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8.4 Hermitian Manifolds and Hermitian Differential Geometry 281 8.5 Kähler Manifolds and Kähler Differential Geometry 287 8.6 Harmonic Forms and ¶-cohomology * Groups 292 8.7 Almost Complex Manifolds 297 8.8 Orbifolds 300 9 Fibre Bundles 303 9.1 Tangent Bundles 303 9.2 Fibre Bundles 305 9.3 Vector Bundles 313 9.4 Principal Bundles 318 Problems 9 328 10 Connections on Fibre Bundles 329 10.1 Connections on Principal Bundles 329 10.2 Holonomy 339 10.3 Curvature 340 10.4 The Covariant Derivative on Associated Vector Bundles 346 10.5 Gauge Theories 354 10.6 Berry's Phase 364 Problems 10 371 11 Characteristic Classes 373 11.1 Invariant Polynomials and the Chern–Weil Homomorphism 373 11.2 Chern Classes 380 11.3 Chern Characters 385 11.4 Pontrjagin and Euler Classes 389 11.5 Chern–Simons Forms 397 11.6 Stiefel–Whitney Classes 402 12 Index Theorems 406 12.1 Elliptic Operators and Fredholm Operators 406 12.2 The Atiyah–Singer Index Theorem 412 12.3 The De Rham Complex 413 12.4 The Dolbeault Complex 415 12.5 The Signature Complex 417