Differential Geometry for Physicists

The following sections are included:

• Contents

• Preface

The following sections are included:

• Manifold

• Differentiable manifold

• Tangent space and tangent vector field

• Cotangent vector field

• Tensor product, exterior product and various higher order tensor fields

• Exterior differentiation

• Orientation and Stokes formula

• Notations and formulae

• Exercises

The following sections are included:

• Continuous mapping between manifolds and its induced mapping

• Integral submanifold and Frobenius theorem

• Integrability of differential equations and Frobenius theorem in terms of differential forms

• The flow of vector fields, one parameter local Lie transformation groups and Lie derivative

• Lie group, Lie algebra and exponential map

• Lie transformation groups, orbit and the space of orbits

• Notations and Formulae

• Exercises

The following sections are included:

• Moving frame approach to tensor field

• Affine connection and covariant differentiation

• The curvature 2-form and the curvature tensor

• Torsion tensor

• Covariant exterior differential

• Holonomy group of connections

• Berry phase, holonomy in physical system

• Notations and Formulea

• Exercises

The following sections are included:

• Metric tensor field, Hodge star and codifferentiation

• Riemannian connection

• Riemannian curvature

• Bianchi identity and Einstein field equation of gravity

• Isometry, conformal transformation and constant curvature space

• Orthogonal frame field and spin connection

• Surfaces and curves in 3-dimensional Euclidean space

• The computation of Riemannian curvature tensor

• Pseudosphere and Bäcklund transformation

• Notations and Formulae

• Exercises

The following sections are included:

• Symplectic manifold

• Special submanifolds of symplectic manifold

• Symplectic and Hamiltonian vector fields, Poisson bracket

• Poission manifold and symplectic leaves

• Homogeneous symplectic manifold and the reduced phase space

• Contact manifold

• Notations and Formulae

• Exercises

The following sections are included:

• Complex structure of manifolds, almost complex manifolds

• Integrable condition of almost complex structure

• Hermitian manifold

• Kähler manifold

• Connections on complex manifold

• Riemannian symmetric space, its Kähler structure and nonlinear realization

• Nonlinear σ-models, soliton solutions and their geometric meaning

• Notations and Formulae

• Exercises

The following sections are included:

• Homotopic mapping and manifolds with the same homotopy type

• Singular homology group

• General homology group and universal coefficient theorem

• Cohomology theory

• de Rham cohomology theory

• Harmonic forms

• Bi-invariant form on group manifold and invariant form on symmetric space

• G-structure of manifold and its restriction to the homology group of manifold

• Notations and Formulae

• Exercises

The following sections are included:

• Homotopy group of manifold

• Relative homotopy group and exact homotopy sequence

• Relation between homotopy group and homology group

• Fibre bundle

• Principal bundle and associated bundle

• Induced bundle, reduction of fibre bundle

• The homotopy classification of fibre bundles, universal fibre bundle

• Notations and Formulae

• Exercises

The following sections are included:

• Connection and curvature on principal bundle

• Connection on associated vector bundle

• Connection on general vector bundle

• Gauge theory, action and Yang-Mills equation

• Local gauge symmetry and current conservation

• Instanton

• Yang-Mills-Higgs monopole

• Seiberg-Witten monopole equation

• Notations and Formulae

• Exercises

The following sections are included:

• Introduction, Weil homomorphism

• Chern class, the splitting principle

• Pontrjagin class

• Euler class

• Stiefel-Whitney class, orientation and spin structure

• Secondary characteristic class (Chern-Simons form)

• Generalized Chern-Simons forms

• Notations and Formulae

• Exercise

The following sections are included:

• Introduction, Euler number and the associated theorem

• Elliptic differential operator, elliptic complex and its analytic index

• Atiyah-Singer index theorem, the symbol bundle and its topological index

• Other classical elliptic complex

• Twisted elliptic complex

• Brief comment on the proof for index theorem, the heat kernel method

• Some applications in physics

• Notations and Formulae

• Exercise

The following sections are included:

• Introduction

• Index theorem for de Rham complex on manifold with boundary

• APS index theorem

• APS index theorem for spin complex, spectral boundary condition

• Index theorem on open infinite manifold

• Weak local boundary condition for Dirac operator

• Notations and Formulae

• Exercise

The following sections are included:

• Family index theorem of Dirac operator

• Relation among cohomology on orbit space, on connection space and on gauge group

• Topological obstruction of variety degree on gauge group and the Čech-de Rham double complex

• The cocycle density of gauge group and gauge algebra

• Topological properties of 4-dim quantum Yang-Mills theory and θ-vacuum

• 3-dim Yang-Mills theory and topological mass term

• Fermion interaction and quantum anomalies

• Topological interpretation of quantum anomalies

• Notations and Formulae

• Exercise

The following sections are included:

• Introduction

• Linear transformations on the quantum hyperplane, quantum group GL_q(2) and SU_q(2)

• Bicovariant calculus on quantum group SU_q(2)

• Q-gauge theory in terms of q-BRST algebra

• q-Deformed Chern Class

• q-Deformed Chern-Simons

• q-Deformed Cocycle Hierarchy

• Notations and Formulae

• Exercise

The following sections are included:

• Appendix A Simple Introduction to Set Theory
  • Basic definitions and notations
  • Equivalence relations and equivalence classes
  • Partial Ordering and Total Ordering
  • Maps

• Appendix B Preliminary Topology
  • Metric space
  • General topological space
  • Connectedness
  • Compactness
  • Product

• Appendix C Some Basic Algebraic Structures
  • Group, ring and field
  • Vector space, module and algebra
  • Euclidean space
  • Normed algebra and C*-algebra
  • Homogeneous space

• Appendix D Homomorphism of Algebraic Structure and Tensor Algebra
  • Linear functions and dual spaces, dual linear map
  • Bilinear functions and tensor products
  • Direct sum and tensor algebra
  • Derivator algebras

• Appendix E Exact Sequence of Homomorphism

• Appendix F Abelian Group
  • Linear independence and rank
  • Finite generated groups

• Appendix G Stokes' Theorem

• Appendix H Quarternion and Milnor Exotic Spheres
  • Division algebras
  • Quarternion H
  • Milnor exotic spheres

• Appendix I Generalized Kronecker δ notation
  • Definition
  • Contractions of order-p δ notation (p ≤ n)
  • Contraction of δ and tensors
  • Generalized Levi-Civita notation

• Appendix J n-dimensional Sphere Sⁿ

• Appendix K SU₍₂₎ Group Manifold

• Appendix L S² Manifold and Hopf Maps

• Appendix M Clifford Algebra, Spin Group and its Representations
  • Clifford Algebra
  • Spin Group
  • The representation of Spin(n)
  • spin(n) algebra module

• Appendix N Good Cover and the Nerve of a Cover. Čech Cohomology

• Appendix O Symmetric Trace

• Appendix P Cohomology with Compact Support and Cohomology on Fibre Bundle, Thorn Class

• Appendix Q K-theory, Topological Invariance of Elliptic Operator

• Appendix R Cohomology of Lie group and Lie algebra

• Appendix S Some Recursion Relations of Q-BRST Algebra

• References

• Index