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4.7 Examples of Higher Homotopy Groups 119 4.8 Defects in Nematic Liquid Crystals 121 4.9 Textures in Superfluid 3He-A 125 Problems 4 128 5 Manifolds 130 5.1 Manifolds 130 5.2 The Calculus on Manifolds 140 5.3 Flows and Lie Derivatives 150 5.4 Differential Forms 157 5.5 Integration of Differential Forms 164 5.6 Lie groups and Lie Algebras 167 5.7 The Action of Lie groups on Manifolds 177 Problems 5 185 6 De Rham Cohomology Groups 187 6.1 Stokes' Theorem 187 6.2 De Rham Cohomology Groups 191 6.3 Poincaré's Lemma 196 6.4 Structure of De Rham Cohomology Groups 198 7 Riemannian Geometry 204 7.1 Riemannian Manifolds and Pseudo-Riemannian Manifolds 204 7.2 Parallel Transport, Connection and Covariant Derivative 207 7.3 Curvature and Torsion 215 7.4 Levi-Civita Connections 221 7.5 Holonomy 232 7.6 Isometries and Conformal Transformations 234 7.7 Killing Vector Fields and Conformal Killing Vector Fields 239 7.8 Non-Coordinate Bases 243 7.9 Differential Forms and Hodge Theory 249 7.10 Aspects of General Relativity and the Polyakov String 257 Problems 7 263 8 Complex Manifolds 265 8.1 Complex Manifolds 265 8.2 Calculus on Complex Manifolds 273 8.3 Complex Differential Forms 277