Manifolds in Increasing Order
of Complexity 177
2.5.9 Matveev s Complexity of 3 Manifolds
and Simplex Glueings 184
2.5.10 Visual Material 188
XIV Table of Contents
3 Visual Symplectic Topology
and Visual Hamiltonian Mechanics
3.1 Some Concepts of Hamiltonian Geometry 193
3.1.1 Hamiltonian Systems on Symplectic Manifolds 193
3.1.2 Involutive Integrals and Liouville Tori 196
3.1.3 Momentum Mapping of an Integrable System 200
3.1.4 Surgery on Liouville Tori at Critical Energy Values . . . .200
3.1.5 Visual Material 204
3.2 Qualitative Questions of Geometric Integration
of Some Differential Equations. Classification
of Typical Surgeries of Liouville Tori
of Integrable Systems with Bott Integrals 208
3.2.1 Nondegenerate (Bott) Integrals 208
3.2.2 Classification of Surgeries of Bott Position
on Liouville Tori 210
3.2.3 The Topological Structure of Critical Energy Levels
at a Fixed Second Integral 215
3.2.4 Examples from Mechanics. The Equations of Motion
of a Rigid Body. The Poisson Sphere.
Geometrical Interpretation of Mechanical Systems 216
3.2.5 An Example of an Investigation of a Mechanical System.
The Liouville System on the Plane 219
3.2.6 The Liouville System on the Sphere 220
3.2.7 Inertial Motion of a Gyrostat 221
3.2.8 The Case of Chaplygin Sretensky 223
3.2.9 The Case of Kovalevskaya 224
3.2.10 Visual Material 225
3.3 Three Dimensional Manifolds and Visual Geometry
of Isoenergy Surfaces of Integrable Systems 231
3.3.1 A One Dimensional Graph as a Hamiltonian Diagram . . .231
3.3.2 What Familiar Manifolds Are Encountered
Among Isoenergy Surfaces? 234
Table of Contents XV
3.3.3 The Simplest Isoenergy Surfaces (with Boundary) 243
3.3.4 Any Isoenergy Surface of an Integrable Nondegenerate
System Falls into the Sum of Five (or Two) Types
of Elementary Bricks 244
3.3.5 New Topological Properties of the Isoenergy
Surfaces Class 246
3.3.6 One Example of a Computer Use in Symplectic Topology 249
3.3.7 Visual Material 252
4 Visual Images in Some Other Fields of Geometry
and Its Applications
4.1 Visual Geometry of Soap Films. Minimal Surfaces . . .255
4.1.1 Boundaries Between Physical Media. Minimal Surfaces . .255
4.1.2 Some Examples of Minimal Surfaces 258
4.1.3 Visual Material 260
4.2 Fractal Geometry and Homeomorphisms 264
4.2.1 Various Concepts of Dimension 264
4.2.2 Fractals 267
4.2.3 Homeomorphisms 268
4.2.4 Visual Material 276
4.3 Visual Computer Geometry in the Number Theory . . 284
Appendix 1 Visual Geometry of Some Natural
and Nonholonomic Systems
1.1 On Projection of Liouville Tori in Systems
with Separation of Variables 293
1.2 What Are Nonholonomic Constraints? 295
1.3 The Variety of Manifolds in the Suslov Problem 297
XVI Table of Contents
Appendix 2 Visual Hyperbolic Geometry
2.1 Discrete Groups and Their Fundamental Region 300
2.2 Discrete Groups Generated by Reflections in the Plane . . 301
2.3 The Gram Matrix and the Coxeter Scheme 303
2.4 Reflection Generated Discrete Groups in Space 303
2.5 A Model of the Lobachevskian Plane 306
2.6 Convex Polygons on the Lobachevskian Plane 308
2.7 Coxeter Polygons on the Lobachevskian Plane 309
2.8 Coxeter Polyhedra in the Lobachevskian Space 310
2.9 Discrete Groups of Motions of Lobachevskian Space
and Groups of Integer Valued Automorphisms
of Hyperbolic Quadratic Forms 314
2.10 Reflection Generated Discrete Groups
in High Dimensional Lobachevskian Spaces 315
References 317
|
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| id | DE-604.BV007749911 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T17:08:50Z |
| institution | BVB |
| isbn | 3540533613 0387533613 |
| language | English Russian |
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| physical | XVI, 324 S. Ill., graph. Darst. |
| publishDate | 1994 |
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| spelling | Fomenko, Anatolij Timofeevič 1945- Verfasser (DE-588)119092689 aut Nagljadnaja geometrija i topologija Visual geometry and topology Anatolij Fomenko Berlin u.a. Springer 1994 XVI, 324 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Aus dem Russ. übers. Meetkunde gtt Topologie gtt Geometry Topology Geometrie (DE-588)4020236-7 gnd rswk-swf Visualisierung (DE-588)4188417-6 gnd rswk-swf Topologie (DE-588)4060425-1 gnd rswk-swf Geometrie (DE-588)4020236-7 s Visualisierung (DE-588)4188417-6 s DE-604 Topologie (DE-588)4060425-1 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005094246&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Fomenko, Anatolij Timofeevič 1945- Visual geometry and topology Meetkunde gtt Topologie gtt Geometry Topology Geometrie (DE-588)4020236-7 gnd Visualisierung (DE-588)4188417-6 gnd Topologie (DE-588)4060425-1 gnd |
| subject_GND | (DE-588)4020236-7 (DE-588)4188417-6 (DE-588)4060425-1 |
| title | Visual geometry and topology |
| title_alt | Nagljadnaja geometrija i topologija |
| title_auth | Visual geometry and topology |
| title_exact_search | Visual geometry and topology |
| title_full | Visual geometry and topology Anatolij Fomenko |
| title_fullStr | Visual geometry and topology Anatolij Fomenko |
| title_full_unstemmed | Visual geometry and topology Anatolij Fomenko |
| title_short | Visual geometry and topology |
| title_sort | visual geometry and topology |
| topic | Meetkunde gtt Topologie gtt Geometry Topology Geometrie (DE-588)4020236-7 gnd Visualisierung (DE-588)4188417-6 gnd Topologie (DE-588)4060425-1 gnd |
| topic_facet | Meetkunde Topologie Geometry Topology Geometrie Visualisierung |
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