The Geometry of Physics - An Introduction


Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)


physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics

This page provides a hyperlinked index for the book

  • Theodore Frankel,

    The Geometry of Physics - An introduction

    Cambridge University Press, 1997, 2004, 2012

    website (with errata and preface for 3rd edition)

on methods of differential geometry and their meaning and use in physics, especially gravity and gauge theory.

Among the nice aspects of the book are

Related books are


I Manifolds, Tensors and Exterior Forms

1 Manifolds and Vector Fields

1.1 Submanifolds of Euclidean space

1.1a Submanifolds of N\mathbb{R}^N

1.1b The Geometry of Jacobian Matrices: The “Differential”

1.1c The main theorem on submanifolds of N\mathbb{R}^N

1.1d A Nontrivial Example: The Configuration Space of a Rigid Body

1.2 Manifolds

1.3 Tangent Vectors and Mappings

1.4 Vector Fields and Flows

2 Tensors and Exterior Forms

2.1 Covectors and Riemannian Metrics

2.2 The Tangent Bundle

2.3 The Cotangent Bundle and Phase Space

2.4 Tensors

2.5 The Grassmann or Exterior Algebra

2.6 Exterior Differentiation

2.7 Pull-Backs

2.8 Orientation and Pseudoforms

2.9 Interior Products and Vector Analysis

2.10 Dictionary

3 Integration of Differential Forms

3.1 Integration over a Parameterized Subset

3.2 Integration over Manifolds with Boundary

3.3 Stokes’ Theorem

3.4 Integration of Pseudoforms

3.5 Maxwell’s Equations

4 The Lie Derivative

4.1 The Lie Derivative of a Vector Field

4.2 The Lie Derivative of a Form

4.3 Differentiation of Integrals

4.4 A Problem Set on Hamiltonian Mechanics

5 The Poincaré Lemma and Potentials

6 Holonomic and Nonholonomic Constraints

II Geometry and Topology

7 3\mathbb{R}^3 and Minkowski Space

7.1 Curvature and Special Relativity

7.2 Electromagnetism in Minkowski Space

8 The Geometry of Surfaces in 3\mathbb{R}^3

8.1 The First and Second Fundamental Form

8.2 Gaussian and Mean Curvature

8.3 The Brouwer Degree of a Map: A Problem Set

8.4 Area, Mean Curvature, and Soap Bubbles

8.5 Gauss’ Theorema Egregium

  • theorema egregium?

8.6 Geodesics

8.7 The Parallel Displacement of Levi-Civita

9 Covariant Differentiation and Curvature

9.1 Covariant Differentiation

9.2 The Riemannian Connection

9.3 Cartan’s Exterior Covariant Differential

9.4 Change of Basis anf Gauge Transformations

9.5 The Curvature Forms in a Riemannian Manifold

9.6 Parallel Displacement and Curvature on a Surface

9.7 Riemann’s Theorem and Horizontal Distribution

10 Geodesics

10.1 Geodesics and Jacobian Fields

10.2 Variational Principles in Mechanics

10.3 Geodesics, Spiders, and the Universe

11 Relativity, Tensors, and Curvature

11.1 Heuristic’s of Einstein’s Theory

11.2 Tensor analysis

11.3 Hilbert’s Action Principle

11.4 The Second Fundamental Form in the Riemannian Case

11.5 The Geometry of Einstein’s Equations

12 Curvature and Topology: Synge’s Theorem

13 Betti Numbers and De Rham’s Theorem

13.1 Singular Chains and Their Boundaries

13.2 The Singular Homology Groups

13.3 Homology Groups of Familiar Manifolds

13.4 De Rham’s Theorem

14 Harmonic Forms

14.1 The Hodge Operators

14.2 Harmonic Forms

14.3 Boundary Values, Relative Homology, and Morse Theory

II Lie Groups, Bundles and Chern Forms

15 Lie Groups

15.1 Lie Groups, Invariant Vector Fields and Forms

15.2 One Parameter Subgroups

15.3 The Lie Algebra of a Lie Group

15.4 Subgroups and Subalgebras

16 Vector Bundles in Geometry and Physics

16.1 Vector Bundles

16.2 Poincaré’s Theorem and the Euler Characteristic

16.3 Connections in a Vector Bundle

16.4 The Electromagnetic Connection

Fiber Bundles, Gauss-Bonnet, and Topological Quantization

17.1 Fiber Bundles and Principal Bundles

17.2 Coset Spaces

17.3 Chern’s Proof of the Gauss-Bonnet-Poincaré Theorem

17.4 Line Bundles, Topological Quantization, and Berry Phase

18 Connections and Associated Bundles

18.1 Forms with Values in a Lie Algebra

18.2 Associated Bundles and Connections

18.3 rr-Form Sections of a Vector Bundle: Curvature

19 The Dirac Equation

19.1 The Groups SO(3)SO(3) and SU(2)SU(2)

19.2 Hamilton, Clifford, and Dirac

19.3 The Dirac Algebra

19.4 The Dirac Operator in Minkowski Space

19.5 The Dirac Operator in Curved Space-Time

20 Yang-Mills Fields

20.1 Noether’s Theorem for Internal Symmetries

20.2 Weyl’s Gauge Invariance Revisited

20.3 The Yang-Mills Nucleon

  • nucleon?

20.4 Compact Groups and Yang-Mills Action

20.5 The Yang-Mills Equation

20.6 The Yang-Mills Instanton

21 Betti Numbers and Covering Spaces

21.1 Bi-invariant Forms on Compact Groups

21.2 The Fundamental Group and Covering Spaces

21.3 The Theorem of S.B. Myers: A Problem Set

21.4 The Geometry of a Lie Group

22 Chern Forms and Homotopy Groups

22.1 Chern-Forms and Winding Numbers

22.2 Homotopies and Extensions

22.3 The Higher Homotopy Groups π k(M)\pi_k(M)

22.4 Some Computations of Homotopy Groups

22.5 Chern Forms as Obstructions

Appendix A. Forms in Continuum Mechanics

Appendix B. Harmonic Chains and Kirchhoff’s Circuit Law

Appendix C. Symmetries, Quarks, and Meson Masses

Appendix D. Representations and Hyperelastic Bodies

Appendix E. Orbits and Morse-Bott Theory in Compact Lie Groups

category: reference

Revised on September 8, 2015 06:42:57 by Urs Schreiber (