几何物理学导论

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