Modern geometry: methods and applications 2 The geometry and topology of manifolds
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York u.a.
Springer
1985
|
| Series: | Graduate texts in mathematics
104 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 430 S. graph. Darst. |
| ISBN: | 0387961623 |
Staff View
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| 240 | 1 | 0 | |a Sovremennaja geometrija |
| 245 | 1 | 0 | |a Modern geometry |b methods and applications |n 2 |p The geometry and topology of manifolds |c B. A. Dubrovin ; A. T. Fomenko ; S. P. Novikov |
| 264 | 1 | |a New York u.a. |b Springer |c 1985 | |
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| 337 | |b n |2 rdamedia | ||
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Record in the Search Index
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Contents
CHAPTER
1
Examples of Manifolds
1
§1.
The concept of a manifold
1
1.1.
Definition of a manifold
1
1.2.
Mappings of manifolds; tensors on manifolds
5
1.3.
Embeddings and immersions of manifolds. Manifolds with
boundary
9
§2.
The simplest examples of manifolds
10
2.1.
Surfaces in Euclidean space. Transformation groups as manifolds
10
2.2.
Projectíve
spaces
15
2.3.
Exercises
19
§3.
Essential facts from the theory of Lie groups
20
3.1.
The structure of a neighbourhood of the identity of a Lie group.
The Lie algebra of a Lie group. Semisimplicity
20
3.2.
The concept of a linear representation. An example of a
non-matrix Lie group
28
§4.
Complex manifolds
31
4.1.
Definitions and examples
31
4.2.
Riemann surfaces as manifolds
37
§5.
The simplest homogeneous spaces
41
5.1.
Action of a group on a manifold
41
5.2.
Examples of homogeneous spaces
42
5.3.
Exercises
46
§6.
Spaces of constant curvature (symmetric spaces)
46
6.1.
The concept of a symmetric space
46
6.2.
The isometry group of a manifold. Properties of its Lie algebra
49
6.3.
Symmetric spaces of the first and second types
51
6.4.
Lie groups as symmetric spaces
53
6.5.
Constructing symmetric spaces. Examples
55
6.6.
Exercises
58
xii Contents
§7.
Vector
bundles on a manifold
59
7.1.
Constructions involving tangent vectors
59
7.2.
The normal vector bundle on a submanifold
62
CHAPTER
2
Foundational Questions. Essential Facts Concerning Functions
on a Manifold. Typical Smooth Mappings
65
§8.
Partitions of unity and their applications
65
8.1.
Partitions of unity
66
8.2.
The simplest applications of partitions of unity. Integrals over a
manifold and the general Stokes formula
69
8.3.
Invariant metrics
74
§9.
The realization of compact manifolds as surfaces in
¥Ѓ
76
§10.
Various properties of smooth maps of manifolds
77
10.1.
Approximation of continuous mappings by smooth ones
77
10.2.
Sard's theorem
79
10.3.
Transversal regularity
83
10.4.
Morse functions
86
§11.
Applications of Sard's theorem
90
11.1.
The existence of embeddings and immersions
90
11.2.
The construction of Morse functions as height functions
93
11.3.
Focal points
95
CHAPTER
3
The Degree of a Mapping. The Intersection Index of Submanifolds.
Applications
99
§12.
The concept of homotopy
99
12.1.
Definition of homotopy. Approximation of continuous maps
and homotopies by smooth ones
99
12.2.
Relative homotopies
102
§13.
The degree of a map
102
13.1.
Definition of degree
102
13.2.
Generalizations of the concept of degree
104
13.3.
Classification of homotopy classes of maps from an arbitrary
manifold to a sphere
106
13.4.
The simplest examples
108
§14.
Applications of the degree of a mapping
110
14.1.
The relationship between degree and integral
110
14.2.
The degree of a vector field on a hypersurface
112
14.3.
The Whitney number. The Gauss-Bonnet formula
114
14.4.
The index of a singular point of a vector field
118
14.5.
Transverse surfaces of a vector field. The
Poincaré-Bendixson
theorem
122
§15.
The intersection index and applications
125
15.1.
Definition of the intersection index
125
15.2.
The total index of a vector field
127
Contents
x¡ji
15.3.
The signed number of fixed points of a self-map (the Lefschetz
number). The
Brouwer
fixed-point theorem
130
15.4.
The linking coefficient
133
CHAPTER
4
Orientability of Manifolds. The Fundamental Group.
Covering Spaces (Fibre Bundles with Discrete Fibre)
135
§16.
Orientability and homotopies of closed paths
135
16.1.
Transporting an orientation along a path
135
16.2.
Examples of non-orientable manifolds
137
§17.
The fundamental group
139
17.1.
Definition of the fundamental group
139
17.2.
The dependence on the base point
141
17.3.
Free homotopy classes of maps of the circle
142
17.4.
Homotopic equivalence
143
17.5.
Examples
144
17.6.
The fundamental group and orientability
147
§18.
Covering maps and covering homotopies
148
18.2.
The definition and basic properties of covering spaces
148
18.2.
The simplest examples. The universal covering
150
18.3.
Branched coverings. Ricmann surfaces
153
18.4.
Covering maps and discrete groups of transformations
156
§19.
Covering maps and the fundamental group. Computation of the
fundamental group of certain manifolds
157
19.1.
Monodromy
157
19.2.
Covering maps as an aid in the calculation of fundamental
groups
160
19.3.
The simplest of the homology groups
164
19.4.
Exercises
166
§20.
The discrete groups of motions of the Lobachevskian plane
166
CHAPTER
5
Homotopy Groups
185
§21.
Definition of the absolute and relative homotopy groups. Examples
185
21.1.
Basic definitions
185
21.2.
Relative homotopy groups. The exact sequence of a pair
189
§22.
Covering homotopies. The homotopy groups of covering spaces
and loop spaces
193
221.
The concept of a fibre space
193
22.2.
The homotopy exact sequence of a fibre space
195
22.3.
The dependence of the homotopy groups on the base point
198
22.4.
The case of Lie groups
201
22.5.
Whitehcad multiplication
204
§23.
Facts concerning the homotopy groups of spheres. Framed normal
bundles. The
Hopf
invariant
207
23.1.
Framed normal bundles and the homotopy groups of spheres
207 |
| any_adam_object | 1 |
| author | Dubrovin, Boris Anatol'evič 1950-2019 Fomenko, Anatolij Timofeevič 1945- Novikov, Sergej P. 1938-2024 |
| author_GND | (DE-588)115874417X (DE-588)119092689 (DE-588)118786490 |
| author_facet | Dubrovin, Boris Anatol'evič 1950-2019 Fomenko, Anatolij Timofeevič 1945- Novikov, Sergej P. 1938-2024 |
| author_role | aut aut aut |
| author_sort | Dubrovin, Boris Anatol'evič 1950-2019 |
| author_variant | b a d ba bad a t f at atf s p n sp spn |
| building | Verbundindex |
| bvnumber | BV005835292 |
| classification_rvk | SK 350 SK 370 SK 380 |
| ctrlnum | (OCoLC)214329115 (DE-599)BVBBV005835292 |
| discipline | Mathematik |
| format | Book |
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| language | English |
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| physical | XV, 430 S. graph. Darst. |
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| spelling | Dubrovin, Boris Anatol'evič 1950-2019 Verfasser (DE-588)115874417X aut Sovremennaja geometrija Modern geometry methods and applications 2 The geometry and topology of manifolds B. A. Dubrovin ; A. T. Fomenko ; S. P. Novikov New York u.a. Springer 1985 XV, 430 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 104 Graduate texts in mathematics ... Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf Geometrie (DE-588)4020236-7 gnd rswk-swf Topologie (DE-588)4060425-1 gnd rswk-swf Variationsrechnung (DE-588)4062355-5 gnd rswk-swf Differentialgeometrie (DE-588)4012248-7 s DE-604 Topologie (DE-588)4060425-1 s Geometrie (DE-588)4020236-7 s Variationsrechnung (DE-588)4062355-5 s Fomenko, Anatolij Timofeevič 1945- Verfasser (DE-588)119092689 aut Novikov, Sergej P. 1938-2024 Verfasser (DE-588)118786490 aut (DE-604)BV005835289 2 Graduate texts in mathematics 104 (DE-604)BV000000067 104 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003651683&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Dubrovin, Boris Anatol'evič 1950-2019 Fomenko, Anatolij Timofeevič 1945- Novikov, Sergej P. 1938-2024 Modern geometry methods and applications Graduate texts in mathematics Differentialgeometrie (DE-588)4012248-7 gnd Geometrie (DE-588)4020236-7 gnd Topologie (DE-588)4060425-1 gnd Variationsrechnung (DE-588)4062355-5 gnd |
| subject_GND | (DE-588)4012248-7 (DE-588)4020236-7 (DE-588)4060425-1 (DE-588)4062355-5 |
| title | Modern geometry methods and applications |
| title_alt | Sovremennaja geometrija |
| title_auth | Modern geometry methods and applications |
| title_exact_search | Modern geometry methods and applications |
| title_full | Modern geometry methods and applications 2 The geometry and topology of manifolds B. A. Dubrovin ; A. T. Fomenko ; S. P. Novikov |
| title_fullStr | Modern geometry methods and applications 2 The geometry and topology of manifolds B. A. Dubrovin ; A. T. Fomenko ; S. P. Novikov |
| title_full_unstemmed | Modern geometry methods and applications 2 The geometry and topology of manifolds B. A. Dubrovin ; A. T. Fomenko ; S. P. Novikov |
| title_short | Modern geometry |
| title_sort | modern geometry methods and applications the geometry and topology of manifolds |
| title_sub | methods and applications |
| topic | Differentialgeometrie (DE-588)4012248-7 gnd Geometrie (DE-588)4020236-7 gnd Topologie (DE-588)4060425-1 gnd Variationsrechnung (DE-588)4062355-5 gnd |
| topic_facet | Differentialgeometrie Geometrie Topologie Variationsrechnung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003651683&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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