.

0 (hodnocen0 x )

BK

3th ed.

Cambridge : Cambridge University, 2012

lxii, 686 s. : il. ; 25 cm

ISBN 978-1-107-60260-1 (brož.)

Obsahuje bibliografii na s. 671-673, rejstřík

000202043

This book is intended to provide // a working knowledge of topics // in exterior differential forms, // differential geometry, algebraic // and differential topology, Lie // groups, fiber and vector bundles, // and Chern forms, that are helpful // for a deeper understanding of // classical and modem physics and // engineering. // This Third Edition includes // a new overview of Cartan’s // exterior differential forms. It // previews many of the geometric // concepts developed in the text // and illustrates their applications // to a single extended problem in // engineering; namely, the Cauchy // stresses created by a small twist // of an elastic cylindrical rod about // its axis. // "... highly readable and enjoyable // ... The book will make an excellent // course text or self-study manual // for this interesting subject.” // Physics Today // "If you’re looking for a well-written // and well-motivated introduction // to differential geometry, this one // looks hard to beat.” // MAA OnLine, Mathematical // Association of America // "... this book should not be missing // in any physics or mathematics // library.” // Zentralblatt MATH // Theodore Frankel received // his Ph.D. from the University // of California, Berkeley. He is // currently Emeritus Professor of // Mathematics at the University of // California, San Diego. // A Few Questions Considered in // The Geometry of Physics ’ // 1. How is the air pressure in an irregular soap // bubble

related to its curvature? Which // curvature? (p. 227) // 2. How does the observed fact that there are // nearby thermodynamic states that cannot be // connected adiabatically imply the existence of // entropy, and why does entropy increase? // (p. 183) // 3. How does special relativity show that the // magnetic flux law div B=0 implies Faraday’s // law, and that Gauss’ law implies AmpereMaxwell’s? (p.200) // 4. How does Weyl’s "principle of gauge // invariance” lead to the conservation of electric // charge in quantum theory? (p. 536) // 5. How does algebraic topology influence whether // one can maintain an electric current in a closed // wire loop that sits in a curved three-dimensional // space? (p.122), and how does the topology of a // configuration space influence the existence of // periodic motions in a dynamical system? // (pp. 284 and 331) // 6. Gauss invented "intrinsic” curvature (p. 232) // and equated it to his "extrinsic” curvature for a // surface in Euclidean space; how does Einstein’s // general relativity generalize this? (p. 318) // 7. How are properties of fluid flows (Euler’s // equations, circulation, vorticity, Woltjer’s // theorem of magnetohydrodynamics) described // via the Lie derivative? (p.144) // 8. In what sense is a full rotation about an axis // "something,” whereas two full rotations is // "nothing,” and how is this related to Dirac’s // equation? (pp.499 and

517) // 9. What was the original quark model of // elementary particle physics and how does Lie // group theory relate the masses of the pion, eta, // and kaon mesons in this model? (p.651) // Cambridge // UNIVERSITY PRESS // www.cambridge.org // 8 // § // q // I // o // è // 8 // 9781107602601 // sO // s // -J // 00 // o // Preface to the Third Edition // Preface to the Second Edition // Preface to the Revised Printing // Preface to the First Edition // Overview. An Informal Overview of Cartan’s Exterior Differential // Forms, Illustrated with an Application to Cauchy’s Stress Tensor // Introduction // O.a. Introduction // Vectors, 1-Forms, and Tensors // C.b. Two Kinds of Vectors // O.c. Superscripts, Subscripts, Summation Convention // O.d. Riemannian Metrics // O.e. Tensors // Integrals and Exterior Forms • // 2 // 6 // o // Line Integrals // Exterior 2-Forms // Exterior p-Forms and Algebra in R" // The Exterior Differential d // The Push-Forward of a Vector and the Pull-Back of a Form // Surface Integrals and "Stokes’ Theorem” // Electromagnetism, or, Is it a Vector or a Form? // Interior Products // Volume Forms and Cartan’s Vector Valued Exterior Forms // Magnetic Field for Current in a Straight Wire // Elasticity and Stresses // 6 // Cauchy Stress, Floating Bodies, Twisted Cylinders, and Strain // Energy // Sketch of Cauchy’s "First Theorem” // Sketch of Cauchy’s "Second Theorem,” Moments as Generators

// of Rotations // A Remarkable Formula for Differentiating Line, Surface, // and... Integrals // page xix // xxi // xxiii // xxv // xxix // xxix // xxix // XXX // XXX // xxxiii // xxxiv // xxxvii // xxxvii // xxxvii // xxxix // xl // xli // xlii // xliv // xlvi // xlvii // xlviii // l // li // a // li // ivii // lix // ixi // I Manifolds, Tensors, and Exterior Forms // 1 Manifolds and Vector Fields // 1.1. Submanifolds of Euclidean Space // 1.1a. Submanifolds of RN 1.1b. The Geometry of Jacobian Matrices: The "Differential” // 1.1c. The Main Theorem on Submanifolds of RN // l.ld. A Nontrivial Example: The Configuration Space of a // Rigid Body // 1.2. Manifolds // 8 // // Some Notions from Point Set Topology // The Idea of a Manifold // A Rigorous Definition of a Manifold // Complex Manifolds: The Riemann Sphere // Tangent Vectors and Mappings // 1.3a. // 1.3b. // Tangent or "Contravariant” Vectors // Vectors as Differential Operators // The Tangent Space to M" at a Point // Mappings and Submanifolds of Manifolds // Change of Coordinates // Vector Fields and Flows // 1.4b // Vector Fields and Flows on R" // Vector Fields on Manifolds // Straightening Flows // // 11 // 11 // 13 // 19 // 21 // 22 // 23 // 24 // 25 // 26 // 29 // 30 // 30 // 34 // 2 Tensors and Exterior Forms // 2.1. Covectors and Riemannian Metrics // 2.1a. Linear Functionals and the Dual Space // 2.1b. The Differential of a Function // Scalar Products in Linear Algebra // 37 // 37 // 37 // 40 // 42

// $ // 2.1d. Riemannian Manifolds and the Gradient Vector // 2.1e. Curves of Steepest Ascent // The Tangent Bundle // 2.2a. // 2.2b. // The Tangent Bundle // The Unit Tangent Bundle // The Cotangent Bundle and Phase Space // 2.3a. The Cotangent Bundle // 2.3b. The Pull-Back of a Covector // 2.3c. The Phase Space in Mechanics // 2.3d. The Poincaré 1 -Form // 2.4. Tensors // 2.4a. Covariant Tensors // 2.4b. Contravariant Tensors // 2.4c. Mixed Tensors // 2.4d. Transformation Properties of Tensors // 2.4e. Tensor Fields on Manifolds // 45 // 46 // 48 // 48 // 50 // 52 // 52 // 52 // 54 // 56 // 58 // 58 // 59 // 60 // 62 // 63 // CONTENTS // The Grassmann or Exterior Algebra // 2.5a. The Tensor Product of Covariant Tensors // 6 d S a // u // ri ri ri H // The Grassmann or Exterior Algebra // The Geometric Meaning of Forms in Rn // Special Cases of the Exterior Product // Computations and Vector Analysis // 66 // 66 // 66 // 70 // 70 // 71 // 73 // Exterior Differentiation // 2.6a. The Exterior Differential 73 // 2.6b. Examples in R3 75 // 2.6c. A Coordinate Expression for J 76 // 2.7. Pull-Backs . 77 // 2.7 a. The Pull-Back of a Covariant Tensor 77 // 2.7 b. The Pull-Back in Elasticity 80 // 2.8. Orientation and Pseudoforms 82 // 2.8a. Orientation of a Vector Space • 82 // 2.8b. Orientation of a Manifold 83 // 2.8c. Orientability and 2-Sided Hypersurfaces 84 // 2.8d. Projective Spaces 85 // 2.8e. Pseudoforms and the Volume Form 85 // 2.8f. The Volume Form in a Riemannian

Manifold 87 // 2.9. Interior Products and Vector Analysis // 2.9a. Interior Products and Contractions // 2.9b. Interior Product in R3 // 2.9c. Vector Analysis in R3 // 2.10. Dictionary // / // 3 Integration of Differential Forms // 3.1. // Integration over a Parameterized Subset // 3.1a. Integration of a p-Form in RP // 3.1b. // Integration over Parameterized Subsets // €) // 3.1c. Line Integrals // 3.1d. Surface Integrals // 3.1e. Independence of Parameterization // 3.1f. Integrals and Pull-Backs // 3.1g. Concluding Remarks // 3.2. // Integration over Manifolds with Boundary // 3.2a. Manifolds with Boundary // 3.2b. Partitions of Unity // 3.2c. Integration over a Compact Oriented Submanifold // 3.2d. Partitions and Riemannian Metrics // 89 // 89 // 90 // 92 // 94 // 95 // 95 // 95 // 96 // 97 // 99 // 101 // 102 // 102 // 104 // 105 // 106 // 108 // 109 // 3.3. Stokes’s Theorem // 110 // 3.4. // 3.3a. // Orienting the Boundary // 3.3b. Stokes’s Theorem // 110 // 111 // Integration of Pseudoforms // 3.4a. Integrating Pseudo-n -Forms on an n-Manifold // 3.4b. Submanifolds with Transverse Orientation // 114 // 115 // 115 // tu // ел // 3.4c. Integration over a Submanifold with Transverse // Orientation // 3.4d. Stokes’s Theorem for Pseudoforms // Maxwell’s Equations // Charge and Current in Classical Electromagnetism // 3.5b. The Electric and Magnetic Fields // 3.5c. Maxwell’s Equations // 3.5d. Forms and Pseudoforms // 4 The Lie Derivative

// 4.1. The Lie Derivative of a Vector Field // 4.1 a. The Lie Bracket // 4.1 b. Jacobi’s Variational Equation // 4.1 c. The Flow Generated by [X, Y] // 4.2. The Lie Derivative of a Form // 4.2a. Lie Derivatives of Forms // 4.2b. // 4.2c. // Formulas Involving the Lie Derivative // Vector Analysis Again // // Differentiation of Integrals // 4.3a. The Autonomous (Time-Independent) Case // 4.3b. Time-Dependent Fields // 4.3c. // Differentiating Integrals // 4.4. // A Problem Set on Hamiltonian Mechanics // 4.4a. Time-Independent Hamiltonians // 4.4b. Time-Dependent Hamiltonians and Hamilton’s // Principle // 4.4c. Poisson brackets // 5 The Poincaré Lemma and Potentials // 5.1. A More General Stokes’s Theorem // . 5.2. Closed Forms and Exact Forms // 5.3. Complex Analysis // 5.4. The Converse to the Poincaré Lemma // 5.5. Finding Potentials // 6 Holonomic and Nonholonomic Constraints // The Frobenius Integrability Condition // 6.1a. Planes in R3 // 6.1b. Distributions and Vector Fields // 6.1c. Distributions and 1-Forms // 6.1d. The Frobenius Theorem // 6.2. Integrability and Constraints // 6.2a. Foliations and Maximal Leaves // 6.2b. Systems of Mayer-Lie // 6.2c. Holonomic and Nonholonomic Constraints // 116 // 117 // 118 // 118 // 119 // 120 // 122 // 125 // 125 // 125 // 127 // 129 // 132 // 132 // 134 // 136 // 138 // 138 // 140 // 142 // 145 // 147 // 151 // 154 // 155 // 155 // 156 // 158 // 160 // 162 // 165 // 165 // 165 // 167 // 167 // 169 // 172

// 172 // 174 // 175 // CONTENTS // -9 // vS // Heuristic Thermodynamics via Caratheodory // 63a. Introduction // 63b. The First Law of Thermodynamics // Some Elementary Changes of State // 63d. The Second Law of Thermodynamics // 63e. Entropy // OS // Increasing Entropy // 6.3g. Chow ’s Theorem on Accessibility // 178 // 178 // 179 // 180 // 181 // 183 // 185 // 187 // n Geometry and Topology // { // 7 R3 and Minkowski Space // 7.1. // Curvature and Special Relativity // 7.1a. Curvature of a Space Curve in R3 // 7.1b. Minkowski Space and Special Relativity // 7.1c. Hamiltonian Formulation // 7.2. // Electromagnetism in Minkowski Space // 7.2a. // Minkowski’s Electromagnetic Field Tensor // 7.2b. Maxwell’s Equations // 8 The Geometry of Surfaces in R3 // 8.1. The First and Second Fundamental Forms // 8.1a. The First Fundamental Form, or Metric Tensor // 8.2. // 58 // 8.1b. The Second Fundamental Form // Gaussian and Mean Curvatures // 8.2a. Symmetry and Self-Adjointness // 8.2b. Principal Normal Curvatures // 8.2c. Gauss and Mean Curvatures: The Gauss Normal Map // The Brouwer Degree of a Map: A Problem Set // 6 $ // 00 00 // 83d. // 8.3e. // 8.4b. // The Brouwer Degree // Complex Analytic (Holomorphic) Maps // The Gauss Normal Map Revisited: The Gauss-Bonnet // Theorem // The Kronecker Index of a Vector Field // The Gauss Looping Integral // Mean Curvature, and Soap Bubbles // The First Variation of Area // Soap Bubbles and Minimal Surfaces // Gauss’s

Theorema Egregium // 8.5a. The Equations of Gauss and Codazzi // 8.5b. The Theorema Egregium // Geodesics // 8.6a. The First Variation of Arc Length // 8.6b. The Intrinsic Derivative and the Geodesic Equation // The Parallel Displacement of Levi-Civita // 191 // 191 // 191 // 192 // 196 // 196 // 196 // 198 // 201 // 201 // 201 // 203 // 205 // 205 // 206 // 207 // 210 // 210 // 214 // 215 // 215 // 218 // 221 // 221 // 226 // 228 // 228 // 230 // 232 // 232 // 234 // 236 // CONTENTS // • • // xii // 9 Covariant Differentiation and Curvature // 9.1. Covariant Differentiation // 9.1b. // Covariant Derivative // Curvature of an Affine Connection // 9.1c. Torsion and Symmetry // The Riemannian Connection // Cartan’s Exterior Covariant Differential // 9.3a. // 9.3b. // Vector-Valued Forms // The Covariant Differential of a Vector Field // Cartan’s Structural Equations // The Exterior Covariant Differential of a Vector-Valued // Form // The Curvature 2-Forms // 241 // 241 // 241 // 244 // 245 // 246 // 247 // 247 // 248 // 249 // Change of Basis and Gauge Transformations // 9.4a. Symmetric Connections Only // 9.4b. Change of Frame // The Curvature Forms in a Riemannian Manifold // 9.5a. // 9.5b. // The Riemannian Connection // Riemannian Surfaces M2 // 9.5c. An Example // Parallel Displacement and Curvature on a Surface // Riemann’s Theorem and the Horizontal Distribution // 9.7a. // 9.7b. // Flat metrics // The Horizontal Distribution of an Affine

Connection // Riemann’s Theorem // 250 // 251 // 253 // 253 // 253 // 255 // 255 // 257 // 257 // 259 // ’ 263 // 263 // 263 // 266 // 10 Geodesics // 10.1. // Geodesics and Jacobi Fields // 10.1a. // 10.1b. // 10.1c. // lO.ld. // Vector Fields Along a Surface in Mn // Geodesics // Jacobi Fields // Energy // 10.2. Variational Principles in Mechanics // 10.2a. // 10.2b. // 10.2c. // 10.2d. // Hamilton’s Principle in the Tangent Bundle // Hamilton’s Principle in Phase Space // Jacobi’s Principle of "Least” Action // Closed Geodesics and Periodic Motions // Geodesics, Spiders, and the Universe // 10.3a. // 10.3b. // Gaussian Coordinates // Normal Coordinates on a Surface // 10.3c. Spiders and the Universe // 269 // 269 // 269 // 271 // 272 // 274 // 275 // 275 // 277 // 278 // 281 // 284 // 284 // 287 // 288 // 11 Relativity, Tensors, and Curvature 291 // 11.1. Heuristics of Einstein’s Theory 291 // 11.1a. The Metric Potentials 291 // 11.1b. Einstein’s Field Equations 293 // 11.1c. Remarks on Static Metrics 296 // • • • // Xlll // CONTENTS // 11.2. Tensor Analysis // 11.2a. Covariant Differentiation of Tensors // 11.2b. Riemannian Connections and the Bianchi // Identities // 11.2c. Second Covariant Derivatives: The Ricci // Identities // 113. // Hilbert’s Action Principle // 113a. Geodesics in a Pseudo-Riemannian Manifold // 11.3b. // Normal Coordinates, the Divergence and Laplacian // 113c. Hilbert’s

Variational Approach to General // Relativity // 11.4. // The Second Fundamental Form in the Riemannian Case // 11.4a. The Induced Connection and the Second Fundamental // Form // 11.4b. The Equations of Gauss and Codazzi // 11.4c. The Interpretation of the Sectional Curvature // 11.4d. Fixed Points of Isometries // 11.5. The Geometry of Einstein’s Equations // 113a. The Einstein Tensor in a (Pseudo-)Riemannian // Space-Time // 11.5b. The Relativistic Meaning of Gauss’s Equation // 11.5c. The Second Fundamental Form of a Spatial Slice // 113d. The Codazzi Equations // 11.5e. Some Remarks on the Schwarzschild Solution // 12 Curvature and Topology: Synge’s Theore // n // 12.1. // Syn // e’s Formula for Second Variation // 12.1a. The Second Variation of Arc Length // 12.1b. Jacobi Fields // 12.2. // Curvature and Simple Connectivity // 12.2a. // Syn; // e’s Theorem // 12.2b. Orientability Revisited // 13 Betti Numbers and De Rham’s Theorem // 13.1. Singular Chains and Their Boundaries // 13.1a. Singular Chains // 13.1b. // Some 2-Dimensional Examples // The Singular Homology Groups // 13.2a. // Coefficient Fields // Finite Simplicial Complexes // 13.2 c. Cycles, Boundaries, Homology and Betti Numbers // Homology Groups of Familiar Manifolds // 13.3 a. Some Computational Tools // 13.3 b. Familiar Examples // De Rham’s Theorem // 13.4a. // The Statement of de Rham’s Theorem // 13.4b. Two Examples // 298 // 298 // 299 // 301 // 303

303 // 303 // 305 // 309 // 309 // 311 // 313 // 314 // 315 // 315 // 316 // 318 // 319 // 320 // 323 // 324 // 324 // 326 // 329 // 329 // 331 // 333 // 333 // 333 // 338 // 342 // 342 // 343 // 344 // 347 // 347 // 350 // 355 // 355 // 357 // xiv // CONTENTS // 14 Harmonic Forms // 14.1. The Hodge Operators // 14.1a. The * Operator // 14.1b. The Codifferential Operator 8 = d* // 14.1c. Maxwell’s Equations in Curved Space-Time M4 // 14.1d. The Hilbert Lagrangian // 14.2. Harmonic Forms // 14.2a. The Laplace Operator on Forms // 14.2b. The Laplacian of a 1 -Form // 14.2c. Harmonic Forms on Closed Manifolds // 14.2d. Harmonic Forms and de Rham’s Theorem // 14.2e. Bochner’s Theorem // // Boundary Values, Relative Homology, and Morse Theory // 14.3a. // 14.3b. // 14.3c. // 14.3d. // 14.3e. // Tangential and Normal Differential Forms // ( // Hodge’s Theorem for Tangential Forms // Relative Homology Groups // Hodge’s Theorem for Normal Forms // Morse’s Theory of Critical Points // 361 // 361 // 361 // 364 // 366 // 367 // 368 // 368 // 369 // 370 // 372 // 374 // 375 // 376 // 377 // 379 // 381 // 382 // III // / / // Lie Groups, Bundles, and Chern Forms // 15 Lie Groups // 15.1. Lie Groups, Invariant Vector Fields and Forms // 15.1a. Lie Groups // 15.1b. Invariant Vector Fields and Forms // One Parameter Subgroups // The Lie Algebra of a Lie Group // 15.3a. // 15.3b. // The Lie Algebra // The Exponential Map // Examples of Lie Algebras

Do the 1 -Parameter Subgroups Cover G? // Subgroups and Subalgebras // 15.4a. // 15.4b. // Ui // è // Left Invariant Fields Generate Right Translations // Commutators of Matrices // Right Invariant Fields // Subgroups and Subalgebras // 16 Vector Bundles in Geometry and Physics // 16.1. Vector Bundles // 16.1 a. Motivation by Two Examples // 16.1 b. Vector Bundles // 16.1 c. Local Trivializations // 16. Id. The Normal Bundle to a Submanifold // 16.2. Poincare’s Theorem and the Euler Characteristic // 16.2 a. Poincare’s Theorem // 16.2 b. The Stiefel Vector Field and Euler’s Theorem // 391 // 391 // 391 // 395 // 398 // 402 // 402 // 403 // 404 // 405 // 407 // 407 // 408 // 409 // 410 // 413 // 413 // 413 // 415 // 417 // 419 // 421 // 422 // 426 // CONTENTS // XV // 16.3. // Connections in a Vector Bundle // 16.3a. Connection in a Vector Bundle // 16.3b. Complex Vector Spaces // 16.3c. The Structure Group of a Bundle // 16.3d. Complex Line Bundles // 16.4. // The Electromagnetic Connection // 428 // 428 // 431 // 433 // 433 // 435 // 16.4a. // 16.4b. // 16.4c. // 16.4d. // Lagrange’s Equations Without Electromagnetism // The Modified Lagrangian and Hamiltonian // Schrodinger’s Equation in an Electromagnetic Field // Global Potentials // 16.4e. The Dirac Monopole // 16.4f. The Aharonov-Bohm Effect // 435 // 436 // 439 // 443 // 444 // 446 // 17 Fiber Bundles, Gauss-Bonnet, and Topological Quantization // 17.1. Fiber Bundles and Principal Bundles

// 17.1a. Fiber Bundles // 17.1b. Principal Bundles and Frame Bundles // 17.1c. Action of the Structure Group on a Principal Bundle // 17.2. Coset Spaces // 17.2a. Cosets // 17.2b. Grassmann Manifolds // 17.3. // Chern’s Proof of the Gauss-Bonnet-Poincare Theorem // 17.3a. A Connection in the Frame Bundle of a Surface // 17.3b. The Gauss_Bonnet_Poincaré Theorem // 17.3c. Gauss-Bonnet as an Index Theorem // 17.4 // Line Bundles, Topological Quantization, and Berry Phase // 17.4a. A Generalization of Gauss-Bonnet // 17.4b. Berry Phase // 17.4c. Monopoles and the Hopf Bundle // 451 // 451 // 451 // 453 // 454 // 456 // 456 // 459 // 460 // 460 // 462 // 465 // 465 // 465 // 468 // 473 // 18 Connections and Associated Bundles // 18.1 // Forms with Values in a Lie Algebra // €7 // 18.1a // 18.1b // 18.1c. // The Maurer-Cartan Form // g-Valued p-Forms on a Manifold // Connections in a Principal Bundle // OO // b // Associated Bundles and Connections // 18.2a // 18.2b // 18.2c. // Associated Bundles // Connections in Associated Bundles // The Associated Ad Bundle // 18.3. r-Form Sections of a Vector Bundle: Curvature // 18.3a. r-Form sections of E // 18.3b. Curvature and the Ad Bundle // 475 // 475 // 475 // 477 // 479 // 481 // 481 // 483 // 485 // 488 // 488 // 489 // 19 The Dirac Equation 491 // 19.1. The Groups So(3) and SU (2) 491 // 19.1a. The Rotation Group S O (3) of R3 492 // 19.1b. SU(2): The Lie algebra au(2) 493 // xvi // CONTENTS // 19.2. // 19.3. // 19.4.

// 19.5. // 20 Yang // 20.1. // 20.2. // 20.3. // ♦ // 20.4. // 20.5. // 19.1 c. SU (2) is Topologically the 3-Sphere // 19.1 d. Ad : SU(2) - SO(3) in More Detail // Hamilton, Clifford, and Dirac // 19.2 a. Spinors and Rotations of R3 // 19.2 b. Hamilton on Composing Two Rotations // 19.2 c. Clifford Algebras // 19.2 d. The Dirac Program: The Square Root of the // d’Alembertian // The Dirac Algebra // 19.3 a. The Lorentz Group // 19.3 b. The Dirac Algebra // The Dirac Operator in Minkowski Space // 19.4 a. Dirac Spinors // 19.4 b. The Dirac Operator // The Dirac Operator in Curved Space-Time // 19.5 a. The Spinor Bundle // 19.5 b. The Spin Connection in SM // Mills Fields // Noether’s Theorem for Internal Symmetries // 20.1a. The Tensorial Nature of Lagrange’s Equations // 20.1b. Boundary Conditions // 20.1c. Noether’s Theorem for Internal Symmetries // 20.1d. Noether’s Principle // Weyl’s Gauge Invariance Revisited // 20.2a. // 20.2b. // 5 // 2 // The Dirac Lagrangian // Weyl’s Gauge Invariance Revisited // The Electromagnetic Lagrangian // Quantization of the A Field: Photons // The Yang-Mills Nucleon // 20.3a. The Heisenberg Nucleon // 20.3b. The Yang-Mills Nucleon // 20.3c. A Remark on Terminology // Compact Groups and Yang-Mills Action // 20.4a. // 20.4b. // 20.4c. // The Unitary Group Is Compact // Averaging over a Compact Group // Compact Matrix Groups Are Subgroups of Unitary // Groups // Ad Invariant Scalar Products

in the Lie Algebra of a // Compact Group // b // è // 20. // . The Yang-Mills Action // The Yang-Mills Equation // 20.5a. // 20.5b. // 20.5c. // The Exterior Covariant Divergence V* // The Yang-Mills Analogy with Electromagnetism // Further Remarks on the Yang-Mills Equations // 495 // 496 // 497 // 497 // 499 // 500 // 502 // 504 // 504 // 509 // 511 // 511 // 513 // 515 // 515 // 518 // 523 // 523 // 523 // 526 // 527 // 528 // 531 // 531 // 533 // 534 // 536 // 537 // 537 // 538 // 540 // 541 // 541 // 541 // 542 // 543 // 544 // 545 // 545 // 547 // 548 // // CONTENTS // ( // 20.6. Yang-Mills Instantons // 20.6a. Instantons // 20.6b. Chern ’s Proof Revisited // 20.6c. Instantons and the Vacuum // 21 Betti Numbers and Covering Spaces // 21.1. // Bi-invariant Forms on Compact Groups // 21.1a. Bi-invariant p-Forms // 21.1b. The Cartan p-Forms // 21.1c. Bi-invariant Riemannian Metrics // 21.1d. Harmonic Forms in the Bi-invariant Metrie // 21.1e. Weyl and Cartan on the Betti Numbers of G // 21.2. // The Fundamental Group and Coverin // Spaces // 21.3. // 21.4. // 21.2a. // 21.2b. // 21.2c. // 21.2d. // Poincare’s Fundamental Group TT1 (M) // The Concept of a Covering Space // The Universal Covering // The Orientable Coverin // 4 // Ja? // 21.2e. Lifting Paths // 21.2f. Subgroups of T1 (M) // 21.2g. The Universal Covering Group // The Theorem of S. B. Myers: A Problem Set // The Geometry of a Lie Group // 21.4a. The Connection of a Bi-invariant Metric // 21.4b.

The Flat Connections // 22 Chern Forms and Homotopy Groups // 22.1. Chern Forms and Winding Numbers // 22.1a. // The Yang-Mills "Winding Number” // 22.2. // 22.1 b. Winding Number in Terms of Field Strength // 22.1 c. The Chern Forms for a U (ri) Bundle // Homotopies and Extensions // 22.2 a. Homotopy // 22.2b. // Covering Homotopy // 22.3. // 22.2c. Some Topology of S U (n) // The Higher Homotopy Groups 7k (M) // 22.3a. // 223b. Homotopy Groups of Spheres // 223c. Exact Sequences of Groups // 223d. The Homotopy Sequence of a Bundle // 22.3e. The Relation Between Homotopy and Homology // Groups // 22.4. // Some Computations of Homotopy Groups // 22.4a. Lifting Spheres from M into the Bundle P // 22.4b. SU (ri) Again // 22.4c. The Hopf Map and Fibering // xvii // 550 // 550 // 553 // 557 // 561 // 561 // 561 // 562 // 563 // 564 // 565 // 567 // 567 // 569 // 570 // 573 // 574 // 575 // 575 // 576 // 580 // 580 // 581 // 583 // 583 // 583 // 585 // 587 // 591 // 591 // 592 // 594 // 596 // 596 // 597 // 598 // 600 // 603 // 605 // 605 // 606 // 606 // xviii // CONTENTS // 22.5. Chern Forms as Obstructions // 608 // 22.5a. The Chern Forms cr for an SU(n) Bundle Revisited 608 // 22.5b. // 22.5c. // 22.5d. // 22.5e. // C2 as an "Obstruction Cocycle” // The Meaning of the Integer j (A4) // Chern’s Integral // Concluding Remarks // 609 // 612 // 612 // 615 // Appendix A. Forms in Continuu // 11 // Mechanics // 617 // A.a. The Equations of Motion of a Stressed Body

// A.b. Stresses are Vector Valued (n - l) Pseudo-Forms // 617 // 618 // A.c. The Piola-Kirchhoff Stress Tensors S and P // 619 // Strain Energy Rate // Some Typical Computations Using Forms // Concluding Remarks // 620 // 622 // 627 // Appendix B. Harmonic Chains and Kirchhoff’s Circuit Laws // B.a. Chain Complexes // B.b. // Cochains and Cohomology // e // "O a // » 03 // Transpose and Adjoint // Laplacians and Harmonic Cochains // Kirchhoff’s Circuit Laws // 628 // 628 // 630 // 631 // 633 // 635 // Appendix C. Symmetries, Quarks, and Meson Masses 640 // C.a. Flavored Quarks 640 // C.b. Interactions of Quarks and Antiquarks 642 // C.c. The Lie Algebra of SU(3) 644 // C.d. Pions, Kaons, and Etas 645 // C.e. A Reduced Symmetry Group 648 // C.f. Meson Masses 650 // Appendix D. Representations and Hyperelastic Bodies 652 // D.a Hyperelastic Bodies 652 // D.b. Isotropic Bodies 653 // D.c. Application of Schur’s Lemma 654 // D.d. Frobenius-Schur Relations 656 // D.e. The Symmetric Traceless 3x3 Matrices Are Irreducible 658 // Appendix E. Orbits and Morse-Bott Theory in Compact Lie Groups 662 // E.a. The Topology of Conjugacy Orbits 662 // E.b. Application of Bott’s Extension of Morse Theory 665 // References // Index // 671 // 675