An Introduction to Geometrical Physics

The following sections are included:

• FOREWORD

• TABLE OF CONTENTS

The simplest geometrical setting used — consciously or not — by physicists in their everyday work is the 3-dimensional euclidean space E³. It consists of the set R³ of ordered triples of real numbers such as p = (p¹, p², p³), q = (q¹, q², q³), etc, and is endowed with a very special characteristic, a metric defined by the distance function

The following sections are included:

• INTRODUCTORY COMMENTS

• TOPOLOGICAL SPACES

• KINDS OF TEXTURE

• FUNCTIONS

• QUOTIENTS AND GROUPS
  • Quotient spaces
  • Topological groups

The following sections are included:

• GRAPHS
  • Graphs, first way
  • Graphs, second way

• THE FIRST TOPOLOGICAL INVARIANTS
  • Simplexes, complexes & all that
  • Topological numbers

The following sections are included:

• GENERAL HOMOTOPY

• PATH HOMOTOPY
  • Homotopy of curves
  • The Fundamental group
  • Some calculations

• COVERING SPACES
  • Multiply-connected Spaces
  • Covering Spaces

• HIGHER HOMOTOPY

The following sections are included:

• MANIFOLDS
  • Topological manifolds
  • Dimensions, integer and other

• CHARTS AND COORDINATES

The following sections are included:

• DEFINITION AND OVERLOOK

• SMOOTH FUNCTIONS

• DIFFERENTIABLE SUBMANIFOLDS

The following sections are included:

• INTRODUCTION

• TANGENT SPACES

• TENSORS ON MANIFOLDS

• FIELDS & TRANSFORMATIONS
  • Fields
  • Transformations

• FRAMES

• METRIC & RIEMANNIAN MANIFOLDS

The following sections are included:

• INTRODUCTION

• EXTERIOR DERIVATIVE

• VECTOR-VALUED FORMS

• DUALITY AND CODERIVATION

• INTEGRATION AND HOMOLOGY
  • Integration
  • Cohomology of differential forms

• ALGEBRAS, ENDOMORPHISMS AND DERIVATIVES

The following sections are included:

• LIE GROUPS

• TRANSFORMATIONS ON MANIFOLDS

• LIE ALGEBRA OF A LIE GROUP

• THE ADJOINT REPRESENTATION

The following sections are included:

• INTRODUCTION

• VECTOR BUNDLES

• THE BUNDLE OF LINEAR FRAMES

• LINEAR CONNECTIONS

• PRINCIPAL BUNDLES

• GENERAL CONNECTIONS

• BUNDLE CLASSIFICATION

The following sections are included:

• QUANTUM GROUPS — A PEDESTRIAN OUTLINE

• QUANTUM GEOMETRY

The following sections are included:

• GROUPS AND LESSER STRUCTURES
  • Definitions
  • Transformation group
  • Representations
  • Groupoids, monoids, semigroups
  • Subgroups

• RINGS AND FIELDS
  • Rings
  • Fields
  • Ring of a group

• MODULES AND VECTOR SPACES
  • Modules
  • Vector spaces
  • The notion of action
  • Dimension
  • Dual space
  • Inner product
  • Endomorphisms and projectors
  • Tensor product

• ALGEBRAS
  • Algebras
  • Kinds of algebras
  • Lie algebra
  • Enveloping algebra
  • Algebra of a group
  • Dual algebra
  • Derivation
  • Coalgebras
  • Bialgebras or Hopf algebras
  • R-matrices

The following sections are included:

• DISCRETE GROUPS
  • Words and free groups
  • Presentations
  • Cyclic groups
  • The group of permutations

• BRAIDS
  • Geometrical braids
  • Braid groups
  • Braids in everyday life
  • Braids presented
  • Braid statistics
  • Direct product representations
  • The Yang-Baxter equation

• KNOTS AND LINKS
  • Knots
  • Links
  • Knot groups
  • Links and braids
  • Invariant polynomials

The following sections are included:

• MEASURE SPACES
  • The algebra of subsets
  • Measurable space
  • Borel algebra
  • Measure and probability
  • Partition of identity
  • Riemannian metric
  • Measure and Integration

• ERGODISM
  • Types of flow
  • The ergodic problem

The following sections are included:

• Inner product space

• Norm

• Normed vector spaces

• Hilbert space

• Banach space

• Topological vector spaces

• Function spaces

The following sections are included:

• Quantization

• Banach algebras

• *-algebras and C*-algebras

• From Geometry to Algebra

• von Neumann algebras

• The Jones polynomials

The following sections are included:

• Introduction

• LINEAR REPRESENTATIONS
  • Generalities
  • Dimension
  • Unitary representations
  • Equivalent representations
  • Characters
  • Irreducible representations
  • Tensor products

• REGULAR REPRESENTATION
  • Invariant spaces
  • Invariant measures
  • Generalities
  • Relation to von Neumann algebras

• FOURIER EXPANSIONS
  • The standard cases
  • Pontryagin duality
  • Noncommutative harmonic analysis
  • The Peter-Weyl theorem
  • Tanaka-Krein duality
  • Quantum groups

The following sections are included:

• CURVES
  • Variation of a curve
  • Variation fields
  • Path functionals
  • Functional differentials
  • Second-variation

• GENERAL FUNCTIONALS
  • Functionals
  • Linear functionals
  • Operators
  • Derivatives – Fréchet and Gateaux

The following sections are included:

• Introduction

• EXTERIOR VARIATIONAL CALCULUS
  • Lagrangian density
  • Variations and differentials
  • The action functional
  • Variational derivative
  • Euler Forms
  • Higher order Forms
  • Relation to operators
  • Continuum Einstein convention

• EXISTENCE OF A LAGRANGIAN
  • Inverse problem of variational calculus
  • Helmholtz-Vainberg theorem
  • Equations with no lagrangian
    • Navier-Stokes equation
    • Korteweg-de Vries equation

• BUILDING LAGRANGIANS
  • The homotopy formula
  • Examples
    • The Helmholtz-Korteweg lagrangian
    • Born-Infeld electrodynamics
    • Einstein's equations
    • Electrodynamics
    • Complex scalar field
    • Second order fermion equation
  • Symmetries of equations

The following sections are included:

• Index of a curve

• Index of a singular point

• Relation to topology

• Basic two-dimensional singularities

• Critical points

• Morse lemma

• Morse indices and topology

• Catastrophes

The following sections are included:

• Introduction

• STRUCTURE EQUATIONS
  • Moving frames
  • The Cartan lemma
  • Adapted frames
  • Second quadratic form
  • First quadratic form

• RIEMANNIAN STRUCTURE
  • Curvature
  • Connection
  • Gauss, Ricci and Codazzi equations
  • Riemann tensor

• GEOMETRY OF SURFACES
  • Gauss Theorem

• RELATION TO TOPOLOGY
  • The Gauss-Bonnet theorem
  • The Chern theorem

The following sections are included:

• The old controversy

• The curvature of a metric space

• The spherical case

• The Boliyai-Lobachevsky case

• On the geodesic curves

• The Poincaré space

The following sections are included:

• Introduction

• SELF-PARALLEL CURVES
  • In General Relativity
  • In Optics
  • As a character of a connection
  • The absolute derivative
  • Self-parallelism
  • Complete spaces
  • Fermi transport

• CONGRUENCES OF CURVES
  • Jacobi equation
  • Vorticity, shear and expansion
  • Landau-Raychaudhury equation

The following sections are included:

• Introduction

• Symplectic structure

• Time evolution

• Canonical transformations

• Phase spaces as bundles

• The algebraic structure

• Relations between Lie algebras

• Liouville integrability

The following sections are included:

• HAMILTON-JACOBI
  • Hamiltonian structure
  • Hamilton-Jacobi equation

• THE LAGRANGE DERIVATIVE
  • The Lagrange derivative as a covariant object

• THE RIGID BODY
  • Frames
  • The configuration space
  • The phase space
  • Dynamics
  • The "space" and the "body" derivatives
  • The reduced phase space
  • Moving frames
  • The rotation group
  • Left- and right-invariant fields
  • The Poinsot construction

The following sections are included:

• Introduction

• STATISTICAL MECHANICS
  • General overview

• LATTICE MODELS
  • The Ising model
  • Spontaneous breakdown of symmetry
  • The Potts model
  • Cayley tree and Bethe lattice
  • The four-color problem

• ELASTICITY
  • Regularity and defects
  • Classical elasticity
  • Nematic systems
  • The Franck index

The following sections are included:

• Characteristics

• Partial differential equations

• Maxwell's equations in a medium

• The eikonal equation

The following sections are included:

• Introduction

• The light ray equation

• Hamilton's point of view

• Relation to geodesies

• The Fermat principle

• Maxwell's fish-eye

• Fresnel's ellipsoid

The following sections are included:

• THE FUNDAMENTAL FIELDS
  • Introduction

• SPACETIME TRANSFORMATIONS
  • The Poincaré group
  • The basic cases

• INTERNAL TRANSFORMATIONS
  • Global and local gauge transformations

• LAGRANGIAN FORMALISM
  • The Euler-Lagrange Equation
  • First Noether's theorem
  • Minimal Coupling Prescription
  • Local phase transformations
  • Second Noether's theorem
  • Using general frames

The following sections are included:

• THE GAUGE TENETS
  • Introduction
  • Electromagnetism
  • Nonabelian theories
  • The gauge prescription
  • Hamiltonian approach
  • Exterior differential formulation

• FUNCTIONAL DIFFERENTIAL APPROACH
  • Functional Forms
  • The space of gauge potentials
  • Gauge conditions
  • Gauge anomalies
  • BRST symmetry

• CHIRAL FIELDS
  • Some comments on chiral fields

The following sections are included:

• Einstein's equation

• The equivalence principle

• Spinors and torsion

The following sections are included:

• General characteristics

• Curvature

• Geodesics and Jacobi equations

• Some qualitative aspects

• Wigner-Inönü contraction

The following sections are included:

• Symmetries and anomalies

• The Souriau momentum

• The Kirillov form

• Integrability revisited

• Classical Yang-Baxter equation

The following sections are included:

• GLOSSARY

• REFERENCES

• ALPHABETIC INDEX