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Preface page xi Introduction 1 1 The concept of a manifold 4 1.1 Topology and continuous maps 4 1.2 Classes of smoothness of maps of Cartesian spaces 6 1.3 Smooth structure, smooth manifold 7 1.4 Smooth maps of manifolds 11 1.5 A technical description of smooth surfaces in Rn 16 Summary of Chapter 1 20 2 Vector and tensor fields 21 2.1 Curves and functions on M 22 2.2 Tangent space, vectors and vector fields 23 2.3 Integral curves of a vector field 30 2.4 Linear algebra of tensors (multilinear algebra) 34 2.5 Tensor fields on M 45 2.6 Metric tensor on a manifold 48 Summary of Chapter 2 53 3 Mappings of tensors induced by mappings of manifolds 54 3.1 Mappings of tensors and tensor fields 54 3.2 Induced metric tensor 60 Summary of Chapter 3 63 4 Lie derivative 65 4.1 Local flow of a vector field 65 4.2 Lie transport and Lie derivative 70 4.3 Properties of the Lie derivative 72 4.4 Exponent of the Lie derivative 75 4.5 Geometrical interpretation of the commutator [V,W], non-holonomic frames 77 4.6 Isometries and conformal transformations, Killing equations 81 Summary of Chapter 4 91 v vi Contents 5 Exterior algebra 93 5.1 Motivation: volumes of parallelepipeds 93 5.2 p-forms and exterior product 95 5.3 Exterior algebra L∗ 102 5.4 Interior product iv 105 5.5 Orientation in L 106 5.6 Determinant and generalized Kronecker symbols 107 5.7 The metric volume form 112 5.8 Hodge (duality) operator ∗ 118 Summary of Chapter 5 125 6 Differential calculus of forms 126 6.1 Forms on a manifold 126 6.2 Exterior derivative 128 6.3 Orientability, Hodge operator and volume form on M 133 6.4 V-valued forms 139 Summary of Chapter 6 143 7 Integral calculus of forms 144 7.1 Quantities under the integral sign regarded as differential forms 144 7.2 Euclidean simplices and chains 146 7.3 Simplices and chains on a manifold 149 7.4 Integral of a form over a chain on a manifold 150 7.5 Stokes’ theorem 151 7.6 Integral over a domain on an orientable manifold 153 7.7 Integral over a domain on an orientable Riemannian manifold 159 7.8 Integral and maps of manifolds 161 Summary of Chapter 7 163 8 Particular cases and applications of Stokes’ theorem 164 8.1 Elementary situations 164 8.2 Divergence of a vector field and Gauss’ theorem 166 8.3 Codifferential and Laplace–deRham operator 171 8.4 Green identities 177 8.5 Vector analysis in E3 178 8.6 Functions of complex variables 185 Summary of Chapter 8 188 9 Poincar´e lemma and cohomologies 190 9.1 Simple examples of closed non-exact forms 191 9.2 Construction of a potential on contractible manifolds 192 9.3∗ Cohomologies and deRham complex 198 Summary of Chapter 9 203 10 Lie groups: basic facts 204 10.1 Automorphisms of various structures and groups 204 Contents vii 10.2 Lie groups: basic concepts 210 Summary of Chapter 10 213 11 Differential geometry on Lie groups 214 11.1 Left-invariant tensor fields on a Lie group 214 11.2 Lie algebra G of a group G 222 11.3 One-parameter subgroups 225 11.4 Exponential map 227 11.5 Derived homomorphism of Lie algebras 230 11.6 Invariant integral on G 231 11.7 Matrix Lie groups: enjoy simplifications 232 Summary of Chapter 11 243 12 Representations of Lie groups and Lie algebras 244 12.1 Basic concepts 244 12.2 Irreducible and equivalent representations, Schur’s lemma 252 12.3 Adjoint representation, Killing–Cartan metric 259 12.4 Basic constructions with groups, Lie algebras and their representations 269 12.5 Invariant tensors and intertwining operators 278 12.6∗ Lie algebra cohomologies 282 Summary of Chapter 12 287 13 Actions of Lie groups and Lie algebras on manifolds 289 13.1 Action of a group, orbit and stabilizer 289 13.2 The structure of homogeneous spaces, G/H 294 13.3 Covering homomorphism, coverings SU(2) → SO(3) and SL(2,C) → L↑ + 299 13.4 Representations of G and G in the space of functions on a G-space, fundamental fields 310 13.5 Representations of G and G in the space of tensor fields of type ρˆ 319 Summary of Chapter 13 325 14 Hamiltonian mechanics and symplectic manifolds 327 14.1 Poisson and symplectic structure on a manifold 327 14.2 Darboux theorem, canonical transformations and symplectomorphisms 336 14.3 Poincar´e–Cartan integral invariants and Liouville’s theorem 341 14.4 Symmetries and conservation laws 346 14.5∗ Moment map 349 14.6∗ Orbits of the coadjoint action 354 14.7∗ Symplectic reduction 360 Summary of Chapter 14 368 15 Parallel transport and linear connection on M 369 15.1 Acceleration and parallel transport 369 15.2 Parallel transport and covariant derivative 372 15.3 Compatibility with metric, RLC connection 382 15.4 Geodesics 389 viii Contents 15.5 The curvature tensor 401 15.6 Connection forms and Cartan structure equations 406 15.7 Geodesic deviation equation (Jacobi’s equation) 418 15.8∗ Torsion, complete parallelism and flat connection 422 Summary of Chapter 15 428 16 Field theory and the language of forms 429 16.1 Differential forms in the Minkowski space E1,3 430 16.2 Maxwell’s equations in terms of differential forms 436 16.3 Gauge transformations, action integral 441 16.4 Energy–momentum tensor, space-time symmetries and conservation laws due to them 448 16.5∗ Einstein gravitational field equations, Hilbert and Cartan action 458 16.6∗ Non-linear sigma models and harmonic maps 467 Summary of Chapter 16 476 17 Differential geometry on TM and T ∗M 478 17.1 Tangent bundle TM and cotangent bundle T ∗M 478 17.2 Concept of a fiber bundle 482 17.3 The maps T f and T ∗ f 485 17.4 Vertical subspace, vertical vectors 487 17.5 Lifts on TM and T ∗M 488 17.6 Canonical tensor fields on TM and T ∗M 494 17.7 Identities between the tensor fields introduced here 497 Summary of Chapter 17 497 18 Hamiltonian and Lagrangian equations 499 18.1 Second-order differential equation fields 499 18.2 Euler–Lagrange field 500 18.3 Connection between Lagrangian and Hamiltonian mechanics, Legendre map 505 18.4 Symmetries lifted from the base manifold (configuration space) 508 18.5 Time-dependent Hamiltonian, action integral 518 Summary of Chapter 18 522 19 Linear connection and the frame bundle 524 19.1 Frame bundle π : LM → M 524 19.2 Connection form on LM 527 19.3 k-dimensional distribution D on a manifoldM 530 19.4 Geometrical interpretation of a connection form: horizontal distribution on LM 538 19.5 Horizontal distribution on LM and parallel transport on M 543 19.6 Tensors on M in the language of LM and their parallel transport 545 Summary of Chapter 19 550 20 Connection on a principal G-bundle 551 20.1 Principal G-bundles 551 Contents ix 20.2 Connection form ω ∈ 1(P, Ad) 559 20.3 Parallel transport and the exterior covariant derivative D 563 20.4 Curvature form ∈ 2(P, Ad) and explicit expressions of D 567 20.5∗ Restriction of the structure group and connection 576 Summary of Chapter 20 585 21 Gauge theories and connections 587 21.1 Local gauge invariance: “conventional” approach 587 21.2 Change of section and a gauge transformation 594 21.3 Parallel transport equations for an object of type ρ in a gauge σ 600 21.4 Bundle P ×ρ V associated to a principal bundle π : P → M 606 21.5 Gauge invariant action and the equations of motion 607 21.6 Noether currents and Noether’s theorem 618 21.7∗ Once more (for a while) on LM 626 Summary of Chapter 21 633 22∗ Spinor fields and the Dirac operator 635 22.1 Clifford algebras C(p, q) 637 22.2 Clifford groups Pin (p, q) and Spin (p, q) 645 22.3 Spinors: linear algebra 650 22.4 Spin bundle π : SM → M and spinor fields on M 654 22.5 Dirac operator 662 Summary of Chapter 22 670 Appendix A Some relevant algebraic structures 673 A.1 Linear spaces 673 A.2 Associative algebras 676 A.3 Lie algebras 676 A.4 Modules 679 A.5 Grading 680 A.6 Categories and functors 681 Appendix B Starring 683 Bibliography 685 Index of (frequently used) symbols 687 Index 690 |
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