Variational problems in topology: the geometry of length, area and volume
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English Russian |
| Published: |
New York u.a.
Gordon and Breach Science Publ.
1990
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Aus d. Russ. übers. |
| Physical Description: | X, 225 S. Ill. |
| ISBN: | 2881247407 |
Staff View
MARC
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| 245 | 1 | 0 | |a Variational problems in topology |b the geometry of length, area and volume |c A. T. Fomenko |
| 264 | 1 | |a New York u.a. |b Gordon and Breach Science Publ. |c 1990 | |
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Record in the Search Index
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| adam_text | CONTENTS
Preface jx
Chapter I. PRELIMINARIES 1
1. Singular and Cellular Homology Croups 1
1. Singular chains and homology groups 1
2. Cell complexes and barycentric subdivisions 4
3. Cellular homology and the computation of singular
homology of the sphere 9
4. Theorem on the coincidence of singular homology
with the cellular homology of a finite cell complex 11
5. Geometric definition of cellular homology groups 13
6. Simplest examples of the computation of cellular
homology groups 15
2. Cohomology Groups and Obstructions to Extending
Mappings 17
1. Singular cochains and the operator S 17
2. The problem of the extension of a continuous mapping
from a subspace to the whole space 19
3. The obstruction to extending mappings 19
4. Cases of the retraction of a space on a subspace
homeomorphic to a sphere 23
3. Fibrations 28
1. Defining a locally trivial fibration 28
2. Examples of fibrations 29
3. The Hopf bundle geometry 30
4. Geometry of the fibration of unit vectors tangent to
a sphere and the index of the vector field 34
Chapter II. FUNCTIONS ON MANIFOLDS 41
1. Exact Morse Functions 41
2. Multidimensional Analog of Morse Theory 47
Chapter III.MANIFOLDS OF SMALL DIMENSIONS 55
1. Homeomorphisms of Two Dimensional Surfaces 55
1. Homcotopy group and classification of
three dimensional manifolds of genus one 55
2. A system of homeotopy group generators of
a complete pretzel 61
V
vi CONTENTS
2. An Algorithm Recognizing the Standard
Three Dimensional Sphere in the Class of Heegard
Diagrams of Genus Two 63
1. The Whitehead graphs encoding three dimensional
manifolds. Operations of indices one and two 63
2. Waves separating the vertices and the operation of
reducing Whitehead graphs 69
3. The existence of a separating vertex on the Whitehead
graphs of genus two, corresponding to the standard
three dimensional sphere. An algorithm recognizing
a sphere 72
3. On Solving the Four Dimensional Poincare Problem: Any
Four Dimensional Homotopy Sphere is Homeomorphic to
the Standard Sphere 74
Chapter IV. MINIMAL SURFACES 93
1. Simplest Properties of Minimal Surfaces 93
1. Plateau physical experiments and methods of
obtaining soap films 93
2. Physical principles underlying the formation of soap
films 96
3. Extremal properties of soap films and minimality of
their area—Properties of the surface of separation
between two media 98
4. The surface of separation between two media which
are in equilibrium is the surface of constant mean
curvature 100
5. Soap films of constant positive curvature and 103
constant zero curvature
6. Stable and unstable surfaces 104
7. Plateau experiments on the stability of a liquid column 105
8. Physical realization of a helicoid 109
9. Physical realization of a catenoid and its surgery with
changing boundary frame 113
10. The change of the topological type of minimal surfaces
depending on their stability or instability 116
11. Two dimensional minimal surfaces in
a three dimensional space and the first Plateau
principle 122
12. Area functional, Dirichlet functional, harmonic
mappings, and conformal coordinates 124
13. Singular points of minimal surfaces and three Plateau
principles 137
14. Realization of minimal surfaces in nature 144
CONTENTS vii
2. Topological Properties of Minimal Surfaces 146
1. On various aproaches to the concepts of surface and
boundary 146
2. Homological boundary of a surface and the role of the
coefficient group 148
3. Curious examples of physical stable minimal surfaces,
nevertheless retracting on their boundaries 151
3* Realization of the Bing house in the form of
a minimal surface in R3 159
4. When does the soap film spanning a boundary
frame not contain closed soap bubbles? 166
5. Minimal cones related to singular points of minimal
surfaces 169
6. Multidimensional minimal cones 172
7. Minimal surfaces invariant under the action of Lie
groups 175
8. Fermat principle, minimal cones, and light rays 179
9. The S. N. Bernshtein problem 187
10. Complex submanifolds as minimal surfaces 189
11. Integral currents 190
12. Spectral bordisms and the multidimensional Plateau
problem 193
13. Existence of a minimum in each homotopy class of
multivarifolds 197
14. Cases where the Dirichlet problem has no solution for
the minimal surface equation of large codimension 198
3. Geometry of Volume Functional and Dirichlet Functional
Extremals 203
1. Lower estimate of the volume of minimal surfaces 203
2. Vector field deformation coefficient 206
3. Closed surfaces of non trivial topological type and
least volume 207
4. The cases where there are no local minima of the
Dirichlet functional in the class of mappings of
homogeneous spaces to an arbitrary Riemannian
manifold 210
5. Cases where there are no Dirichlet functional local
minima in the class of mappings of an arbitrary
Riemannian manifold to a homogeneous space 213
References 216
Index 223
|
| any_adam_object | 1 |
| author | Fomenko, Anatolij Timofeevič 1945- |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 514 - Topology |
| dewey-raw | 514 |
| dewey-search | 514 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV004239726 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T16:10:15Z |
| institution | BVB |
| isbn | 2881247407 |
| language | English Russian |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-002637560 |
| oclc_num | 20826023 |
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| owner | DE-12 DE-824 DE-384 DE-83 DE-188 |
| owner_facet | DE-12 DE-824 DE-384 DE-83 DE-188 |
| physical | X, 225 S. Ill. |
| publishDate | 1990 |
| publishDateSearch | 1990 |
| publishDateSort | 1990 |
| publisher | Gordon and Breach Science Publ. |
| record_format | marc |
| spelling | Fomenko, Anatolij Timofeevič 1945- Verfasser (DE-588)119092689 aut Topologičeskie variacionnye zadači Variational problems in topology the geometry of length, area and volume A. T. Fomenko New York u.a. Gordon and Breach Science Publ. 1990 X, 225 S. Ill. txt rdacontent n rdamedia nc rdacarrier Aus d. Russ. übers. Inégalités variationnelles Inéquations variationnelles (Mathématiques) ram Topologie Topologie ram Topology Variational inequalities (Mathematics) Topologie (DE-588)4060425-1 gnd rswk-swf Variationsungleichung (DE-588)4187420-1 gnd rswk-swf Topologie (DE-588)4060425-1 s Variationsungleichung (DE-588)4187420-1 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002637560&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Fomenko, Anatolij Timofeevič 1945- Variational problems in topology the geometry of length, area and volume Inégalités variationnelles Inéquations variationnelles (Mathématiques) ram Topologie Topologie ram Topology Variational inequalities (Mathematics) Topologie (DE-588)4060425-1 gnd Variationsungleichung (DE-588)4187420-1 gnd |
| subject_GND | (DE-588)4060425-1 (DE-588)4187420-1 |
| title | Variational problems in topology the geometry of length, area and volume |
| title_alt | Topologičeskie variacionnye zadači |
| title_auth | Variational problems in topology the geometry of length, area and volume |
| title_exact_search | Variational problems in topology the geometry of length, area and volume |
| title_full | Variational problems in topology the geometry of length, area and volume A. T. Fomenko |
| title_fullStr | Variational problems in topology the geometry of length, area and volume A. T. Fomenko |
| title_full_unstemmed | Variational problems in topology the geometry of length, area and volume A. T. Fomenko |
| title_short | Variational problems in topology |
| title_sort | variational problems in topology the geometry of length area and volume |
| title_sub | the geometry of length, area and volume |
| topic | Inégalités variationnelles Inéquations variationnelles (Mathématiques) ram Topologie Topologie ram Topology Variational inequalities (Mathematics) Topologie (DE-588)4060425-1 gnd Variationsungleichung (DE-588)4187420-1 gnd |
| topic_facet | Inégalités variationnelles Inéquations variationnelles (Mathématiques) Topologie Topology Variational inequalities (Mathematics) Variationsungleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002637560&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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