DIFFERENTIAL GEOMETRY, ANALYSIS AND PHYSICS
CONTENTS PRELIMINARIES AND LOCAL THEORY CALCULUS CHAIN RULE, PRODUCT RULE AND TAYLOR’S THEOREM LOCAL THEORY OF MAPS DI?ERENTIABLE MANIFOLDS ROUGH IDEAS I TOPOLOGICAL MANIFOLDS DI?ERENTIABLE MANIFOLDS AND DI?ERENTIABLE MAPS PSEUDO-GROUPS AND MODELS SPACES SMOOTH MAPS AND DI?EOMORPHISMS COVERINGS AND DISCRETE GROUPS COVERING SPACES AND THE FUNDAMENTAL GROUP DISCRETE GROUP ACTIONS GRASSMANNIAN MANIFOLDS PARTITIONS OF UNITY MANIFOLDS WITH BOUNDARY THE TANGENT STRUCTURE ROUGH IDEAS II TANGENT VECTORS INTERPRETATIONS THE TANGENT MAP THE TANGENT AND COTANGENT BUNDLES TANGENT BUNDLE THE COTANGENT BUNDLE IMPORTANT SPECIAL SITUATIONS SUBMANIFOLD, IMMERSION AND SUBMERSION SUBMANIFOLDS SUBMANIFOLDS OF RN REGULAR AND CRITICAL POINTS AND VALUES IMMERSIONS IMMERSED SUBMANIFOLDS AND INITIAL SUBMANIFOLDS SUBMERSIONS MORSE FUNCTIONS PROBLEM SET LIE GROUPS I DE?NITIONS AND EXAMPLES LIE GROUP HOMOMORPHISMS FIBER BUNDLES AND VECTOR BUNDLES I TRANSITIONS MAPS AND STRUCTURE USEFUL WAYS TO THINK ABOUT VECTOR BUNDLES SECTIONS OF A VECTOR BUNDLE SHEAVES,GERMS AND JETS JETS AND JET BUNDLES VECTOR FIELDS AND 1-FORMS DE?NITION OF VECTOR ?ELDS AND 1-FORMS PULL BACK AND PUSH FORWARD OF FUNCTIONS AND 1-FORMS FRAME FIELDS LIE BRACKET LOCALIZATION ACTION BY PULLBACK AND PUSH-FORWARD FLOWS AND VECTOR FIELDS LIE DERIVATIVE TIME DEPENDENT FIELDS LIE GROUPS II SPINORS AND ROTATION MULTILINEAR BUNDLES AND TENSORS FIELDS MULTILINEAR ALGEBRA CONTRACTION OF TENSORS ALTERNATING MULTILINEAR ALGEBRA ORIENTATION ON VECTOR SPACES MULTILINEAR BUNDLES TENSOR FIELDS TENSOR DERIVATIONS 10 DI?ERENTIAL FORMS 10 1 PULLBACK OF A DI?ERENTIAL FORM 10 2 EXTERIOR DERIVATIVE 10 3 MAXWELL’S EQUATIONS 10 4 LIE DERIVATIVE, INTERIOR PRODUCT AND EXTERIOR DERIVATIVE 10 5 TIME DEPENDENT FIELDS (PART II) 10 6 VECTOR VALUED AND ALGEBRA VALUED FORMS 10 7 GLOBAL ORIENTATION 10 8 ORIENTATION OF MANIFOLDS WITH BOUNDARY 10 9 INTEGRATION OF DI?ERENTIAL FORMS 10 10STOKES’ THEOREM 10 11VECTOR BUNDLE VALUED FORMS 11 DISTRIBUTIONS AND FROBENIUS’ THEOREM 11 1 DE?NITIONS 11 2 INTEGRABILITY OF REGULAR DISTRIBUTIONS 11 3 THE LOCAL VERSION FROBENIUS’ THEOREM 11 4 FOLIATIONS 11 5 THE GLOBAL FROBENIUS THEOREM 11 6 SINGULAR DISTRIBUTIONS 12 CONNECTIONS ON VECTOR BUNDLES 12 1 DE?NITIONS 12 2 LOCAL FRAME FIELDS AND CONNECTION FORMS 12 3 PARALLEL TRANSPORT 12 4 CURVATURE 13 RIEMANNIAN AND SEMI-RIEMANNIAN MANIFOLDS 13 1 THE LINEAR THEORY 13 1 1 SCALAR PRODUCTS 13 1 2 NATURAL EXTENSIONS AND THE STAR OPERATOR 13 2 SURFACE THEORY 13 3 RIEMANNIAN AND SEMI-RIEMANNIAN METRICS 13 4 THE RIEMANNIAN CASE (POSITIVE DE?NITE METRIC) 13 5 LEVI-CIVITA CONNECTION 13 6 COVARIANT DI?ERENTIATION OF VECTOR ?ELDS ALONG MAPS 13 7 COVARIANT DI?ERENTIATION OF TENSOR ?ELDS 13 8 COMPARING THE DI?ERENTIAL OPERATORS 14 FORMALISMS FOR CALCULATION 14 1 TENSOR CALCULUS 14 2 COVARIANT EXTERIOR CALCULUS, BUNDLE-VALUED FORMS 15 TOPOLOGY 15 1 ATTACHING SPACES AND QUOTIENT TOPOLOGY 15 2 TOPOLOGICAL SUM 15 3 HOMOTOPY 15 4 CELL COMPLEXES 16 ALGEBRAIC TOPOLOGY 16 1 AXIOMS FOR A HOMOLOGY THEORY 16 2 SIMPLICIAL HOMOLOGY 16 3 SINGULAR HOMOLOGY 16 4 CELLULAR HOMOLOGY 16 5 UNIVERSAL COE?CIENT THEOREM 16 6 AXIOMS FOR A COHOMOLOGY THEORY 16 7 DE RHAM COHOMOLOGY 16 8 TOPOLOGY OF VECTOR BUNDLES 16 9 DE RHAM COHOMOLOGY 16 10THE MEYER VIETORIS SEQUENCE 16 11SHEAF COHOMOLOGY 16 12CHARACTERISTIC CLASSES 17 LIE GROUPS AND LIE ALGEBRAS 17 1 LIE ALGEBRAS 17 2 CLASSICAL COMPLEX LIE ALGEBRAS 17 2 1 BASIC DE?NITIONS 17 3 THE ADJOINT REPRESENTATION 17 4 THE UNIVERSAL ENVELOPING ALGEBRA 17 5 THE ADJOINT REPRESENTATION OF A LIE GROUP 18 GROUP ACTIONS AND HOMOGENOUS SPACES 18 1 OUR CHOICES 18 1 1 LEFT ACTIONS 18 1 2 RIGHT ACTIONS 18 1 3 EQUIVARIANCE THE ACTION OF DI?(M) AND MAP-RELATED VECTOR ?ELDS 18 1 5 LIE DERIVATIVE FOR EQUIVARIANT BUNDLES 18 2 HOMOGENEOUS SPACES 19 FIBER BUNDLES AND CONNECTIONS 19 1 DE?NITIONS 19 2 PRINCIPAL AND ASSOCIATED BUNDLES 20 ANALYSIS ON MANIFOLDS 20 1 BASICS 20 1 1 STAR OPERATOR II 20 1 2 DIVERGENCE, GRADIENT, CURL 20 2 THE LAPLACE OPERATOR 20 3 SPECTRAL GEOMETRY 20 4 HODGE THEORY 20 5 DIRAC OPERATOR 20 5 1 CLI?ORD ALGEBRAS 20 5 2 THE CLI?ORD GROUP AND SPINOR GROUP THE STRUCTURE OF CLI?ORD ALGEBRAS 20 6 1 GAMMA MATRICES 20 7 CLI?ORD ALGEBRA STRUCTURE AND REPRESENTATION 20 7 1 BILINEAR FORMS 20 7 2 HYPERBOLIC SPACES AND WITT DECOMPOSITION 20 7 3 WITT’S DECOMPOSITION AND CLI?ORD ALGEBRAS 20 7 4 THE CHIRALITY OPERATOR 20 7 5 SPIN BUNDLES AND SPIN-C BUNDLES 20 7 6 HARMONIC SPINORS 21 COMPLEX MANIFOLDS 21 1 SOME COMPLEX LINEAR ALGEBRA 21 2 COMPLEX STRUCTURE 21 3 COMPLEX TANGENT STRUCTURES 21 4 THE HOLOMORPHIC TANGENT MAP 21 5 DUAL SPACES 21 6 EXAMPLES 21 7 THE HOLOMORPHIC INVERSE AND IMPLICIT FUNCTIONS THEOREMS 22 CLASSICAL MECHANICS 22 1 PARTICLE MOTION AND LAGRANGIAN SYSTEMS 22 1 1 BASIC VARIATIONAL FORMALISM FOR A LAGRANGIAN 22 1 2 TWO EXAMPLES OF A LAGRANGIAN 22 2 SYMMETRY, CONSERVATION AND NOETHER’S THEOREM 22 2 1 LAGRANGIANS WITH SYMMETRIES 22 2 2 LIE GROUPS AND LEFT INVARIANTS LAGRANGIANS 22 3 THE HAMILTONIAN FORMALISM 23 SYMPLECTIC GEOMETRY 23 1 SYMPLECTIC LINEAR ALGEBRA 23 2 CANONICAL FORM (LINEAR CASE) 23 3 SYMPLECTIC MANIFOLDS 23 4 COMPLEX STRUCTURE AND K¨AHLER MANIFOLDS 23 5 SYMPLECTIC MUSICAL ISOMORPHISMS 23 6 DARBOUX’S THEOREM 23 7 POISSON BRACKETS AND HAMILTONIAN VECTOR ?ELDS 23 8 CON?GURATION SPACE AND PHASE SPACE 23 9 TRANSFER OF SYMPLECTIC STRUCTURE TO THE TANGENT BUNDLE 23 10COADJOINT ORBITS 23 11THE RIGID BODY 23 11 1THE CON?GURATION IN R3N 23 11 2MODELLING THE RIGID BODY ON SO(3) 23 11 3THE TRIVIAL BUNDLE PICTURE 23 12THE MOMENTUM MAP AND HAMILTONIAN ACTIONS 24 POISSON GEOMETRY 24 1 POISSON MANIFOLDS 25 QUANTIZATION 25 1 OPERATORS ON A HILBERT SPACE 25 2 C*-ALGEBRAS 25 2 1 MATRIX ALGEBRAS 25 3 JORDAN-LIE ALGEBRAS 26 APPENDICES A PRIMER FOR MANIFOLD THEORY 26 1 1 FIXING A PROBLEM 26 2 B TOPOLOGICAL SPACES 26 2 1 SEPARATION AXIOMS 26 2 2 METRIC SPACES 26 3 C TOPOLOGICAL VECTOR SPACES 26 3 1 HILBERT SPACES 26 3 2 ORTHONORMAL SETS 26 4 D OVERVIEW OF CLASSICAL PHYSICS 26 4 1 UNITS OF MEASUREMENT 26 4 2 NEWTON’S EQUATIONS 26 4 3 CLASSICAL PARTICLE MOTION IN A CONSERVATIVE ?ELD 26 4 4 SOME SIMPLE MECHANICAL SYSTEMS 26 4 5 THE BASIC IDEAS OF RELATIVITY 26 4 6 VARIATIONAL ANALYSIS OF CLASSICAL FIELD THEORY 26 4 7 SYMMETRY AND NOETHER’S THEOREM FOR ?ELD THEORY 26 4 8 ELECTRICITY AND MAGNETISM 26 4 9 QUANTUM MECHANICS 26 5 E CALCULUS ON BANACH SPACES 26 6 CATEGORIES 26 7 DI?ERENTIABILITY 26 8 CHAIN RULE, PRODUCT RULE AND TAYLOR’S THEOREM 26 9 LOCAL THEORY OF MAPS 26 9 1 LINEAR CASE 26 9 2 LOCAL (NONLINEAR) CASE 26 10THE TANGENT BUNDLE OF AN OPEN SUBSET OF A BANACH SPACE 26 11PROBLEM SET 26 11 1EXISTENCE AND UNIQUENESS FOR DI?ERENTIAL EQUATIONS 26 11 2DI?ERENTIAL EQUATIONS DEPENDING ON A PARAMETER 26 12MULTILINEAR ALGEBRA 26 12 1SMOOTH BANACH VECTOR BUNDLES 26 12 2FORMULARY 26 13CURVATURE 26 14GROUP ACTION 26 15NOTATION AND FONT USAGE GUIDE