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

-closure of $\textrm{Ham}^c(\mathbb{R}^{2n})$       325
-closure of Symp(M)      3 33 312 377
$\alpha$ -generating function      302
$\mathbb{C}^n$       see “Euclidean space”
$\mathbb{R}^{2n}$       see “Euclidean space”
$\textrm{Symp}^c(\mathbb{R}^4)$ is contractible      325
1-form generating symplectomorphism      302
Abraham      265
Abreu      334
action      12 16—17 280
Action and Calabi invariant      331
Action and contact flow      111
Action as generating function      295 359—362
Action as Lagrangian      290
Action discrete      288 291
Action form      16
Action integral      12 16
Action of fixed point      299
Action of fixed point as area swept out      301
Action of group      396; see “Group action on loop space”
Action used to calculate Hofer length      393
Action, minima of      284
Action, principle of least action      13 16
Action, spectrum      298—300 302 386
Action-angle variables      24
Adjunction formula      139
Adjunction formula, generalized      453
Aebisher      29
Ahara      433
Almost complex structure      37 117;
Almost complex structure and symplectic structure      257
Almost complex structure examples      118—120
Almost complex structure homotopic to nondegenerate form      118 257
Almost complex structure integrable      123
Almost complex structure invariant      185
Almost complex structure on hypersurfaces in       119
Almost complex structure on open manifold homotopic to symplectic structure      258 259
Almost complex structure on spheres      118 120
Almost complex structure on symplectic manifold      3
Almost complex structure on torus      444
Almost complex structure, $\omega$ -compatible      117
Almost complex structure, compatible with metric      118
Almost complex structure, regular      142
Almost symplectic manifold      86
Alvarez      26
Amman      402
Andreotti      453
Annulus invariant      269
Annulus twist map      269
Annulus, action for map of      300
Annulus, flux for      332
Annulus, symplectic maps of      268
Anti-symplectic      56
Archimedes      82
Area swept out as action of fixed point      301
Area swept out as flux      317
Area-preserving diffeomorphisms      265—279
Area-preserving monotone      270
Area-preserving strongly monotone      275
Area-preserving twist map      269—274
Arnold      20 25 47 66 84 95 105 275 343 357
Arnold’s conjecture      2 35 99 280 339—346 402
Arnold’s conjecture for Lagrangian intersections      357
Arnold’s conjecture for nbhd of $\mathbb{1}$       344
Arnold’s conjecture or cotangent bundles      308 358
Arnold’s conjecture proof for torus      345
Arnold’s conjecture, cohomological version      343
Atiyah      170 179 180 182 188 192 193
Aubry — Mather theory      277
Audin      65 100 178 252 433
Ball and blowing up      239—241
Ball constructing embedding      3 387
Ball symplectic shapes      1 3 31
Ball uniqueness of structure in dimension      4 430
Ball volume-preserving shapes      31
Ball, displacement energy of      385 386
Ball, embedding spaces of      246
Banyaga      97 248 311 324 328 329
Barth      136 446
Basic class      444 453
Basis $\omega$ -standard      40
Basis $\omega$ -standard and g-orthogonal      67 68
Basis, symplectic      39
Bates, L.      424
Bates, S.      5 295 298 325 376
Beltrami equation      127
Bennequin      252
Berline      193
Bers      128
Betti numbers       133
Betti numbers and Crit(M)      343
Betti numbers of Calabi — Yau manifold      139
Betti numbers of K3 surfaces      139
Betti numbers of Kaehler manifold      89 132
Betti numbers, alternating sum of      36 343
Bhupal      309 340 358 359 369
Bialy      300 391 392 401
Bilinear form, nondegeneracy condition      38
Bilinear form, skew      38
Bilinear form, skew geometry of      37
Bilinear form, skew rank      42
Bilinear form, skew standard basis      42
Bilinear form, symmetric      63
Bilinear form, symmetric and skew      37 57
Billiard problem      277—279
Biran      249 450
Birkhoff      269 272 339
Blowing up and down      233—251
Blowing up and down along submanifold      238 250
Blowing up and down and embeddings of balls      246
Blowing up and down and moment maps      242
Blowing up and down and symplectic reduction      251
Blowing up and down as fibre sum      254
Blowing up and down complex      233
Blowing up and down full fillings      249 450
Blowing up and down in dimension 4      248
Blowing up and down Kaehler      243 248
Blowing up and down on complex manifold      234
Blowing up and down radius of ball and cohomology class      239 241
Blowing up and down rational      254
Blowing up and down resolving singularities      238
Blowing up and down symplectic      239—251
Blowing up and down symplectic precise version      245
Blowing up and down uniqueness up to isotopy      246
Blowing up and down, packing problem      249
Blowing up and weight      241
Bott      6 96 109 120 133 155 170 179 183 192 193 245 320
Boundary value problems      42
Brown      272
Bundle      see “Fibration”
Calabi      119 329
Calabi group      321
Calabi group volume-preserving case      329
Calabi group, discreteness of      323 324
Calabi homomorphism      311 328—332
Calabi homomorphism as average symplectic action      331
Calabi homomorphism for general manifolds      332
Calabi homomorphism has simple kernel      332
Calabi — Yau manifold      139
Candelas      139
Canonical 1-form      90; see “Cotangent bundle”
Canonical 1-form on cotangent bundle      90
Canonical bundle      134
Canonical class      132 133
Canonical transformation      18
Capacity      3 32 371
Capacity applications      377
Capacity characterizes symplectomorphisms      377
Capacity conformality axiom      373
Capacity displacement energy as      384

Capacity Ekeland — Hofer      375
Capacity examples      374
Capacity from generating functions      401
Capacity higher      400
Capacity higher of open ellipsoid      428
Capacity Hofer — Zehnder      377 384;
Capacity intrinsic      375
Capacity linear      56
Capacity monotonicity axiom      373
Capacity nonintrinsic      375
Capacity nontriviality axiom      373
Capacity of ellipsoids      375
Capacity relative      375
Capacity relative conformality axiom      375
Capacity relative monotonicity axiom      375
Capacity relative nontriviality axiom      375
Capacity symplectic      373
Capacity symplectic for subsets      374
Capacity width      56 374
Cartan      312
Cartan differential form of      267 385
Castelnuovo — Enriques criterion      134 239
Category      see “Ljusternik — Schnirelmann theory
Cauchy — Riemann equations      142 368
Cayley numbers      119
Cayley transform      66
Chaperon      3 288 309 340 344 369
Characteristic foliation      29 87 99
Characteristics      29 128
Chekanov      309 358 359 369
Chern class      see “First Chern class”
Chern number      446
Chern — Weil theory      77
Christoffel symbols      26
Cieliebak      429 430
Circle action      151—161
Circle action condition to be Hamiltonian      154
Classifying space      69 203
Clean intersection      174
Cohomologically trivial open set      353
Coisotropic submanifold      99 104 173—175 307
Coisotropic submanifold, isotropic foliation of      100
Coisotropic submanifold, reduction of      173
Coisotropic submanifold, regular      173
Coisotropic subspace      38 40 42
Compatible metric      129
Compatible symplectic form on fibration      198
Compatible triple      121
Complex projective plane      79
Complex projective plane, blow-up of      249
Complex projective plane, conies in      144
Complex projective plane, curves in      146
Complex projective plane, Gromov invariants of      442
Complex projective plane, uniqueness      147
Complex projective space      131
Complex projective space action of U(n)      167
Complex projective space as reduced manifold      152
Complex projective space blown up at a point      236 242
Complex projective space canonical line bundle      80 234
Complex projective space hypersurfaces in      136 139
Complex projective space image of moment map      180
Complex projective space symplectic structures on $\mathbb{C}P^2$       432
Complex structure      see “almost complex structure”
Complex structure $\omega$ -compatible      63—65 67
Complex structure $\omega$ -tame      65
Complex structure on manifold      123
Complex structure on vector bundles      69
Complex structure on vector bundles $\omega$ -compatible      69
Complex structure on vector bundles equivalent to symplectic      69
Complex structure on vector spaces      61—68
Conformally equivalent metrics      129
Conley      288 293 339 340 344 347
Conley index      346—349
Conley — Betti numbers      350 353
Conley — Zehnder index of periodic point      293
Conley — Zehnder theorem      2 288 345
Connected sum and blowing up      235
Connected sum and K3 surfaces      254
Connected sum choice of framing      253
Connected sum fibrewise      253
Connected sum fibrewise not complex construction      254
Connected sum fibrewise twisted      445
Connected sum of almost complex manifolds      252
Connected sum of contact manifolds      252
Connected sum pointwise      251
Connected sum symplectic      251—257
Connected sum symplectic structures on      419
Connection      207
Connection 1-form      208
Connection 2-form      213
Connection and first Chern class      109 157
Connection condition to be symplectic      211
Connection fat      226
Connection flat      171
Connection Hamiltonian      226
Connection holonomy      208
Connection symplectic      207—215
Contact form      105
Contact geometry      81
Contact isotopy      109 112
Contact manifold      105—116
Contact manifold symplectically tillable      115
Contact manifold, Darboux’s theorem for      112
Contact manifold, examples      106 108 109
Contact manifold, stability theorem for      112
Contact manifold, symplectization of      113 426
Contact structure      105
Contact structure and surgery      252
Contact structure as equivalence class of forms      106
Contact structure dominated by symplectic form      115
Contact structure flexibility of      419
Contact structure on classification      108
Contact structure on fillable      422
Contact structure on $S^{2n+1}$       108
Contact structure on $S^{2n+1}$ and $R^{2n+1}$       116
Contact structure on $\mathbb{R}^{2n+1}$       106 '
Contact structure on $\mathbb{R}^{2n+1}$ classification      426
Contact structure on 1-jet bundle      107
Contact structure on manifolds of low dimension      252
Contact structure on unit cosphere bundle      107
Contact structure tight      419
Contact vector field      110
Contactomorphism      109
Convexity of image of moment map      179
Cordero      90
Cotangent bundle      2 90—93
Cotangent bundle Arnold’s conjecture      308 358
Cotangent bundle canonical 1-form      90
Cotangent bundle Floer theory in      309
Cotangent bundle of $\mathbb{R}^{2n}$ as product      297
Cotangent bundle vertical      160
Cotangent bundle, action of G on T*G      167
Cotangent bundle, exact symplectomorphisms of      315
Cotangent bundle, rigidity in      364—366
Coupling form      216 226 391
Courant      128
Critical fibre critical points      306
Critical level      355
Critical manifold      183
Critical manifold index      183
Critical point      355
Critical point existence because of linking      411
Critical point geometrically distinct      346
Critical point may be saddle point      13
Critical point, number of      35
Crossing form      50 54
Crossing index      50 54
Cuplength      343 354
Curvature      208
Curvature as moment map      171 172

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