McDuff D., Salamon D. — Introduction to Symplectic Topology :: Электронная библиотека попечительского совета мехмата МГУ

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McDuff D., Salamon D. — Introduction to Symplectic Topology

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Название: Introduction to Symplectic Topology

Авторы: McDuff D., Salamon D.

Аннотация:

Symplectic structures underlie the equations of classical mechanics and their properties are reflected in the behavior of a wide range of physical systems. Over the last few years powerful new methods in analysis and topology have led to the development of the modern global theory of symplectic topology, including several striking and important results. At its publication in 1995, Introduction to Symplectic Topology was the first comprehensive introduction to the subject and it has since become an established text in this fast-developing branch of mathematics. This second edition has been significantly revised and expanded, with new references and additional examples and theorems. It includes a section on new developments and an expanded discussion of Taubes and Donaldson's recent results.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 2nd edition

Год издания: 1998

Количество страниц: 486

Добавлена в каталог: 01.07.2006

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Предметный указатель

Curvature identity      222
Curvature of symplectic connection      221 222
Darboux chart      96 104
Darboux’s theorem      1 28 37 81 95
Darboux’s theorem equivariant      104
Darboux’s theorem for 2-manifolds      98
Darboux’s theorem for contact structures      112
Deformation equivalent fibrations      437
Deformation equivalent nonisotopic 6-manifolds      436
Deformation nonequivalent 6-manifolds      436
Dehn twist generalized      213—215 335
Dehn twist positive      455
Delzant      181 242
Diffeomorphism area-preserving      see “Area-preserving diffeomorphism”
Diffeomorphism symplectic      see “Symplectomorphism”
Diffeomorphism symplectic iff preserves capacity      377
Discrete Floer theory      370
Discrete Hamiltonian mechanics      288—293
Discrete holomorphic curve      370
Discrete loop space      408
Discrete method versus $\infty$ -dim method      416
Discrete path      291 292
Discrete symplectic action      291 345 346 396 406—408
Discrete symplectic action Morse theory      351
Discrete symplectic action number of fixed points      356
Discrete variational problem      277 286
Disjunction of Lagrangian submanifolds      423
Displacement energy and Hofer metric      382
Displacement energy and nonsqueezing      383
Displacement energy as relative capacity      384
Displacement energy of ball      385 386
Distribution contact      105
Distribution contact vector fields tangent to      110
Distribution horizontal      207 210
Distribution isotropic      100
Dolgachev surface      445
Donaldson      3 4 34 72 83 124 138 139 172 179 207 418 434 446 450 452
Duistermaat      159 191 192
Duistermaat — Heckman formula      191—196
Dynamics      see “Hamiltonian dynamics”
Ehresmann      120
Ekeland      2 3 32 47 371 373 377 400 402
Ekeland — Hofer capacity      375
Eliashberg      2 32 108 116 252 262 339 366 377 383 419 422 426 430 432
Ellipsoid      31 58
Ellipsoid, capacity of      375
Ellipsoid, closed classification of      430
Ellipsoid, open classification of      428
Elliptic methods      3
Elliptic surface and fibre connected sum      253
Elliptic surface as blow-up      237
Embedding      see “Symplectic embedding”
Embeddings of balls      246
Embeddings of balls construction      387
Energy      26
Energy, conservation of      22
Energy, displacement      see “Displacement energy”
Energy, symplectic      2
Energy-capacity inequality      384—394
Enriques      134
Epstein      328
Equivariant cohomology      193
Equivariant cohomology localization theorem      194
Euclidean space      2
Euclidean space $Ham^c(\mathbb{R}^{2n})$       325
Euclidean space $Symp^c(\mathbb{R}^4)$ is contractible      325
Euclidean space as Kaehler manifold      130
Euclidean space as symplectic manifold      82
Euclidean space as symplectic vector space      38 39
Euclidean space, billiards in      277
Euclidean space, exotic symplectic structure on      100 424
Euclidean space, generating functions on      280
Euclidean space, group actions on      165
Euclidean space, hamiltonian mechanics in      13—18
Euclidean space, hofer metric on      391
Euclidean space, mechanics in      12
Euclidean space, no exact Lagrangians      423
Euclidean space, reduces to $\mathbb{C}P^{2n}$ "      152
Euclidean space, symplectic geometry of      18—25
Euclidean space, symplectic invariants for subsets      3 375
Euclidean space, symplectic topology of      28—36
Euclidean space, twist maps in      269
Euclidean space, vector products on      118
Euler class      79
Euler — Lagrange equation      12 13 15 16 277 285 287
Evaluation map $\pi_1(Ham(M))$ to $\pi_1$ (M)      327 369
Evaluation map $\pi_k (Ham(M))$ to $\pi_k (M)$       334
Exact symplectic action      162
Examples almost complex manifolds      118—120
Examples contact structure on $\mathbb{R}^3$       116
Examples cotangent bundle      90
Examples Duistermaat — Heckman formula      191
Examples exact Lagrangian immersion in $\mathbb{R}^{2n}$       423
Examples existence problem      419
Examples exotic structure on $\mathbb{R}^4$       424
Examples first Chern class      79 80
Examples fixed points for maps of       342
Examples flux for annulus      332
Examples flux for torus      325
Examples generating function of type V      289
Examples Gromov width      432
Examples Hamiltonian flow on $S^{2n-1}$       22
Examples harmonic oscillator      26
Examples invariants for subsets of $\mathbb{R}^{2n}$       428
Examples Kaehler manifolds      130 131 135
Examples large ball in $T^2\times \mathbb{R^{2n}}$       373
Examples nonKaehler symplectic manifold      89 251
Examples of blowing up      236—238
Examples of contact manifolds      106 108 109
Examples of convexity theorem      180—183
Examples of moment maps      165—170
Examples of nonequivalent symplectic structures      435
Examples of symplectic blowing up      242
Examples of symplectic manifolds      89 91—93
Examples ruled surface      248 249 442
Examples ruled surfaces      439
Examples Siegel upper half space      48
Examples symplectic fibrations      202
Examples toric manifolds      433
Examples, area form on       82
Examples, capacities      374
Examples, capacity of 2-manifold      400
Examples, exactness for       314
Exceptional divisor      234 239
Exceptional divisor as ball      239
Exceptional divisor Poincare dual of      235
Exceptional sphere      147 241 249
Exponential map      27
Fermat’s principle of least time      12
Fernandez      90
Feynman path integral      290
Fibration -bundles      156—159 202—206
Fibration locally trivial      197
Fibration symplectic      197—202
Fibration symplectic coupling form      216
Fibration symplectic singular      213
Fibration, classification of -bundles      203
Fibration, compatible symplectic form on      198
Fibration, compatible symplectic form on deformation equivalence      439
Fibration, constructing symplectic -bundles constructing symplectic form      199
Fibration, elliptic      237
Fibration, Hamiltonian      226—232
Fibration, Hopf      153
Fibration, Lefschetz      455 456
Fibration, structure group of      198
Fibre critical points      306
Filling problem      115 419
Fintushel      446
First Chern class      37 71 73—80
First Chern class and connections      109 157

First Chern class as obstruction      79
First Chern class axiomatic definition      74
First Chern class of K3 surface      254
First Chern class on Riemann surface      74
Fixed extrema of isotopy      392
Fixed point degenerate      340
Fixed point existence for twist maps      270
Fixed point geometrically distinct      270
Fixed point hyperbolic      348
Fixed point nondegenerate      340
Fixed point set as symplectic submanifold      186
Fixed point set normally hyperbolic      183
Fixed point theorems      34—36
Fixed point theorems of Lefschetz      343
Fixed point, number of      1
flexibility      2 33
Flexibility for open symplectic manifolds      257
Flexibility of contact structures      419
Floer      4 340 356 402 429 432
Floer homology      336 351 366—370
Floer theory      3 309
Floer theory discrete      370
Floer’s proof of Arnold’s conjectures      357
Floquet multipliers      266
Flux group      216 229 321
Flux group, discreteness of      323 324
Flux homomorphism      315—328
Flux homomorphism calculation for torus      325
Flux homomorphism exact case      320
Flux homomorphism geometric interpretation      317
Flux homomorphism has simple kernel      329
Flux homomorphism is surjective      318
Flux homomorphism is well defined      316
Flux homomorphism volume-preserving case      318
Fogerty      179
Foliation characteristic      29 87 99
Foliation with Reeb component      419
For linear map      289
Form global angular      109
Form primitive for $\omega$       424
Fortune      369
Frankel      154 453
Franks      26 272
Fujiki      172
Fukaya      35 141 146 309 369 373
Gauge transformation      208
Gauge transformation, symplectic      219
Geiges      252 432
Gelfand — Robbin quotient      42 43
Generalized adjunction formula      453
Generating function      3 275—277
Generating function determined by H      283
Generating function for Lagrangians      303—310
Generating function for nbhd of $\mathbb{1}$ in Ham(M)      302
Generating function for symplectomorphism of torus      341
Generating function in billiard problem      278
Generating function of strong monotone map      275
Generating function of type S      280—283 292
Generating function of type V      289 297 406
Generating function on Ham(M)      293—303
Generating function quadratic at infinity      358 360
Generating function via discrete action      359—362
Geodesic      26 93
Geodesic broken      344
Geodesic flow      26
Geodesic for the Hofer metric      392
Geodesic local minimum of energy      27 286
Ginzburg      29 30 155 399
Givental      47 66 95 105 178 288 340 344 346 369
Global angular form      109
Goldman      179
Gole      277 288 339
Gompf      3 5 90 101 140 233 252—254 256 445
Gotay      90 201 211
Gottlieb      335
Gradient-like flow      355
Gradient-like flow number of fixed points      355
Graph of 1-form      99
Graph of isotopy      385 388
Graph of isotopy characteristics on      386
Graph of symplectomorphism      99 296
gray      90
Gray’s stability theorem      112
Greene      427
Griffiths      136 154 237 238 445 450
Gromov      2 4 31 33 65 81 100 147 233 237 251 252 257 262 312 325 358 377 401 417 419 423 424 426 430 452
Gromov invariants      434 437
Gromov invariants and Seiberg — Witten invariants      439—450
Gromov width      373 374 384
Gromov width upper bound for      432
Gromov — Witten invariants      232 436
Gromov’s compactness theory      144
Gromov’s flexibility theorem      257—261
Gromov’s nonsqueezing theorem      see “Nonsqueezing theorem”
Group action condition to be Hamiltonian      162
Group action exact      162
Group action Hamiltonian      162
Group action of       151
Group action of classification      433
Group action of condition to be Hamiltonian      154
Group action semi-free      155 156
Group action weakly Hamiltonian      162
Group of contactomorphisms as Lie group      312
Group of diffeomorphisms algebraic structure      328
Group of diffeomorphisms as Lie group      312
Group of diffeomorphisms commutator subgroup is simple      328
Group of diffeomorphisms is perfect      328
Group of Hamiltonian symplectomorphisms      see “Ham(M)”
Group of symplectomorphisms      20 33;
Group of volume-preserving diffeomorphisms      21 33
Group of volume-preserving diffeomorphisms algebraic structure      328
Group of volume-preserving diffeomorphisms as Lie group      312
Group, representations      5
Guillemin      5 151 159 162 179 180 185 197 207 210 215 226 242 250
Haar measure      46
Haefliger      262
Hall      265 275
Ham(M) $\pi_1$ of      322
Ham(M) algebraic structure when M noncompact      329
Ham(M) as kernel of flux      318
Ham(M) closure of      323 324
Ham(M) for closed M, simplicity of      329
Ham(M) for noncompact manifolds      314 320
Ham(M) fragmentation lemma      329
Ham(M) juxtaposition of paths      313
Ham(M) nbhd of identity      302
Hamilton — Jacobi equation      280 283—286
Hamilton — Jacobi equation in contact geometry      111
Hamiltonian action      162
Hamiltonian action weak      162
Hamiltonian admissible      394
Hamiltonian circle action      151
Hamiltonian connection      226
Hamiltonian difference equations      290
Hamiltonian dynamics      24 25 165 265
Hamiltonian dynamics and Hamilton — Jacobi equation      284
Hamiltonian fibration      226
Hamiltonian flow      1 18 34 85
Hamiltonian flow, invariant tori for      284
Hamiltonian for symplectomorphism of cotangent bundle      92
Hamiltonian function      14 85
Hamiltonian function, extension of      402—405
Hamiltonian group action      161
Hamiltonian group action examples      166
Hamiltonian holonomy      215 226
Hamiltonian isotopy      88 294 295 313
Hamiltonian mechanics      12—27
Hamiltonian mechanics discrete      288
Hamiltonian quadratic growth      403
Hamiltonian symplectomorphism      88 294—303 313

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