This book grew out of courses given at the Instituto de Fisica Teorica for many years. As the title announces， it is intended as a first， elementary approach to "Geometrical Physics" ： to be understood as a chapter of Mathematical Physics. Mathematical Physics is a moving subject， and has moved faster in recent times. From the study of differential equations and related special functions， it has migrated to the more qualitative realms of topology and algebra. The bridge has been the framework of geometry. The passage supposes an acquaintance with concepts and terms of a new kind， to which this text is a tentative introduction. In its technical uses， the word "geometry" has since long lost its metric etymological meaning. It is the science of space， or better， of spaces. Thus， the name should be understood as a study of those spaces which are of interest in Physics. This emphasis on the notion of space has dominated the choice of topics - they will have in common the use of "spaces". Some may seem less geometric than others， but a space is always endowed with a few basic， irreducible properties enabling some kind of analysis， allowing a discussion of relations between its different parts.
作者簡(jiǎn)介
暫缺《幾何物理學(xué)導(dǎo)論》作者簡(jiǎn)介
圖書(shū)目錄
0 SPACE AND GEOMETRY PART I MANIFOLDS 1 GENERAL TOPOLOGY 1.0 INTRODUCTORY COMMENTS 1.1 TOPOLOGICAL SPACES 1.2 KINDS OF TEXTURE 1.3 FUNCTIONS 1.4 QUOTIENTS AND GROUPS 1.4.a Quotient spaces 1.4.b Topological groups 2 HOMOLOGY 2.1 GRAPHS 2.1.a Graphs, first way 2.1.b Graphs, second way 2.2 THE FIRST TOPOLOGICAL INVARIANTS 2.2.a Simplexes, complexes & all that 2.2.b Topological numbers 3 HOMOTOPY 3.0 GENERAL HOMOTOPY 3.1 PATH HOMOTOPY 3.1.a Homotopy of curves 3.1.b The Fundamental group 3.1.c Some calculations 3.2 COVERING SPACES 3.2.a Multiply-connected Spaces 3.2.b Coveting Spaces 3.3 HIGHER HOMOTOPY 4 MANIFOLDS & CHARTS 4.1 MANIFOLDS 4.1.a Topological manifolds 4.1.b Dimensions, integer and other 4.2 CHARTS AND COORDINATES 5 DIFFERENTIABLE MANIFOLDS 5.1 DEFINITION AND OVERLOOK 5.2 SMOOTH FUNCTIONS 5.3 DIFFERENTIABLE SUBMANIFOLDS PART II DIFFERENTIABLE STRUCTURE 6 TANGENT STRUCTURE 6.1 INTRODUCTION 6.2 TANGENT SPACES 6.3 TENSORS ON MANIFOLDS 6.4 FIELDS & TRANSFORMATIONS 6.4.a Fields 6.4.b Transformations 6.5 FRAMES 6.6 METRIC & RIEMANNIAN MANIFOLDS 7 DIFFERENTIAL FORMS 7.1 INTRODUCTION 7.2 EXTERIOR DERIVATIVE 7.3 VECTOR-VALUED FORMS 7.4 DUALITY AND CODERIVATION 7.5 INTEGRATION AND HOMOLOGY 7.5.a Integration 7.5.b Cohomology of differential forms 7.6 ALGEBRAS, ENDOMORPHISMS AND DERIVATIVES 8 SYMMETRIES 8.1 LIE GROUPS 8.2 TRANSFORMATIONS ON MANIFOLDS 8.3 LIE ALGEBRA OF A LIE GROUP 8.4 THE ADJOINT REPRESENTATION 9 FIBER BUNDLES 9.1 INTRODUCTION 9.2 VECTOR BUNDLES 9.3 THE BUNDLE OF LINEAR FRAMES 9.4 LINEAR CONNECTIONS 9.5 PRINCIPAL BUNDLES 9.6 GENERAL CONNECTIONS 9.7 BUNDLE CLASSIFICATION PART III FINAL TOUCH 10 NONCOMMUTATIVE GEOMETRY 10.1 QUANTUM GROUPS -- A PEDESTRIAN OUTLINE 10.2 QUANTUM GEOMETRY PART IV MATHEMATICAL TOPICS Math.1 THE BASIC ALGEBRAIC STRUCTURES A GROUPS AND LESSER STRUCTURES 1 Definitions 2 Transformation group 3 Representations 4 Groupoids, monoids, semigroups 5 Subgroups B RINGS AND FIELDS 6 Rings 7 Fields 8 Ring of a group C MODULES AND VECTOR SPACES 9 Modules 10 Vector spaces 11 The notion of action 12 Dimension 13 Dual space 14 Inner product 15 Endomorphisms and projectors 16 Tensor product D ALGEBRAS 17 Algebras 18 Kinds of algebras 19 Lie algebra 20 Enveloping algebra 21 Algebra of a group 22 Dual algebra 23 Derivation 24 Coalgebras 25 Bialgebras or Hopf algebras 26 R-matrices Math.2 DISCRETE GROUPS. BRAIDS AND KNOTS A DISCRETE GROUPS 1 Words and free groups 2 Presentations 3 Cyclic groups 4 The group of permutations B BRAIDS 5 Geometrical braids 6 Braid groups 7 Braids in everyday life 8 Braids presented 9 Braid statistics 10 Direct product representations 11 The Yang-Baxter equation C KNOTS AND LINKS 12 Knots 13 Links 14 Knot groups 15 Links and braids 16 Invariant polynomials Math.3 SETS AND MEASURES A MEASURE SPACES 1 The algebra of subsets 2 Measurable space 3 Boml algebra 4 Measure and probability 5 Partition of identity 6 Riemannian metric 7 Measure and Integration B ERGODISM 8 Types of flow 9 The ergodic problem Math.4 TOPOLOGICAL LINEAR SPACES 1 Inner product space 2 Norm 3 Normed vector spaces 4 Hilbert space 5 Banach space 6 Topological vector spaces 7 Function spaces Math.5 BANACH ALGEBRAS 1 Quantization 2 Banach algebras 3 *-algebras and C*-algebras 4 From Geometry to Algebra 5 yon Neumann algebras 6 The Jones polynomials Math.6 REPRESENTATIONS 0 Introduction A LINEAR REPRESENTATIONS 1 Generalities 2 Dimension 3 Unitary representations 4 Equivalent representations 5 Characters 6 Irreducible representations 7 Tensor products B REGULAR REPRESENTATION 8 Invariant spaces 9 Invariant measures 10 Generalities 11 Relation to yon Neumann algebras C FOURIER EXPANSIONS 12 The standard cases 13 Pontryagin duality 14 Noncommutative harmonic analysis 15 The Peter-Weyl theorem 16 Tanaka-Krein duality 17 Quantum groups Math.7 VARIATIONS & FUNCTIONALS A CURVES 1 Variation of a curve 2 Variation fields 3 Path functionals 4 Functional differentials 5 Second-variation B GENERAL FUNCTIONALS 6 Functionals 7 Linear functionals 8 Operators 9 Derivatives - Frechet and Gateaux Math.8 FUNCTIONAL FORMS 0 Introduction A EXTERIOR VARIATIONAL CALCULUS 1 Lagrangian density 2 Variations and differentials 3 The action functional 4 Variational derivative 5 Euler Forms 6 Higher order Forms 7 Relation to operators 8 Continuum Einstein convention B EXISTENCE OF A LAGRANGIAN 9 Inverse problem of variational calculus 10 Helmholtz-Vainberg theorem 11 Equations with no lagrangian a Navier-Stokes equation b Korteweg-de Vries equation C BUILDING LAGRANGIANS 12 The homotopy formula 13 Examples a The Helmholtz-Korteweg lagrangian b Born-lnfeld electrodynamics c Einstein''s equations d Electrodynamics e Complex scalar field f Second order fermion equation 14 Symmetries of equations Math.9 SINGULAR POINTS 1 Index of a curve 2 Index of a singular point 3 Relation to topology 4 Basic two-dimensional singularities 5 Critical points 6 Morse lemma 7 Morse indices and topology 8 Catastrophes Math. 10 EUCLIDEAN SPACES AND SUBSPACES 0 Introduction A STRUCTURE EQUATIONS 1 Moving frames 2 The Caftan lemma 3 Adapted frames 4 Second quadratic form 5 First quadratic form B RIEMANNIAN STRUCTURE 6 Curvature 7 Connection 8 Gauss, Ricci and Codazzi equations. 9 Riemann tensor C GEOMETRY OF SURFACES 10 Gauss Theorem D RELATION TO TOPOLOGY 11 The Gauss-Bonnet theorem 12 The Chem theorem Math. 11 NON-EUCLIDEAN GEOMETRIES 1 The old controversy 2 The curvature of a metric 3 The spherical case 4 The Boliyai-Lobachevsky case 5 On the geodesic curves 6 The Poincare space Math. 12 GEODESICS 0 Introduction A SELF-PARALLEL CURVES 1 In General Relativity 2 In Optics 3 As a character of a connection 4 The absolute derivative 5 Self-parallelism 6 Complete spaces 7 Fermi transport B CONGRUENCES OF CURVES 8 Jacobi equation 9 Vorticity, shear and expansion 10 Landau-Raychaudhury equation PART V PHYSICAL TOPICS Phys.1 HAMILTONLAN MECHANICS 0 Introduction 1 Symplectic structure 2 Time evolution 3 Canonical transformations 4 phase spaces as bundles 5 The algebraic structure 6 Relations between Lie algebras 7 Liouville integrability Phys.2 MORE MECHANICS A HAMILTON-JACOBI 1 Hamiltonian structure 2 Hamnlton-Jacobi equation B THE LAGRANGE DERIVATIVE 3 The Lagrange derivative as a covariant object C THE RIGID BODY 4 Frames 5 The configuration space 6 The phase space 7 Dynamics 8 The space and the body derivatives 9 The reduced phase space 10 Moving frames 11 The rotation group 12 Left-and right-invariant fields 13 The Poinsot construction STATISTICS AND ELASTICITY 0 Introduction A STATISTICAL MECHANICS 1 General overview B LATTICE MODELS 2 The Ising model 3 Spontaneous breakdown of symmetry 4 The Potts model 5 Cayley tree and Bethe lattice 6 The four-color problem C ELASTICITY 7 Regularity and defects 8 Classical elasticity 9 Nematic systems 10 The Franck index PROPAGATION OF DISCONTINUITIES 1 Characteristics 2 Partial differential equations 3 Maxwell''s equations in a medium 4 The eikonal equation Phys.5 GEOMETRICAL OPTICS 0 Introduction 1 The light ray equation 2 Hamilton''s point of view 3 Relation to geodesics 4 The Fermat principle 5 Maxwell''s fish-eye 6 Fresnel''s ellipsoid Phys.6 CLASSICAL RELATIVISTIC FIELDS A THE FUNDAMENTAL FIELDS 0 Introduction B SPACETIME TRANSFORMATIONS 1 The Poincare group 2 The basic cases C INTERNAL TRANSFORMATIONS 3 Global and local gauge transformations D LAGRANGIAN FORMALISM 4 The Euler-Lagrange Equation 5 First Noether''s theorem 6 Minimal Coupling Prescription 7 Local phase transformations 8 Second Noether''s theorem 9 Using general frames Phys.7 GAUGE FIELDS A THE GAUGE TENETS 0 Introduction 1 Electromagnetism 2 Nonabelian theories 3 The gauge prescription 4 Hamiltonian approach 5 Exterior differential formulation B FUNCTIONAL DIFFERENTIAL APPROACH 6 Functional Forms 7 The space of gauge potentials 8 Gauge conditions 9 Gauge anomalies 10 BRST symmetry C CHIRAL FIELDS 11 Some comments on chiral fields Phys.8 GENERAL RELATIVITY 1 Einstein''s equation 2 The equivalence principle 3 Spinors and torsion Phys.9 DE SITTER SPACES 1 General characteristics 2 Curvature 3 Geodesics and Jacobi equations 4 Some qualitative aspects 5 Wigner-Inonu contraction Phys.10 SYMMETRIES ON PHASE SPACE 1 Symmetries and anomalies 2 The Souriau momentum 3 The Kirillov form 4 Integrability revisited 5 Classical Yang-Baxter equation GLOSSARY REFERENCES ALPHABETIC INDEX
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