AN INTRODUCTION TO GEOMETRICAL PHYSICS

By K. KADAR

CONTENTS I MANIFOLDS GENERAL TOPOLOGY INTRODUCTORY COMMENTS TOPOLOGICAL SPACES KINDS OF TEXTURE FUNCTIONS QUOTIENTS AND GROUPS QUOTIENT SPACES TOPOLOGICAL GROUPS HOMOLOGY GRAPHS GRAPHS, ?RST WAY GRAPHS, SECOND WAY THE FIRST TOPOLOGICAL INVARIANTS SIMPLEXES, COMPLEXES & ALL THAT TOPOLOGICAL NUMBERS HOMOTOPY GENERAL HOMOTOPY PATH HOMOTOPY HOMOTOPY OF CURVES THE FUNDAMENTAL GROUP SOME CALCULATIONS COVERING SPACES MULTIPLY-CONNECTED SPACES COVERING SPACES HIGHER HOMOTOPY MANIFOLDS & CHARTS MANIFOLDS TOPOLOGICAL MANIFOLDS DIMENSIONS, INTEGER AND OTHER CHARTS AND COORDINATES DIFFERENTIABLE MANIFOLDS DEFINITION AND OVERLOOK SMOOTH FUNCTIONS DIFFERENTIABLE SUBMANIFOLDS II DIFFERENTIABLE STRUCTURE TANGENT STRUCTURE INTRODUCTION TANGENT SPACES TENSORS ON MANIFOLDS FIELDS & TRANSFORMATIONS FIELDS TRANSFORMATIONS FRAMES METRIC & RIEMANNIAN MANIFOLDS DIFFERENTIAL FORMS EXTERIOR DERIVATIVE VECTOR-VALUED FORMS DUALITY AND CODERIVATION INTEGRATION AND HOMOLOGY INTEGRATION COHOMOLOGY OF DI?ERENTIAL FORMS ALGEBRAS, ENDOMORPHISMS AND DERIVATIVES SYMMETRIES LIE GROUPS TRANSFORMATIONS ON MANIFOLDS LIE ALGEBRA OF A LIE GROUP THE ADJOINT REPRESENTATION FIBER BUNDLES VECTOR BUNDLES THE BUNDLE OF LINEAR FRAMES LINEAR CONNECTIONS PRINCIPAL BUNDLES GENERAL CONNECTIONS BUNDLE CLASSIFICATION III FINAL TOUCH 10 NONCOMMUTATIVE GEOMETRY 10 1 QUANTUM GROUPS — A PEDESTRIAN OUTLINE 10 2 QUANTUM GEOMETRY IV MATHEMATICAL TOPICS THE BASIC ALGEBRAIC STRUCTURES GROUPS AND LESSER STRUCTURES RINGS AND ?ELDS MODULES AND VECTOR SPACES ALGEBRAS COALGEBRAS DISCRETE GROUPS BRAIDS AND KNOTS A DISCRETE GROUPS B BRAIDS C KNOTS AND LINKS SETS AND MEASURES MEASURE SPACES ERGODISM TOPOLOGICAL LINEAR SPACES INNER PRODUCT SPACE NORM NORMED VECTOR SPACES HILBERT SPACE BANACH SPACE TOPOLOGICAL VECTOR SPACES FUNCTION SPACES BANACH ALGEBRAS QUANTIZATION BANACH ALGEBRAS *-ALGEBRAS AND C*-ALGEBRAS FROM GEOMETRY TO ALGEBRA VON NEUMANN ALGEBRAS THE JONES POLYNOMIALS REPRESENTATIONS A LINEAR REPRESENTATIONS B REGULAR REPRESENTATION C FOURIER EXPANSIONS VARIATIONS & FUNCTIONALS A CURVES VARIATION OF A CURVE VARIATION ?ELDS PATH FUNCTIONALS FUNCTIONAL DI?ERENTIALS SECOND-VARIATION B GENERAL FUNCTIONALS FUNCTIONALS LINEAR FUNCTIONALS OPERATORS DERIVATIVES – FR´ECHET AND GATEAUX FUNCTIONAL FORMS A EXTERIOR VARIATIONAL CALCULUS LAGRANGIAN DENSITY VARIATIONS AND DI?ERENTIALS THE ACTION FUNCTIONAL VARIATIONAL DERIVATIVE EULER FORMS HIGHER ORDER FORMS RELATION TO OPERATORS B EXISTENCE OF A LAGRANGIAN INVERSE PROBLEM OF VARIATIONAL CALCULUS HELMHOLTZ-VAINBERG THEOREM EQUATIONS WITH NO LAGRANGIAN C BUILDING LAGRANGIANS THE HOMOTOPY FORMULA EXAMPLES SYMMETRIES OF EQUATIONS SINGULAR POINTS RELATION TO TOPOLOGY BASIC TWO-DIMENSIONAL SINGULARITIES CRITICAL POINTS MORSE LEMMA MORSE INDICES AND TOPOLOGY CATASTROPHES 10 EUCLIDEAN SPACES AND SUBSPACES 10 1 A STRUCTURE EQUATIONS 10 1 1 MOVING FRAMES 10 1 2 THE CARTAN LEMMA 10 1 3 ADAPTED FRAMES 10 1 4 SECOND QUADRATIC FORM 10 1 5 FIRST QUADRATIC FORM 10 2 B RIEMANNIAN STRUCTURE 10 2 1 CURVATURE 10 2 2 CONNECTION 10 2 3 GAUSS, RICCI AND CODAZZI EQUATIONS 10 2 4 RIEMANN TENSOR 10 3 C GEOMETRY OF SURFACES 10 3 1 GAUSS THEOREM 10 4 D RELATION TO TOPOLOGY 10 4 1 THE GAUSS-BONNET THEOREM 10 4 2 THE CHERN THEOREM 11 NON-EUCLIDEAN GEOMETRIES 11 1 THE OLD CONTROVERSY 11 2 THE CURVATURE OF A METRIC SPACE 11 3 THE SPHERICAL CASE 11 4 THE BOLIYAI-LOBACHEVSKY CASE 11 5 ON THE GEODESIC CURVES 11 6 THE POINCAR´E SPACE 12 GEODESICS 12 1 SELF–PARALLEL CURVES 12 1 1 IN GENERAL RELATIVITY 12 1 2 THE ABSOLUTE DERIVATIVE 12 1 3 SELF–PARALLELISM 12 1 4 COMPLETE SPACES 12 1 5 FERMI TRANSPORT 12 1 6 IN OPTICS 12 2 CONGRUENCES 12 2 1 JACOBI EQUATION 12 2 2 VORTICITY, SHEAR AND EXPANSION 12 2 3 LANDAU–RAYCHAUDHURY EQUATION V PHYSICAL TOPICS HAMILTONIAN MECHANICS INTRODUCTION SYMPLECTIC STRUCTURE TIME EVOLUTION CANONICAL TRANSFORMATIONS PHASE SPACES AS BUNDLES THE ALGEBRAIC STRUCTURE RELATIONS BETWEEN LIE ALGEBRAS LIOUVILLE INTEGRABILITY MORE MECHANICS HAMILTON–JACOBI HAMILTONIAN STRUCTURE HAMILTON-JACOBI EQUATION THE LAGRANGE DERIVATIVE THE LAGRANGE DERIVATIVE AS A COVARIANT DERIVATIVE THE RIGID BODY FRAMES THE CON?GURATION SPACE THE PHASE SPACE DYNAMICS THE “SPACE” AND THE “BODY” DERIVATIVES THE REDUCED PHASE SPACE MOVING FRAMES THE ROTATION GROUP LEFT– AND RIGHT–INVARIANT ?ELDS 2 3 10 THE POINSOT CONSTRUCTION STATISTICS AND ELASTICITY A STATISTICAL MECHANICS GENERAL OVERVIEW B LATTICE MODELS THE ISING MODEL SPONTANEOUS BREAKDOWN OF SYMMETRY THE POTTS MODEL CAYLEY TREES AND BETHE LATTICES THE FOUR-COLOR PROBLEM C ELASTICITY REGULARITY AND DEFECTS CLASSICAL ELASTICITY NEMATIC SYSTEMS THE FRANCK INDEX PROPAGATION OF DISCONTINUITIES CHARACTERISTICS PARTIAL DI?ERENTIAL EQUATIONS MAXWELL’S EQUATIONS IN A MEDIUM THE EIKONAL EQUATION GEOMETRICAL OPTICS THE LIGHT-RAY EQUATION HAMILTON’S POINT OF VIEW RELATION TO GEODESICS THE FERMAT PRINCIPLE MAXWELL’S ?SH-EYE FRESNEL’S ELLIPSOID CLASSICAL RELATIVISTIC FIELDS A THE FUNDAMENTAL ?ELDS B SPACETIME TRANSFORMATIONS C INTERNAL TRANSFORMATIONS D LAGRANGIAN FORMALISM GAUGE FIELDS A THE GAUGE TENETS ELECTROMAGNETISM NONABELIAN THEORIES THE GAUGE PRESCRIPTION HAMILTONIAN APPROACH EXTERIOR DI?ERENTIAL FORMULATION B FUNCTIONAL DI?ERENTIAL APPROACH FUNCTIONAL FORMS THE SPACE OF GAUGE POTENTIALS GAUGE CONDITIONS GAUGE ANOMALIES BRST SYMMETRY C CHIRAL ?ELDS GENERAL RELATIVITY EINSTEIN’S EQUATION THE EQUIVALENCE PRINCIPLE SPINORS AND TORSION DE SITTER SPACES GENERAL CHARACTERISTICS CURVATURE GEODESICS AND JACOBI EQUATIONS SOME QUALITATIVE ASPECTS WIGNER-IN¨ON¨U CONTRACTION 10 SYMMETRIES ON PHASE SPACE 10 1 SYMMETRIES AND ANOMALIES 10 2 THE SOURIAU MOMENTUM 10 3 THE KIRILLOV FORM 10 4 INTEGRABILITY REVISITED 10 5 CLASSICAL YANG-BAXTER EQUATION