An Introduction to Geometrical Physics

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.

Contents:

• Manifolds:

  • General Topology
  • Homology
  • Homotopy
  • Manifolds and Charts
  • Differentiable Manifolds

• Differentiable Structure:

  • Tangent Structure
  • Differential Forms
  • Symmetries
  • Fiber Bundles

• Nature's Extreme Geometries:

  • Quantum Geometry
  • Cosmology: the Standard Model
  • Planck Scale Kinematics

• Mathematical Topics:

  • The Basic Algebraic Structures
  • Discrete Groups: Braids and Knots
  • Sets and Measures
  • Topological Linear Spaces
  • Banach Algebras
  • Representations
  • Variations and Functionals
  • Functional Forms
  • Singular Points
  • Euclidian Spaces and Subspaces
  • Non-Euclidian Geometries
  • Geodesics

• Physical Topics:

  • Hamiltonian Mechanics
  • More Mechanics
  • Symmetries in Phase Space
  • Statistics and Elasticity
  • Propagation of Discontinuities
  • Geometrical Optics
  • Classical Relativistic Fields
  • Gauge Fields: Fundamentals
  • More Gauge Fields: Wu-Yang Copies
  • General Relativity
  • Teleparallel Gravity
  • Einstein–Cartan Theory
  • de Sitter Invariant Special Relativity

Readership: Graduate students and researchers.
Key Features:

• In addition to the traditional contents of books on special and general relativities, this book discusses also some recent advances, like for example, the foundations of de Sitter invariant special relativity, intended as describing the kinematics at the Planck scale, and teleparallel gravity, a gravitational theory that is equivalent to general relativity, but completely different from the conceptual point of view. It briefly discusses also some implications of these theories for cosmology