An Introduction to Geometrical Physics 2nd Edition by Aldrovandi, Pereira 9789813146839 9813146834

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ISBN 10: 9813146834
ISBN 13: 9789813146839
Author: Aldrovandi, Pereira

This book focuses on the unifying power of the geometrical language in bringing together concepts from many different areas of physics, ranging from classical physics to the theories describing the four fundamental interactions of Nature — gravitational, electromagnetic, strong nuclear, and weak nuclear.The book provides in a single volume a thorough introduction to topology and differential geometry, as well as many applications to both mathematical and physical problems. It is aimed as an elementary text and is intended for first year graduate students.In addition to the traditional contents of books on special and general relativities, this book discusses also some recent advances such as de Sitter invariant special relativity, teleparallel gravity and their implications in cosmology for those wishing to reach a higher level of understanding.

An Introduction to Geometrical Physics 2nd Table of contents:

Part 1: MANIFOLDS

General Topology

Introductory remarks

Topological spaces

Kinds of texture

Functions

Quotients and groups

Homology

Introductory remarks

Graphs

Graphs, first way

Graphs, second way

The first topological invariants

Simplexes, complexes and all that

Topological numbers

Final remarks

Homotopy

General homotopy

Path homotopy

Homotopy of curves

The fundamental group

Some examples

Covering spaces

Multiply-connected spaces

Riemann surfaces

Higher homotopy

Manifolds and Charts

Manifolds

Topological manifolds

Dimensions, integer and other

Charts and coordinates

Differentiable Manifolds

Definition and overlook

Smooth functions

Differentiable submanifolds

Part 2: DIFFERENTIABLE STRUCTURE

Tangent Structure

Introduction

Tangent spaces

Tensors on manifolds

Fields and transformations

Frames

Metric and riemannian manifolds

Differential Forms

Introduction

Exterior derivative

Vector-valued forms

Duality and coderivation

Integration and homology

Integration

Cohomology of differential forms

Algebras, endomorphisms and derivatives

Symmetries

Lie groups

Transformations on manifolds

Lie algebra of a Lie group

The adjoint representation

Fiber Bundles

Introduction

Vector bundles

The bundle of linear frames

Structure group

Soldering

Orthogonal groups

Reduction

Tetrads

Linear connections

Principal bundles

General connections

Bundle classification

Part 3: NATURE’S EXTREME GEOMETRIES

Quantum Geometry

Quantum goups: a pedestrian outline

Noncommutative geometry

Cosmology: the Standard Model

The geometrical perspective

The physical content

The cosmos curvature

Planck Scale Kinematics

Part 4: MATHEMATICAL TOPICS

The Basic Algebraic Structures

Some general concepts

Groups and lesser structures

Rings, ideals and fields

Modules and vector spaces

Algebras

Coalgebras

Discrete Groups: Braids and Knots

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

Sets and Measures

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

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

Introductory remarks

Linear representations

Regular representation

Fourier expansions

Variations and Functionals

Curves

Variation of a curve

Variation fields

Path functionals

Functional differentials

Second variation

General functionals

Functionals

Linear functionals

Operators

Derivatives: Fréchet and Gateaux

Functional Forms

Introductory remarks

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

Building lagrangians

The homotopy formula

Examples

Symmetries of equations

Final remarks

Singular Points

Index of a curve

Index of a singular point

Relation to topology

Basic two-dimensionals singularities

Critical points

Morse lemma

Morse indices and topology

Catastrophes

Euclidean Spaces and Subspaces

Introductory remarks

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

Non-Euclidean Geometries

The age-old controversy

The curvature of a metric space

The spherical case

The Boliyai–Lobachevsky case

On the geodesic curves

The Poincaré space

Geodesics

Introductory remarks

Self-parallel curves

In General Relativity

In optics

The absolute derivative

Self-parallelism

Complete spaces

Fermi transport

Congruences

Jacobi equation

Vorticity, shear and expansion

Landau–Raychaudhury equation

Part 5: 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 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

Symmetries on Phase Space

Symmetries and anomalies

The Souriau momentum

The Kirillov form

Integrability revisited

Classical Yang–Baxter equation

Statistics and Elasticity

Statistical mechanics

Introduction

General overview

Lattice models

The Ising model

Spontaneous breakdown of symmetry

The Potts model

Cayley trees and Bethe lattices

The four-color problem

Elasticity

Regularity and defects

Classical elasticity

Nematic systems

The Franck index

Propagation of Discontinuities

Characteristics

Partial differential equations

Maxwell’s equations in a medium

The eikonal equation

Geometrical Optics

Introduction

The light-ray equation

Hamilton’s point of view

Relation to geodesics

The Fermat principle

Maxwell’s fish-eye

Fresnel’s ellipsoid

Classical Relativistic Fields

The fundamental fields

Spacetime transformations

The Poincaré group

The basic cases

Internal transformations

Lagrangian formalism

The Euler–Lagrange equation

First Noether’s theorem

Minimal coupling prescription

Local phase transformations

Second Noether’s theorem

Gauge Fields: Fundamentals

Introductory remarks

The gauge tenets

Electromagnetism

Non-abelian theories

The gauge prescription

Hamiltonian approach

Exterior differential formulation

Functional differential approach

Functional forms

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