The geometry of physics by Theodore Frankel in 2012

differential geometry, physics, mathematics, theoretical physics

0. Overview: An informal overview of Cartan’s exterior differential forms, illustrated with an application to Cauchy’s stress tensor

Introduction

• 0.a Introduction

Vectors, 1-forms, and tensors

• 0.b Two kinds of vectors

  • \[ \mathbf{v} = \mathbf{\partial} v\]

    • \[\mathbf{\partial} = \left( \mathbf{\partial}_1 , \ldots , \mathbf{\partial}_n \right)\]
    • \[v = \left( v^1 , \cdots ; v^n \right)^T\]

• 0.c Superscripts, subscripts, summation convention

• 0.d Riemmanian metrics progress indicators

• 0.e Tensors

Integrals and exterior forms

• 0.f Line integrals

• 0.g Exterior 2-forms

• 0.h Exterior p-forms and algebra in \[\mathbb{R}^n\]

• 0.i The exterior differential \[d\]

• 0.j The push-forward of a vector and the pull-back of a form

• 0.k Surface integrals and “Stokes’ Theorem”

• 0.l Electromagnetism, or, Is it a vector or a form?

• 0.m Interior products

• 0.n Volume forms and Cartan’s vector valued exterior forms

• 0.o Magnetic field for current in a straight wire

Electricity and stresses

• 0.p Cauchy stress, floating bodies, twisted cylinders, and strain energy

• 0.q Sketch of Cauchy’s “First theorem”

• 0.r Sketch of Cauchy’s “Second theorem” moments as generators of rotations

• 0.s A remarkable formula for differentiating Line, Surface, and …, Integrals

I. Manifolds, tensors, and exterior forms

1. Manifolds and vector fields

• 1.1 Submanifolds of Euclidean space

  • 1.1a Submanifolds of \[\mathbb{R}^N\]
  • 1.1b The geometry of Jacobian matrices: The “Differential”
  • 1.1c The main theorem on submanifolds of \[\mathbb{R}^N\]
  • 1.1d A nontrivial example: the configuration space of a rigid body

• 1.2 Manifolds

  • a. Some notions from point set topology
  • b. The idea of a manifold
  • c. A rigorous definition of a manifold
  • d. Complex manifolds: the Riemann sphere

• 1.3 Tangent vectors and mappings

  • a. Tangent or contravariant vectors
  • b. Vectors as differential operators
  • c. The tangent space to M^n at a point
  • d. Mappings and submanifolds of manifolds
  • e. Change of coordinates

• 1.4 Vector fields and flows

  • a. Vector fields and flows on \[\mathbb{R}^n\]
  • b. Vector fields on manifolds
  • c. Straightening 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 parametrized subset

• 3.2 Integration over manifolds with boundary

• 3.3 Stokes’s 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 Poincare lemma and potentials

• 5.1 A more general Stokes’s theorem

• 5.2 Closed forms and exact forms

• 5.3 Complex analysis

• 5.4 The converse to the Poincare lemma

• 5.5 Finding potentials

6. Holonomic and nonholonomic constraints

• 6.1 The Frobenius integrability condition

• 6.2 Integrability and constraints

• 6.3 Heuristic thermodynamics via Caratheodory

II. Geometry and topology

7. \[\mathbb{R}^3\] and Minkowski space

• 7.1 Curvature and special relativity

  • a. Curvature of a space curve in \[\mathbb{R}^3\]
  • b. Minkowski space and special relativity
  • c. Hamiltonian formulation

• 7.2 Electromagnetism in Minkowski space

  • a. Minkowski’s electromagnetic field tensor
  • b. Maxwell’s equations

8. The geometry of surfaces in \[\mathbb{R}^3\]

• 8.1 The first and second fundamental forms

  • a. The first fundamental form, or metric tensor
  • b. The second fundamental form

• 8.2 Gaussian and mean curvatures

  • a. Symmetry and self-adjointness
  • b. Principal normal curvatures
  • c. Gauss and mean curvatures: the Gauss normal map

• 8.3 The Brouwer degree of a map: a problem set

  • a. The Brouwer degree
  • b. Complex analytic (Holomorphic) maps
  • c. The Gauss normal map revisited: the Gauss-Bonnet theorem
  • d. The Kronecker Index of a vector field
  • e. The Gauss looping integral

• 8.4 Area, mean curvature, and soap bubbles

  • a. The first variation of area
  • b. Soap bubbles and minimal surfaces

• 8.5 Gauss’s theorema egregium

  • a. The equations of Gauss and Codazzi
  • b. The Theorema Egregium

• 8.6 Geodesics

  • a. The first variation of arc length
  • b. The intrinsic derivative and the geodesic equation

• 8.7 The parallel displacement of Levi-Civita

9. Covariant differentiation and curvature

• 9.1 Covariant differentiation

  • a. Covariant derivative
  • b. Curvature of an affine connection
  • c. Torsion and symmetry

• 9.2 The Riemmannian connection

• 9.3 Cartan’s exterior covariant differential

  • a. Vector-valued forms
  • b. The Covariant differential of a vector field
  • c. Cartan’s structural equations
  • d. The exterior covariant differential of a vector-valued form
  • e. The curvature 2-forms

• 9.4 Change of basis and gauge transformations

  • a. Symmetric connections only
  • b. Change of frame

• 9.5 The curvature forms in a Riemannian manifold

  • a. The Riemmannian connection
  • b. Riemannian surfaces \[M^2\]

• 9.6 Parallel displacement and curvature on a surface

• 9.7 Riemann’s theorem and the horizontal distribution

  • a. Flat metrics
  • b. The horizontal distribution of an affine connection
  • c. Riemann’s theorem

10. Geodesics

• 10.1 Geodesics and Jacobi fields

  • a. Vector fields along a surface in \[M^n\]
  • b. Geodesics
  • c. Jacobi fields
  • d. Energy

• 10.2 Variational principles in mechanics

  • a. Hamilton’s principle in the tangent bundle
  • b. Hamilton’s principle in phase space
  • c. Jacobi’s principle of least action
  • d. Closed geodesics and periodic motions

• 10.3 Geodesics, spiders, and the universe

  • a. Gaussian coordinates
  • b. Normal coordinates on a surface
  • c. Spiders and the universe

11. Relativity, tensors, and curvature

• 11.1 Heuristics of Einstein’s theory

  • a. The metric potentials
  • b. Einstein’s field equations
  • c. Remarks on static metrics

• 11.2 Tensor analysis

  • a. Covariant differentiation of tensors
  • b. Riemannian connections and the Bianchi identities
  • c. Second covariant derivatives: the Ricci identities

• 11.3 Hilbert’s action principle

  • a. Geodesics in a pseudo-Riemannian manifold
  • b. Normal coordinates, the divergence and Laplacian
  • c. Hilbert’s variational approach to general relativity

• 11.4 The second fundamental form in the Riemannian case

  • a. The induced connection and the second fundamental form
  • b. The equations of Gauss and Codazzi
  • c. The interpretation of the sectional curvature
  • d. Fixed points of isometries

• 11.5 The geometry of Einstein’s equations

  • a. The Einstein tensor in a pseudo Riemannian space-time
  • b. The relativistic meaning of Gauss’ equation
  • c. The second fundamental form of a spatial slice
  • d. The Codazzi equations
  • e. Some remarks on the Schwarzschild solution

12. Curvature and topology: Synge’s theorem

• 12.1 Synge’s formula for second variation

  • a. The second variation of arc length
  • b. Jacobi fields

• 12.2 Curvature and simple connectivity

  • a. Synge’s theorem
  • b. Orientability revisited

13. Betti numbers and De Rham’s theorem

• 13.1 Singular chains and their boundaries

  • a. Singular chains
  • b. Some 2-dimensional examples

• 13.2 The singular homology groups

  • a. Coefficient fields
  • b. Finite simplicial complexes
  • c. Cycles, boundaries, homology and Betti numbers

• 13.3 Homology groups of familiar manifolds

  • a. Some computational tools
  • b. Familiar examples

• 13.4 De Rham’s theorem

  • a. The statement of de Rham’s theorem
  • b. Two examples

14. Harmonic forms

• 14.1 The Hodge operators

  • a. The * operator
  • b. The codifferential operator \[\delta = d^*\]
  • c. Maxwell’s equations in curved space-time \[M^4\]
  • d. The Hilbert Lagrangian

• 14.2 Harmonic forms

  • a. The Laplace operator on forms
  • b. The Laplacian of a 1-form
  • c. Harmonic forms on closed manifolds
  • d. Harmonic forms and de Rham’s theorem
  • e. Bochner’s theorem

• 14.3 Boundary values, relative homology, and Morse theory

  • a. Tangential and normal differential forms
  • b. Hodge’s theorem for tangential forms
  • c. Relative homology groups
  • d. Hodge’s theorem for normal forms
  • e. Morse’s theory of critical points

III. Lie groups, bundles, and Chern forms

15. Lie groups

• 15.1 Lie groups, invariant vector fields and forms

  • a. Lie groups
  • b. Invariant vector fields and forms

• 15.2 One parameter subgroups

• 15.3 The Lie algebra of a Lie group

  • a. The Lie algebra
  • b. The exponential map
  • c. Examples of Lie algebras
  • d. Do the 1-parameter subgroups cover G?

• 15.4 Subgroups and subalgebras

  • a. Left invariant fields generate right translations
  • b. Commutators of matrices
  • c. Right invariant fields
  • d. Subgroups and subalgebras

16. Vector bundles in geometry and physics

• 16.1 Vector bundles

  • a. Motivation by two examples
  • b. Vector bundles
  • c. Local trivializations
  • d. The normal bundle to a submanifold

• 16.2 Poincare’s theorem and the Euler characteristic

  • a. Poincare’s theorem
  • b. The Stiefel vector field and Euler’s theorem

• 16.3 Connections in a vector bundle

  • a. Connection in a vector bundle
  • b. Complex vector spaces
  • c. The structure group of a bundle
  • d. Complex line bundles

• 16.4 The electromagnetic connection

  • a. Lagrange’s equations without electromagnetism
  • b. The modified Lagrangian and Hamiltonian
  • c. Schrodinger’s equation in an electromagnetic field
  • d. Global potentials
  • e. The Dirac monopole
  • f. The Aharonov-Bohm effect

17. fiber bundles, Gauss-Bonnet, and topological quantization

• 17.1 fiber bundles and principal bundles

  • a. fiber bundles
  • b. principal bundles and frame bundles
  • c. Action of the structure group on a principal bundle

• 17.2 Coset spaces

  • a. Cosets
  • b. Grassmann manifolds

• 17.3 Chern’s proof of the Gauss-Bonnet-Poincare theorem

  • a. A connection in the frame bundle of a surface
  • b. The Gauss-Bonnet-Poincare theorem
  • c. Gauss-Bonnet as an index theorem

• 17.4 Line bundles, topological quantization and Berry phase

  • a. A generalization of Gauss-Bonnet
  • b. Berry phase
  • c. Monopoles and the Hopf bundle

18. Connections and associated bundles

• 18.1 Forms with values in a Lie algebra

  • a. The Maurer-Cartan form
  • b. g-valued p-forms on a Manifold
  • c. Connections in a principal bundle

• 18.2 Associated bundles and connections

  • a. Associated bundles
  • b. Connections in associated bundles
  • c. The associated \[Ad\] bundle

• 18.3 r-Form sections of a vector bundle: curvature

  • a. r-Form sections of E
  • b. Curvature and the \[Ad\] bundle

19. The Dirac equation

• 19.1 The groups \[SO(3)\] and \[SU(2)\]

  • a. The rotation group \[SO(3)\] of \[\mathbb{R}^3\]
  • b. \[SU(2)\]: the Lie algebra \[su(2)\]
  • c. \[SU(2)\] is topologically the 3-sphere
  • d. \[Ad \colon SU(2) \rightarrow SO(3)\] in more detail

• 19.2 Hamilton, Clifford, and Dirac

  • a. spinors and rotations of \[\mathbb{R}^3\]
  • b. Hamilton on composing two rotations
  • c. Clifford algebras
  • d. The Dirac program: the square root of the d’Alembertian

• 19.3 The Dirac algebra

  • a. The Lorentz algebra
  • b. The Dirac algebra

• 19.4 The Dirac operator \[\partial\] in Minkowski space

  • a. Dirac spinors
  • b. The Dirac operator

• 19.5 The Dirac operator in curved space-time

  • a. The spinor bundle
  • b. The spin connection in SM

20. Yang-Mills fields

• 20.1 Noether’s theorem for internal symmetries

  • a. The tensorial nature of Lagrange’s equations
  • b. Boundary conditions
  • c. Noether’s theorem for internal symmetries
  • d. Noether’s principle

• 20.2 Weyl’s gauge invariance revisited

  • a. The Dirac Lagrangian
  • b. Weyl’s Gauge invariance revisited
  • c. The electromagnetic Lagrangian
  • d. Quantization of the A field: photons

• 20.3 Yang-Mills nucleon

  • a. The Heisenberg nucleon
  • b. The Yang-Mills nucleon
  • c. A remark on terminology

• 20.4 Compact groups and Yang-Mills action

  • a. The unitary group is compact
  • b. Averaging over a compact group
  • c. Compact matrix groups are subgroups of unitary groups
  • d. \[Ad\] invariant scalar products in the Lie algebra of a compact group
  • e. The Yang-Mills action

• 20.5 The Yang-Mills equation

  • a. The exterior covariant divergence \[\Delta^*\]
  • b. The Yang-Mills analogy with electromagnetism
  • c. Further remarks on the Yang-Mills equations

• 20.6 Yang-Mills instantons

  • a. Instantons
  • b. Chern’s proof revisited
  • c. Instantons and the vacuum

21. Betti numbers and covering spaces

• 21.1 Bi-invariant forms on compact groups

  • a. Bi-invariant p-forms
  • b. The Cartan p-forms
  • c. Bi-invariant Riemannian metrics
  • d. Harmonic forms in the bi-invariant metric
  • e. Weyl and Cartan on the Betti numbers of G

• 21.2 The fundamental group and covering spaces

  • a. Poincare’s fundamental group \[\pi_1(M)\]
  • b. The concept of a covering space
  • c. The universal covering
  • d. The orientable covering
  • e. Lifting paths
  • f. Subgroups of \[\pi(M)\]
  • g. The universal covering group

• 21.3 The theoreme of S.B. Myers: A problem set

• 21.4 The geometry of a Lie group

  • a. The connection of a bi-invariant metric
  • b. The flat connections

22. Chern forms and homotopy groups

• 22.1 Chern forms and winding numbers

  • a. The Yang-Mills winding number
  • b. Winding number in terms of field strength
  • c. The Chern forms for a \[U(n)\] bundle

• 22.2 Homotopies and extensions

  • a. Homotopy
  • b. Covering homotopy
  • c. Some topology of \[SU(n)\]

• 22.3 The higher homotopy groups \[\pi_k(M)\]

  • a. \[\pi_k (M)\]
  • b. Homotopy groups of spheres
  • c. Exact sequences of groups
  • d. The homotopy sequence of a bundle
  • e. The relation between homotopy and homology groups

• 22.4 Some computations of homotopy groups

  • a. Lifting spheres from \[M\] into the bundle \[P\]
  • b. \[SU(n)\] again
  • c. The Hopf map and fibering

• 22.5 Chern forms as obstructions

  • a. The Chern forms \[c_r\] for an \[SU(n)\] bundle revisited
  • b. \[c_2\] as an obstruction cocycle
  • c. The meaning of the integer \[j(\Delta_4)\]
  • d. Chern’s integral
  • e. Concluding remarks

Appendices

A. Forms in continuum mechanics

• a The equations of motion of a stressed body

• b. Stresses are vector valued \[(n-1)\] pseudo-forms

• c. The Piola-Kirchoff stress tensors \[S\] and \[P\]

• d. Strain energy rate

• e. Some typical computations using forms

• f. Concluding remarks

B. Harmonic chains and Kirchhoff’s circuit laws

• a. Chain complexes

• b. Cochains and cohomology

• c. Transpose and adjoint

• d. Laplacians and harmonic cochains

• e. Kirchoff’s circuit laws

C. Symmetries, quarks, and meson masses

• a. Flavored quarks

• b. Interactions of quarks and antiquarks

• c. The Lie algebra of \[SU(3)\]

• d. Pions, kaons, and etas

• e. A reduced symmetry group

• f. Meson masses

D. Representations and hyperelastic bodies

• a. Hyperelastic bodies

• b. Isotropic bodies

• c. Application of Schur’s lemma

• d. Frobenius-Schur relations

• e. The symmeteric traceless \[3 \times 3\] matrices are irreducible

E. Orbits and Morse-Bott theory in compact Lie groups

• a. The topology of conjugacy orbits

• b. Application of Bott’s extension of Morse theory