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Isham J. — Modern Differential Geometry for Physics
Isham J. — Modern Differential Geometry for Physics

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Название: Modern Differential Geometry for Physics

Автор: Isham J.

Аннотация:

This edition of the invaluable text Modern Differential Geometry for Physicists contains an additional chapter that introduces some of the basic ideas of general topology needed in differential geometry. A number of small corrections and additions have also been made. These lecture notes are the content of an introductory course on modern, coordinate-free differential geometry which is taken by first-year theoretical physics PhD students, or by students attending the one-year MSc course "Fundamental Fields and Forces" at Imperial College. The book is concerned entirely with mathematics proper, although the emphasis and detailed topics have been chosen bearing in mind the way in which differential geometry is applied these days to modern theoretical physics. This includes not only the traditional area of general relativity but also the theory of Yang-Mills fields, nonlinear sigma models and other types of nonlinear field systems that feature in modern quantum field theory. The volume is divided into four parts: (i) introduction to general topology; (ii) introductory coordinate-free differential geometry; (iii) geometrical aspects of the theory of Lie groups and Lie group actions on manifolds; (iv) introduction to the theory of fibre bundles. In the introduction to differential geometry the author lays considerable stress on the basic ideas of "tangent space structure", which he develops from several different points of view - some geometrical, others more algebraic. This is done with awareness of the difficulty which physics graduate students often experience when being exposed for the first time to the rather abstract ideas of differential geometry.


Язык: en

Рубрика: Математика/Геометрия и топология/Дифференциальная геометрия/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: second edition

Год издания: 2001

Количество страниц: 290

Добавлена в каталог: 19.04.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Accumulation point      47
Adjoint map      162
Anti-homomorphism      152
Associated bundle      233
Associated bundle, automorphism group of      239
Associated bundle, local isomorphism between pair of      237
Associated bundle, map between pair of      237
Associated bundle, trivial      237
Bianchi identity      274
Billiard-ball hairy      101
Boolean algebra      20 42
Boundary of a set      12
Boundary point      12 31
Bundle      202
Bundle, $C^\infty$      204
Bundle, associated      see Associated bundle
Bundle, base space      202
Bundle, composition of pair of      215
Bundle, cross-section of      see Cross-section
Bundle, etale      212
Bundle, fibre      204
Bundle, fibre over x      202
Bundle, G      220
Bundle, isomorphism between a pair of      215
Bundle, local isomorphism between a pair of      215
Bundle, locally trivial      216
Bundle, M-map      215
Bundle, map      214
Bundle, normal      210
Bundle, normal, of an embedding      249
Bundle, of tensors      236
Bundle, principal      see Principal bundle
Bundle, product      204
Bundle, projection map of      202
Bundle, pull-back      217
Bundle, restriction to subset of base space      213
Bundle, sub      213
Bundle, total space      202
Bundle, trivial      216
Bundle, universal      219 228 232
Bundle, vector      see Vector bundle
Cartan structural equation      272
Cartan — Maurer equation      173
Cartan — Maurer form      173 257
Cartesian product      6 36 64
Characteristic class      219 231
Closed set      13 31 35
Closure      53
Collection of subsets coarser than      26
Collection of subsets finer than      26
Commutation relations, affine      111
Commutation relations, canonical      110
Compact space      47
Connection, affine sum of pair of      256
Connection, local representative of      256
Connection, principal bundle in      254
Continuous map      see Map continuous
Coordinate chart      61
Coordinate chart, atlas of      62
Coordinate chart, domain of      61
Coordinate functions      63
coordinates      63
Cotangent bundle      123
Cotangent space      123
Cotangent vector      123
Covariant derivative      269 270
Covering space      210
Cross-section      200 207 241
Cross-section, associated bundle of      246
Cross-section, local representative of      247
Cross-section, principal bundle of      230
Cross-section, product bundle of      207
Cross-section, pull-back of      218
Cross-section, tangent bundle of      98
Curvature two-form      272
Curve, definition of      73
Curve, horizontal lift of      263
Curve, tangent pair of      73
deRham complex      141 171
DeRham's theorem      217
Derivation, at a point      80
Derivation, components of      84
Derivation, of the ring $C^\infty(\mathcal M)$      99
Diffeomorphism, definition of      70
Diffeomorphism, group of      70 224 229
Diffeomorphism, one-parameter group of      111
Differential structure      62
Differentiate manifold      62
Differentiate manifold, complex      63
Differentiate manifold, infinite-dimensional      63
Directed set      46
Directional derivative      75
Dual basis of      121
Dual map between pair of      121
Dual vector space      121
Eilenberg — MacLane space      228
Equivalence relation      see Relation equivalence
Exact sequence      255
Exponential map      165
Exterior derivative, covariant      272
Exterior derivative, function of      131
Exterior derivative, n-form of      see n-form exterior
Exterior of a set      12
Exterior point      12 31
Filter, base      29 47 50
Filter, base, convergence with respect to      30
Filter, definition of      29
Filter, neighbourhood      see Neighbourhood filter
Filter, subbase      31
Filter, ultra      45
First countable space      see Topology first
Function, $C^r$      69
Function, continuous      49
Function, diagonal      95
Function, differentiable      70
Function, distance      5
Function, equivariant      178
Function, homotopic pair of      145
Function, inverse set      48
Function, local representative of      69
Function, smooth      70
G-product      232
Gauge, group      16 224 229 231 258
Gauge, orbit      16
Gauge, transformation      224 226 239 258
Gribov effect      231
Group action      52
Group action, orbit of      52
Group, action on a set      175
Group, action on a set, free      179
Group, action on a set, kernel of      179
Group, action on a set, orbit of      180
Group, action on a set, orbit space of      182
Group, action on a set, stability group of      180
Group, action on a set, transitive      179
Group, adjoint representation of      168
Group, cohomology      201 217 219 229 232 242
Group, general linear, complex      153
Group, general linear, real      152
Group, holonomy      267
Group, homotopy      228 229 242 246
Group, left translation of      150
Group, Lie      see Lie group
Group, Lorentz      157
Group, orthogonal      155
Group, partial      113
Group, right translation of      150
Group, special      157
Group, special linear      154
Group, special unitary      157
Group, spin      225
Group, topological      151
Group, unitary      157
Hausdorff space      see Topology Hausdorff
Heyting algebra      42
Homeomorphism      51
Homotopic functions      see Function
Homotopic pair of Hopf bundle      222
Horizontal subspace      268
Instanton      201 222 226 243
Integral curve      see Vector field integral
Interior of a set      12
Interior point      12 31
Intuitionistic logic      42
Isotropy group      see Group action stability
Jacobi identity      103
Jacobian matrix      88
Kaluza — Klein theory      201
Klein bottle      205
Lattice, a topology as an example      40
Lattice, anti-atomic      45
Lattice, atomic      44
Lattice, complete      19 41
Lattice, definition of      19
Lattice, distributive      19 41
Lattice, example of closed linear subspaces of a Hilbert space      22
Lattice, null element in      19
Lattice, structure on space of topologies      43
Lattice, unit element in      19
Lattice, upper set in      27
Lie algebra      103
Lie algebra, dual      171
Lie algebra, Lie group of      159
Lie algebra, of GL(n, M)      169
Lie algebra, structure constants of      161
Lie derivative, function of      98
Lie derivative, one-form of      130
Lie derivative, vector field of      116
Lie group, action on a manifold      177
Lie group, definition of      150
Lie group, homomorphism between pair of      152
Lie group, one-parameter subgroup of      166
Lie group, subgroup of      150
Limit point      12 14 31 35
Linear frame      223
Little group      see Group action stability
Locale      56
Locally compact space      153
Lower set      37
MAP      see Function
Map, continuous      50
Metric      6
Metric, bounded      11
Metric, Lorentzian      18 224 245
Metric, Riemannian      8 224 229 241 243 244
Metric, space      see Space metric
Metric, stronger than      7 16 26
Metric, weaker than      7
Metrics, equivalent      7 16
Metrics, isometric      7
Metrics, join of      10
Metrics, lattice structure on      11
Metrics, meet of      11
Metrics, partial ordering of      21
Moebius band      205
Moebius transformations      152
n-form      135
n-form, closed      142
n-form, exact      142
n-form, exterior derivative of      137
n-form, exterior product of      135
n-form, left-invariant      171
n-form, right-invariant      171
Neighbourhood, filter      29 34
Neighbourhood, space      29
Neighbourhood, structure      29 34
Neighbourhoods, equivalent families of      26 27
Neighbourhoods, of a point      17 23 27 32
Net      46
Net, convergence of      46
Non-linear $\sigma$-model      240
Norm      8
One form, Lie derivative of      see Lie derivative one
One-form, components of      126
One-form, definition of      123
One-form, pull-back of      127
One-form, V-valued      126
Open set      13 31 35
Open set, pseudo-complement of      41
Orbit space      see Group action
Orbit space on a set, orbit space of overlap function      61
Palais' Theorem      198
Parallel translation      267 268
Partially ordered set      see Poset
Poincare lemma      142
Poset, covering element in      15
Poset, definition of      15
Poset, greatest lower bound in      18
Poset, join operation in      19
Poset, least upper bound in      19
Poset, meet operation in      18
Poset, orthocomplemented      19
Poset, topologies on      36
Pre-order      15
Principal bundle      221
Principal bundle, automorphism group of      226
Principal bundle, connection in      see Connection principal
Principal bundle, G-extension of      241
Principal bundle, H-restriction of      241
Principal bundle, locally trivial      226
Principal bundle, map      225
Principal bundle, structure functions of      239
Principal bundle, structure group of      221
Principal bundle, translation function of      234
Principal bundle, trivial      226
Product topology      see Topology product
Pseudo-metric      6 53
Relation, definition of      15
Relation, equivalence      16 50
Riemann sphere      188
Scalar density      236
Sequence, convergent      3 5 6 14 24 26 29
Sequence, of functions, point wise convergence of      24 31
Sequence, tails of      5 24 26 47
Sequence, tails of, as a filter base      30
Sheaf      212
Space, metric      3 6 31 54
Space, neighbourhood      see Neighbourhood space
Space, topological      see Topology
Spontaneous symmetry breakdown      241
Stability group      see Group action stability
Stereographic projection      68
Submanifold      64
Tangent bundle      74
Tangent bundle, differential structure on      89
Tangent space      74
Tangent space, of a product manifold      92
Tangent vector      74
Tangent vector, push-forward of      78
Tensor      133
Tensor density      236
Tensor, contravariant      133
Tensor, covariant      133
Tensor, field      134 135
Topological space      see Topology
Topology, $T_0$      53
Topology, $T_1$      53
Topology, $T_2$      see Topology Hausdorff
Topology, base of      35
Topology, coarser      see Topology weaker
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