nLab The Geometry of Physics - An Introduction

Contents

Context

Differential geometry

Physics

This page provides a hyperlinked index for the book

Theodore Frankel:

The Geometry of Physics – An introduction

Cambridge University Press (1997, 2004, 2012)

doi:10.1017/CBO9781139061377

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

it discusses pseudoforms on top of ordinary differential forms, instead of just assuming that all manifolds are oriented as often done — and what's more, it explains the physical meaning of this! (But it still uses metrics more than necessary.)

Related books:

Mikio Nakahara, Geometry, Topology and Physics, Graduate Student Series in Physics (2003).
Raoul Bott, Loring Tu, Differential Forms in Algebraic Topology

(more advanced algebraic topology)
Werner Greub, Stephen Halperin, Ray Vanstone, Connections, Curvature, and Cohomology
Chris Isham, Modern Differential Geometry for Physicists

(differential geometry in mathematical physics)
David Hestenes, Space-time Algebra, New York (1966)

(Clifford algebra and spinors)

Contents

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

7.1 Curvature and Special Relativity
7.2 Electromagnetism in Minkowski Space

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

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
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 $r$ -Form Sections of a Vector Bundle: Curvature

19 The Dirac Equation

19.1 The Groups $SO(3)$ and $SU(2)$
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
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 $\pi_k(M)$
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

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 $\mathbb{R}^N$

Whitney embedding theorem

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

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

tangent bundle

2.3 The Cotangent Bundle and Phase Space

2.4 Tensors

tensor

2.5 The Grassmann or Exterior Algebra

2.6 Exterior Differentiation

de Rham differential

2.7 Pull-Backs

pullback of a differential form

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

Stokes' theorem

3.4 Integration of Pseudoforms

3.5 Maxwell’s Equations

Maxwell's equations

4 The Lie Derivative

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

Hamiltonian mechanics

5 The Poincaré Lemma and Potentials

electromagnetic potential

6 Holonomic and Nonholonomic Constraints

II Geometry and Topology

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

7.1 Curvature and Special Relativity

7.2 Electromagnetism in Minkowski Space

electromagnetism

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

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

area
mean curvature?

8.5 Gauss’ Theorema Egregium

theorema egregium?

8.6 Geodesics

geodesic

8.7 The Parallel Displacement of Levi-Civita

9 Covariant Differentiation and Curvature

9.1 Covariant Differentiation

covariant derivative

9.2 The Riemannian Connection

Levi-Civita 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

geodesic

10.2 Variational Principles in Mechanics

10.3 Geodesics, Spiders, and the Universe

observable universe

11 Relativity, Tensors, and Curvature

11.1 Heuristic’s of Einstein’s Theory

general covariance

11.2 Tensor analysis

11.3 Hilbert’s Action Principle

Einstein-Hilbert action

11.4 The Second Fundamental Form in the Riemannian Case

11.5 The Geometry of Einstein’s Equations

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

singular chain complex

13.2 The Singular Homology Groups

singular homology

13.3 Homology Groups of Familiar Manifolds

Betti number

13.4 De Rham’s Theorem

de Rham theorem

14 Harmonic Forms

14.1 The Hodge Operators

Hodge star operator

14.2 Harmonic Forms

harmonic differential form

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

subgroup

16 Vector Bundles in Geometry and Physics

16.1 Vector Bundles

vector bundle

16.2 Poincaré‘s Theorem and the Euler Characteristic

Euler characteristic

16.3 Connections in a Vector Bundle

connection on a vector bundle

16.4 The Electromagnetic Connection

line bundle with connection

Fiber Bundles, Gauss-Bonnet, and Topological Quantization

17.1 Fiber Bundles and Principal Bundles

17.2 Coset Spaces

coset

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

Gauss-Bonnet theorem

17.4 Line Bundles, Topological Quantization, and Berry Phase

18 Connections and Associated Bundles

18.1 Forms with Values in a Lie Algebra

Lie algebra valued 1-form

18.2 Associated Bundles and Connections

18.3 $r$ -Form Sections of a Vector Bundle: Curvature

19 The Dirac Equation

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

19.2 Hamilton, Clifford, and Dirac

19.3 The Dirac Algebra

Lorentz group

19.4 The Dirac Operator in Minkowski Space

Dirac operator

19.5 The Dirac Operator in Curved Space-Time

20 Yang-Mills Fields

20.1 Noether’s Theorem for Internal Symmetries

Noether's theorem

20.2 Weyl’s Gauge Invariance Revisited

gauge invariance

20.3 The Yang-Mills Nucleon

nucleon

20.4 Compact Groups and Yang-Mills Action

20.5 The Yang-Mills Equation

Yang-Mills equation

20.6 The Yang-Mills Instanton

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

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 $\pi_k(M)$

homotopy group

22.4 Some Computations of Homotopy Groups

22.5 Chern Forms as Obstructions

obstruction

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

Last revised on June 14, 2023 at 19:56:23. See the history of this page for a list of all contributions to it.