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Nash C., Sen S. Ч Topology and geometry for physicists
Nash C., Sen S. Ч Topology and geometry for physicists

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Ќазвание: Topology and geometry for physicists

јвторы: Nash C., Sen S.


Applications from condensed matter physics, statistical mechanics and elementary particle theory appear in the book. An obvious omission here is general relativity Ч we apologize for this. We originally intended to discuss general relativity. However, both the need to keep the size of the book within the reasonable limits and the fact that accounts of the topology and geometry of relativity are already available, for example, in The Large Scale Structure of Space-Time by S. Hawking and G. Ellis, made us reluctantly decide to omit this topic.

язык: en

–убрика: ћатематика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 1983

 оличество страниц: 319

ƒобавлена в каталог: 22.12.2013

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
ѕредметный указатель
$\alpha$-planes      285Ч289
$\alpha$-planes and anti-self-dual connections      288Ч289
$\alpha$-planes and complex projective space      287Ч288
$\beta$-planes      285Ч288
Affine transformation law      178 181
Algebraic geometry      289 294
Almost complex structures      170Ч171
Almost Hamiltonian structures      165Ч167 171
Anti-self-duality      259 285Ч297
Anti-self-duality and $\alpha$-planes      286 288Ч289
Anti-self-duality and holomorphic structure      292Ч297
Anti-self-duality, definition of      259
Atiyah Ч Singer index theorem      220 280
Atiyah Ч Singer index theorem and instantons      280
Atlas      26 36Ч37
Betti number      91
Bianchi identities      182Ч183 190 192 257
Biholomorphic      221
Bogomolny equation      298Ч301
Bohm Ч Aharanov effect      301Ч302
Boundary      86
Boundary operator      84 121Ч123
Brouwer fixed point theorem      6
Calculus on manifolds      37Ч48
Canonical transformations      169
Cap product      139
Cauchy's residue theorem      1Ч2
Chain      84
Characteristic classes      see "Names of specific characteristic classes"
Characteristic classes in general      200Ч225
Characteristic classes in terms of curvature and invariant polynomials      206Ч211
Characteristic classes of a manifold      219
Characteristic classes, calculation of      213Ч217
Characteristic classes, formulae obeyed by      219Ч221
Characteristic classes, universal      221
Characteristic numbers of a manifold      219
Chart      26 36Ч37
Chern character      220
Chern classes      204Ч210 213Ч215 217Ч220
Chern classes in terms of curvature      207Ч210
Christoffel symbol      187
Closed form      123
Closed set      13Ч15 17
Cohomology      120Ч139
Cohomology and de Rham cohomology      122Ч138
Cohomology and non-compactness      136Ч137
Cohomology and non-orientability      136Ч137
Cohomology groups      120
Cohomology groups for $S^{n}$      136
Cohomology groups for $T^{n}$      136
Cohomology groups, calculation of      127Ч136
Cohomology groups, de Rham      123Ч139
Cohomology groupsfor a general manifold      136
Cohomology versus homology      138Ч139
Cohomology with real coefficients      121
Compact set      16Ч19
Compactness      16Ч19 22Ч23
Compactness, topological invariance of      22Ч23
Complex orthogonal group      285Ч288
Complex projective space      138 217 287Ч288
Complex projective space and $\alpha$-planes      287Ч288
Complex projective space and spin structures      217
Complex structures      170Ч171 291
Conformal invariance      259Ч260
Conformal invariance of the * operation      260
Conformal invariance of the action S      260
Connected set      19Ч20
Connectedness      19Ч20 22Ч23 53
Connectedness, path      53
Connectedness, topological invariance of      22Ч23
Connection form      177Ч178
Connections      174Ч198 256Ч303
Connections in the tangent bundle      184Ч194
Connections, Levi Ч Civita      191Ч194
Connections, Maxwell      196Ч198 297Ч298
Connections, physical significance of      301Ч302
Connections, Yang Ч Mills      194Ч196 256Ч303
Continuity      9Ч12
Continuous deformation      9 25
Contractibility of Gl(n,C) to U(n)      163Ч164
Contractibility of Gl(n,R) to O(n)      162
Contractibility to a maximal compact subgroup      163
Contractible manifold      125Ч127
Contractible manifold, definition of      125
Contravariant vector      39Ч40
Cotangent bundle      150 166
Covariant derivative      174 179Ч180 183Ч187
Covariant exterior derivative      181Ч182
Covariant vector      39Ч40 42
Covering space      254
Critical point      227
Cup product      137Ч138
Cup product and ring structure of H*(M;R)      138
Curvature      179Ч181
Curvature and lack of commutativity of covariant differentiation      179Ч180
Curvature tensor      180Ч181 188
Curvature, dual of      183
CYCLE      86
defects      244Ч255
Defects, combination rule for line defects      249
Defects, crossing of line defects      251Ч253
Defects, definition of      246
Defects, stability of      246
Deformation      1
Deformation retract      64 135
Dense set      16
Diffeomorphism      49 221
Differentiable structures      49Ч50
Differentiable structures, different      49
Differential      40
Differential form      see "r-form"
Differential geometry      25Ч50 171Ч225
DIMENSION      23Ч24
Dimension, topological invariance of for $R^{n}$      23Ч24
Dimension, topological invariance of for a manifold      26
disconnected set      20
Dual      39Ч40
Dual basis      40
Dual of a linear map      39Ч40
Dual of a vector space      39
Dynamical system      8
Equivalence of T(M) and T*(M)      163
Euler class      204 210Ч211 215Ч217 220
Euler class and Pfaffian      211
Euler Lagrange equations      257
Euler Ч Poincare characteristic      140
Euler Ч Poincare formula      106
Exact form      123
Exact sequences      99 123
Exact sequences of a fibre bundle      253
Exact sequences of homology groups      99
Exact sequences of homotopy groups      115Ч116
Exact sequences, application of exact homology sequence      100Ч104
Exact sequences, theorem for four groups      100
Excision theorem      98
Exterior algebra      42
Exterior derivative      41Ч43 121Ч123
Fibre bundle      140Ч225
Fibre bundle, base space of      142
Fibre bundle, classification of      200Ч212
Fibre bundle, contraction of the base space of      159Ч165
Fibre bundle, definition of      142
Fibre bundle, fibre of      142
Fibre bundle, list of      151
Fibre bundle, local triviality of      142
Fibre bundle, principal      152Ч156 261
Fibre bundle, projection of      142
Fibre bundle, reconstruction from transition functions      144Ч147
Fibre bundle, reduction of the group of      159Ч165
Fibre bundle, section of      152Ч156
Fibre bundle, structure group of      142
Fibre bundle, total space of      142
Fibre bundle, transition functions of      142
Fibre bundle, triviality of      152Ч156
Fibre bundle, triviality of over a contractible base      161
Finitely generated group      90
Fixed points      4Ч8
Fixed points of an operator      6Ч8
Fixed points of the disc $B^{2}$      4Ч7
Flow of a vector field      169
Freely generated group      90
Functions      9Ч12
Functions with discontinuity      10Ч11
Functions, continuous      9
Fundamental group      51Ч78
Fundamental group of $D^{2}$      71
Fundamental group of $S^{1}$      71
Fundamental group of $S^{2}$      72
Fundamental group of a product $X \times Y$      77
Fundamental group of Klein bottle      76
Fundamental group of Moebius band      72
Fundamental group of projective plane      75
Fundamental group of torus $T^{2}$      74
Fundamental group, definition of      56
Fundamental group, dependence on base point      58
Fundamental group, non-abelian nature of      66
Fundamental group, the calculating theorem      68
Fundamental theorem of algebra      3
G-structures      163Ч171
G-structures, topological restrictions to existence of      164 167Ч171
Gauge potential      178 299 see
Gauge potential, static      299
Gauss Ч Bonnet theorem      140 223Ч225
Generators of an Abelian group      90
Geodesics      190Ч191
GL(n,R)      149Ч152
Global invariants and local geometry      221Ч225
Grassmann manifold      201 205Ч206
Grassmann manifold, complex      205
Grassmann manifold, oriented      205
Hamiltonian manifold      166
Hamiltonian structures      166Ч171
Hamiltonian structures and classical phase space      166
Hamiltonian structures and Hamilton's equations      168Ч169
Hausdorff space      27Ч28
hessian      227
Holes in a manifold and cohomology      131
Holomorphic structure      292Ч297
Holomorphic vector bundle      291Ч297
Homeomorphism      20Ч24 26 221Ч222
Homeomorphism and continuous deformation      20
Homeomorphism and equivalence classes      20Ч22
Homeomorphism as a continuous invertible map      23
Homology groups      79Ч108 120
Homology groups of $S^{1}$      89
Homology groups of $S^{n}$      104
Homology groups of a connected polyhedron      89
Homology groups of a contractible space      87
Homology groups of D2      87
Homology groups of projective plane      91
Homology groups, definition of      86
Homology groups, excision theorem of      98
Homology groups, simplicial      79
Homology groups, singular      107
Homology groups, torsion subgroup of      91
Homotopy      21Ч22 128Ч129 135 160Ч161 244Ч255 260Ч262
Homotopy and defects      244Ч255
Homotopy and instantons      260Ч262
Homotopy and pullback bundles      160Ч161
Homotopy group      109Ч119
Homotopy group Abelian nature for n>1      112
Homotopy group of $S^{n}$      118
Homotopy group, definition for n>1      111
Homotopy group, first      see "Fundamental group"
Homotopy of maps      21Ч22 128Ч129 135
Homotopy type      60
Homotopy, invariants      22
Hopf map      225
Horizontal lift      175 184 187 195
Horizontal subspace      175Ч177
Hurewicz Theorem      118
Induced bundle      see "Pullback bundle"
instantons      256Ч297
Instantons and absolute minima      263Ч265
Instantons and finite action      259Ч260
Instantons and holomorphic vector bundles      289Ч294
Instantons and Minkowski space      297
Instantons and the second Chern class      269Ч272
Instantons and twistor methods      283Ч288
Instantons with k=1      265Ч269
Instantons with k>1      272 278Ч282
Instantons, construction of from holomorphic vector bundles      295Ч296
Instantons, topology and boundary conditions      260Ч262
Integral curves      172
Integration of      44Ч48
Integration of r-forms      44Ч49
Invariant polynomials      207Ч210
Invariant polynomials, homogeneity of      207
Invariants      221Ч225
Invariants of complex structure      221Ч222
Invariants of differential structure      221Ч222
Invariants of topological structure      221Ч225
Isotopy      21
Jacobi identity      182
Jacobian determinant      34Ч35 46
Jacobian matrix      150Ч151
k-cell      231
K-theory      139
Killing form      256
Kunneth formula      105
Lagrangian      256
Landau theory      236
Lie derivative      171Ч174
Linear independence and vector fields      157Ч158
Liouvilles theorem      296
Local product      141Ч143
Loop      53
Loop, homotopy of      54
Loop, inverse of      54
Loop, product of      54
Lorentz structures      171
Manifold      25Ч50 137
Manifold, definition of      26
Manifold, infinite dimensional      49
Manifold, orientable      137
Manifold, pseudo-Riemannian      29
Manifold, Riemannian      29 46
Maxwell's equations      197 298
Metric      164
Metric space      28Ч29
Metric, signature of      164
Minkowski space      258 297
Minkowski space, compactified      297
Moebius strip      34Ч35 137 141Ч148 152Ч156
Moebius strip as a fibre bundle      141Ч148
Moebius strip, principal bundle associated with      152Ч156
Monopoles      297Ч303
Monopoles and holomorphic vector bundles      301
Monopoles, Abelian      297Ч298
Monopoles, topological charge of      299Ч300
Morse inequalities      229
Morse inequalities, proof of      233Ч236
Morse lemma      229
Morse theory      49 227Ч243
Multi-instantons      272 278Ч282
N-loop      109
n-loop, homotopy of      110
n-loop, product of      110
Neighbourhoods      13
Nematic liquid crystal      250
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