Modern Differential Geometry for Physicists

The following sections are included:

• Preface

• Contents

The following sections are included:

• Preliminary Remarks
  • Remarks on differential geometry
  • Remarks on topology

• Metric Spaces
  • The simple idea of convergence
  • The idea of a metric space
  • Examples of metric spaces
  • Operations on metrics
  • Some topological concepts in metric spaces

• Partially Ordered Sets and Lattices
  • Partially ordered sets
  • Lattices

• General Topology
  • An example of non-metric convergence
  • The idea of a neighbourhood space
  • Topological spaces
  • Some examples of topologies on a finite set
  • A topology as a lattice
  • The lattice of topologies τ(X) on a set X
  • Some properties of convergence in a general topological space
  • The idea of a compact space
  • Maps between topological spaces
  • The idea of a homeomorphism
  • Separation axioms
  • Frames and locales

The following sections are included:

• Preliminary Remarks

• The Main Definitions
  • Coordinate charts
  • Some examples of differentiable manifolds
  • Differentiable maps

• Tangent Spaces
  • The intuitive idea
  • A tangent vector as an equivalence class of curves
  • The vector space structure on
  • The push-forward of an equivalence class of curves
  • Tangent vectors as derivations
  • The tangent space T_vV of a vector space V
  • A simple example of the push-forward operation
  • The tangent space of a product manifold

The following sections are included:

• Vector Fields
  • The main definition
  • The vector field commutator
  • h-related vector fields

• Integral Curves and Flows
  • Complete vector fields
  • One-parameter groups of diffeomorphisms
  • Local flows
  • Some concrete examples of integral curves and flows

• Cotangent Vectors
  • The algebraic dual of a vector space
  • The main definitions
  • The pull-back of a one-form
  • A simple example of the pull-back operation
  • The Lie derivative

• General Tensors and n-Forms
  • The tensor product operation
  • The idea of an n-form
  • The definition of the exterior derivative
  • The local nature of the exterior derivative

• DeRham Cohomology

The following sections are included:

• The Basic Ideas
  • The first definitions
  • The orthogonal group

• The Lie Algebra of a Lie Group
  • Left-invariant vector fields
  • The completeness of a left-invariant vector field
  • The exponential map
  • The Lie algebra of GL(n, ℝ)

• Left-Invariant Forms
  • The basic definitions
  • The Cartan-Maurer form

• Transformation Groups
  • The basic definitions
  • Different types of group action
  • The main theorem for transitive group actions
  • Some important transitive actions

• Infinitesimal Transformations
  • The induced vector field
  • The main result

The following sections are included:

• Bundles in General
  • Introduction
  • The definition of a bundle
  • The idea of a cross-section
  • Covering spaces and sheaves
  • The definition of a sub-bundle
  • Maps between bundles
  • The pull-back operation
  • Universal bundles

• Principal Fibre Bundles
  • The main definition
  • Principal bundle maps
  • Cross-sections of a principal bundle

• Associated Bundles
  • The main definition
  • Associated bundle maps
  • Restricting and extending the structure group
  • Riemannian metrics as reductions of
  • Cross-sections as functions on the principle bundle

• Vector Bundles
  • The main definitions
  • Vector bundles as associated bundles

The following sections are included:

• Connections in a Principal Bundle
  • The definition of a connection
  • Local representatives of a connection
  • Local gauge transformations
  • Connections in the frame bundle

• Parallel Transport
  • Parallel transport in a principal bundle
  • Parallel transport in an associated bundle
  • Covariant differentiation
  • The curvature two-form

The following sections are included:

• BIBLIOGRAPHY

• INDEX