Preface |
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xiii | |
List of Notations and Symbols |
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xvii | |
List of Conventions |
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xix | |
Part 1 Basic Concepts |
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Chapter 1 Poisson Brackets |
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3 | (20) |
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3 | (2) |
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5 | (1) |
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1.3 Poisson and Hamiltonian diffeomorphisms |
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6 | (2) |
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8 | (6) |
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1.5 Poisson actions and quotients |
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14 | (5) |
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19 | (4) |
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Chapter 2 Poisson Bivectors |
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23 | (20) |
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2.1 The point of view of bivectors |
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23 | (5) |
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28 | (2) |
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2.3 Poisson maps and bivector fields |
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30 | (1) |
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31 | (9) |
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40 | (3) |
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Chapter 3 Local Structure of Poisson Manifolds |
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43 | (16) |
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3.1 The Weinstein Splitting Theorem |
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43 | (3) |
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46 | (1) |
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47 | (3) |
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3.4 The isotropy Lie algebra |
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50 | (3) |
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3.5 Linearization of Poisson structures |
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53 | (2) |
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55 | (4) |
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Notes and References for Part 1 |
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59 | (4) |
Part 2 Poisson Geometry Around Leaves |
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Chapter 4 Symplectic Leaves and the Symplectic Foliation |
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63 | (22) |
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4.1 The symplectic foliation |
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63 | (6) |
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4.2 Regular Poisson structures |
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69 | (4) |
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4.3 More examples of symplectic foliations |
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73 | (3) |
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4.4 The coupling construction |
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76 | (5) |
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81 | (4) |
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Chapter 5 Poisson Transversals |
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85 | (16) |
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5.1 Slices and Poisson transversals |
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85 | (6) |
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5.2 The transverse Poisson structure to a leaf |
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91 | (4) |
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5.3 Poisson maps and Poisson transversals |
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95 | (2) |
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97 | (4) |
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Chapter 6 Symplectic Realizations |
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101 | (32) |
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101 | (2) |
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103 | (8) |
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6.3 Symplectic realizations of linear Poisson structures |
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111 | (6) |
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6.4 Libermann's Theorem and dual pairs |
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117 | (7) |
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124 | (5) |
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129 | (4) |
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133 | (24) |
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7.1 Constant Dirac structures |
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134 | (3) |
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137 | (5) |
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7.3 Pullbacks of Dirac structures |
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142 | (3) |
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7.4 Pushforwards of Dirac structures |
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145 | (4) |
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149 | (4) |
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153 | (4) |
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Chapter 8 Submanifolds in Poisson Geometry |
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157 | (36) |
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157 | (8) |
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8.2 Poisson-Dirac submanifolds |
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165 | (4) |
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8.3 Coregular Poisson-Dirac submanifolds |
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169 | (3) |
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8.4 Coisotropic submanifolds |
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172 | (8) |
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8.5 Example: Fixed point sets |
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180 | (4) |
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8.6 Pre-Poisson submanifolds |
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184 | (5) |
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189 | (4) |
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Notes and References for Part 2 |
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193 | (4) |
Part 3 Global Aspects |
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Chapter 9 Poisson Cohomology |
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197 | (26) |
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9.1 The cotangent Lie algebroid |
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197 | (3) |
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9.2 The Poisson differential and Poisson cohomology |
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200 | (1) |
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201 | (5) |
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9.4 Shadows of Poisson cohomology |
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206 | (7) |
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9.5 The cohomological obstruction to linearization |
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213 | (5) |
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218 | (5) |
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Chapter 10 Poisson Homotopy |
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223 | (30) |
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223 | (2) |
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225 | (3) |
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10.3 Integration and the contravariant Stokes Theorem |
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228 | (3) |
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10.4 Cotangent path-homotopy |
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231 | (6) |
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10.5 Poisson homotopy and homology groups |
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237 | (8) |
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10.6 Variation of symplectic area |
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245 | (4) |
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249 | (4) |
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Chapter 11 Contravariant Geometry and Connections |
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253 | (24) |
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11.1 Contravariant connections on vector bundles |
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253 | (3) |
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11.2 Parallel transport along cotangent paths |
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256 | (4) |
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11.3 Flat contravariant connections |
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260 | (5) |
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11.4 Geodesics for contravariant connections |
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265 | (4) |
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11.5 Existence of symplectic realizations |
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269 | (4) |
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273 | (4) |
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Notes and References for Part 3 |
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277 | (6) |
Part 4 Symplectic Groupoids |
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Chapter 12 Complete Symplectic Realizations |
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283 | (30) |
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12.1 The infinitesimal action |
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284 | (2) |
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12.2 Case study: Linear Poisson structures |
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286 | (2) |
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12.3 Case study: The zero Poisson structure |
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288 | (2) |
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12.4 Case study: Nondegenerate Poisson structures |
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290 | (3) |
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293 | (6) |
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12.6 The Poisson homotopy groupoid |
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299 | (3) |
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12.7 Lagrangian fibrations |
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302 | (7) |
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309 | (4) |
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Chapter 13 A Crash Course on Lie Groupoids |
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313 | (48) |
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313 | (3) |
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13.2 Lie groupoids: Examples and basic constructions |
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316 | (10) |
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13.3 The Lie algebroid of a Lie groupoid |
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326 | (4) |
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13.4 Lie algebroids: Examples and basic constructions |
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330 | (9) |
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13.5 Duals of Lie algebroids |
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339 | (6) |
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345 | (3) |
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13.7 The non-Hausdorff setting |
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348 | (9) |
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357 | (4) |
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Chapter 14 Symplectic Groupoids |
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361 | (58) |
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14.1 Symplectic groupoids and Poisson structures |
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361 | (7) |
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368 | (9) |
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14.3 Integrability of Poisson structures I |
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377 | (11) |
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14.4 Symplectic groupoid actions |
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388 | (5) |
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14.5 Hausdorffness issues |
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393 | (2) |
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14.6 The Poisson homotopy groupoid |
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395 | (10) |
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405 | (3) |
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14.8 Integrability of Poisson structures II |
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408 | (6) |
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414 | (5) |
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Notes and References for Part 4 |
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419 | (6) |
Part 5 Appendices |
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425 | (12) |
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425 | (3) |
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428 | (5) |
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A.3 Time-dependent vector fields |
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433 | (4) |
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Appendix B. Symplectic Structures |
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437 | (10) |
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437 | (4) |
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B.2 Symplectic and Hamiltonian actions |
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441 | (6) |
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447 | (12) |
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447 | (5) |
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C.2 Foliated differential forms |
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452 | (2) |
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454 | (5) |
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Appendix D. Groupoids: Conventions and Choices |
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459 | (4) |
Bibliography |
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463 | (10) |
Index |
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473 | |