Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts

Cover
Title Page
Copyright Page
Dedication
Table of Contents
Prologue
Acknowledgements
ACT I The Nature of Space
1 Euclidean and Non-Euclidean Geometry
1.1 Euclidean and Hyperbolic Geometry
1.2 Spherical Geometry
1.3 The Angular Excess of a Spherical Triangle
1.4 Intrinsic and Extrinsic Geometry of Curved Surfaces
1.5 Constructing Geodesics via Their Straightness
1.6 The Nature of Space
2 Gaussian Curvature
2.1 Introduction
2.2 The Circumference and Area of a Circle
2.3 The Local Gauss–Bonnet Theorem
3 Exercises for Prologue and Act I
ACT II The Metric
4 Mapping Surfaces: The Metric
4.1 Introduction
4.2 The Projective Map of the Sphere
4.3 The Metric of a General Surface
4.4 The Metric Curvature Formula
4.5 Conformal Maps
4.6 Some Visual Complex Analysis
4.7 The Conformal Stereographic Map of the Sphere
4.8 Stereographic Formulas
4.9 Stereographic Preservation of Circles
5 The Pseudosphere and the Hyperbolic Plane
5.1 Beltrami’s Insight
5.2 The Tractrix and the Pseudosphere
5.3 A Conformal Map of the Pseudosphere
5.4 The Beltrami–Poincaré Half-Plane
5.5 Using Optics to Find the Geodesics
5.6 The Angle of Parallelism
5.7 The Beltrami–Poincaré Disc
6 Isometries and Complex Numbers
6.1 Introduction
6.2 Möbius Transformations
6.3 The Main Result
6.4 Einstein’s Spacetime Geometry
6.5 Three-Dimensional Hyperbolic Geometry
7 Exercises for Act II
ACT III Curvature
8 Curvature of Plane Curves
8.1 Introduction
8.2 The Circle of Curvature
8.3 Newton’s Curvature Formula
8.4 Curvature as Rate of Turning
8.5 Example: Newton’s Tractrix
9 Curves in 3-Space
10 The Principal Curvatures of a Surface
10.1 Euler’s Curvature Formula
10.2 Proof of Euler’s Curvature Formula
10.3 Surfaces of Revolution
11 Geodesics and Geodesic Curvature
11.1 Geodesic Curvature and Normal Curvature
11.2 Meusnier’s Theorem
11.3 Geodesics are “Straight”
11.4 Intrinsic Measurement of Geodesic Curvature
11.5 A Simple Extrinsic Way to Measure Geodesic Curvature
11.6 A New Explanation of the Sticky-Tape Construction of Geodesics
11.7 Geodesics on Surfaces of Revolution
11.7.1 Clairaut’s Theorem on the Sphere
11.7.2 Kepler’s Second Law
11.7.3 Newton’s Geometrical Demonstration of Kepler’s Second Law
11.7.4 Dynamical Proof of Clairaut’s Theorem
11.7.5 Application: Geodesics in the Hyperbolic Plane (Revisited)
12 The Extrinsic Curvature of a Surface
12.1 Introduction
12.2 The Spherical Map
12.3 Extrinsic Curvature of Surfaces
12.4 What Shapes Are Possible?
13 Gauss’s Theorema Egregium
13.1 Introduction
13.2 Gauss’s Beautiful Theorem (1816)
13.3 Gauss’s Theorema Egregium (1827)
14 The Curvature of a Spike
14.1 Introduction
14.2 Curvature of a Conical Spike
14.3 The Intrinsic and Extrinsic Curvature of a Polyhedral Spike
14.4 The Polyhedral Theorema Egregium
15 The Shape Operator
15.1 Directional Derivatives
15.2 The Shape Operator S
15.3 The Geometric Effect of S
15.4 DETOUR: The Geometry of the Singular Value Decomposition and of the Transpose
15.5 The General Matrix of S
15.6 Geometric Interpretation of S and Simplification of [S]
15.7 [S] Is Completely Determined by Three Curvatures
15.8 Asymptotic Directions
15.9 Classical Terminology and Notation: The Three Fundamental Forms
16 Introduction to the Global Gauss–Bonnet Theorem
16.1 Some Topology and the Statement of the Result
16.2 Total Curvature of the Sphere and of the Torus
16.2.1 Total Curvature of the Sphere
16.2.2 Total Curvature of the Torus
16.3 Seeing K(Sg) via a Thick Pancake
16.4 Seeing K(Sg) via Bagels and Bridges
16.5 The Topological Degree of the Spherical Map
16.6 Historical Note
17 First (Heuristic) Proof of the Global Gauss–Bonnet Theorem
17.1 Total Curvature of a Plane Loop: Hopf’s Umlaufsatz
17.2 Total Curvature of a Deformed Circle
17.3 Heuristic Proof of Hopf’s Umlaufsatz
17.4 Total Curvature of a Deformed Sphere
17.5 Heuristic Proof of the Global Gauss–Bonnet Theorem
18 Second (Angular Excess) Proof of the Global Gauss–Bonnet Theorem
18.1 The Euler Characteristic
18.2 Euler’s (Empirical) Polyhedral Formula
18.3 Cauchy’s Proof of Euler’s Polyhedral Formula
18.3.1 Flattening Polyhedra
18.3.2 The Euler Characteristic of a Polygonal Net
18.4 Legendre’s Proof of Euler’s Polyhedral Formula
18.5 Adding Handles to a Surface to Increase Its Genus
18.6 Angular Excess Proof of the Global Gauss–Bonnet Theorem
19 Third (Vector Field) Proof of the Global Gauss–Bonnet Theorem
19.1 Introduction
19.2 Vector Fields in the Plane
19.3 The Index of a Singular Point
19.4 The Archetypal Singular Points: Complex Powers
19.5 Vector Fields on Surfaces
19.5.1 The Honey-Flow Vector Field
19.5.2 Relation of the Honey-Flow to the Topographic Map
19.5.3 Defining the Index on a Surface
19.6 The Poincaré–Hopf Theorem
19.6.1 Example: The Topological Sphere
19.6.2 Proof of the Poincaré–Hopf Theorem
19.6.3 Application: Proof of the Euler–L’Huilier Formula
19.6.4 Poincaré’s Differential Equations Versus Hopf’s Line Fields
19.7 Vector Field Proof of the Global Gauss–Bonnet Theorem
19.8 The Road Ahead
20 Exercises for Act III
ACT IV Parallel Transport
21 An Historical Puzzle
22 Extrinsic Constructions
22.1 Project into the Surface as You Go!
22.2 Geodesics and Parallel Transport
22.3 Potato-Peeler Transport
23 Intrinsic Constructions
23.1 Parallel Transport via Geodesics
23.2 The Intrinsic (aka, “Covariant”) Derivative
24 Holonomy
24.1 Example: The Sphere
24.2 Holonomy of a General Geodesic Triangle
24.3 Holonomy Is Additive
24.4 Example: The Hyperbolic Plane
25 An Intuitive Geometric Proof of the Theorema Egregium
25.1 Introduction
25.2 Some Notation and Reminders of Definitions
25.3 The Story So Far
25.4 The Spherical Map Preserves Parallel Transport
25.5 The Beautiful Theorem and Theorema Egregium Explained
26 Fourth (Holonomy) Proof of the Global Gauss–Bonnet Theorem
26.1 Introduction
26.2 Holonomy Along an Open Curve?
26.3 Hopf’s Intrinsic Proof of the Global Gauss–Bonnet Theorem
27 Geometric Proof of the Metric Curvature Formula
27.1 Introduction
27.2 The Circulation of a Vector Field Around a Loop
27.3 Dry Run: Holonomy in the Flat Plane
27.4 Holonomy as the Circulation of a Metric-Induced Vector Field in the Map
27.5 Geometric Proof of the Metric Curvature Formula
28 Curvature as a Force between Neighbouring Geodesics
28.1 Introduction to the Jacobi Equation
28.1.1 Zero Curvature: The Plane
28.1.2 Positive Curvature: The Sphere
28.1.3 Negative Curvature: The Pseudosphere
28.2 Two Proofs of the Jacobi Equation
28.2.1 Geodesic Polar Coordinates
28.2.2 Relative Acceleration = Holonomy of Velocity
28.3 The Circumference and Area of a Small Geodesic Circle
29 Riemann’s Curvature
29.1 Introduction and Summary
29.2 Angular Excess in an n-Manifold
29.3 Parallel Transport: Three Constructions
29.3.1 Closest Vector on Constant-Angle Cone
29.3.2 Constant Angle within a Parallel-Transported Plane
29.3.3 Schild’s Ladder
29.4 The Intrinsic (aka “Covariant”) Derivative ∇v
29.5 The Riemann Curvature Tensor
29.5.1 Parallel Transport Around a Small “Parallelogram”
29.5.2 Closing the “Parallelogram” with the Vector Commutator
29.5.3 The General Riemann Curvature Formula
29.5.4 Riemann’s Curvature Is a Tensor
29.5.5 Components of the Riemann Tensor
29.5.6 For a Given wo, the Vector Holonomy Only Depends on the Plane of the Loop and Its Area
29.5.7 Symmetries of the Riemann Tensor
29.5.8 Sectional Curvatures
29.5.9 Historical Notes on the Origin of the Riemann Tensor
29.6 The Jacobi Equation in an n-Manifold
29.6.1 Geometrical Proof of the Sectional Jacobi Equation
29.6.2 Geometrical Implications of the Sectional Jacobi Equation
29.6.3 Computational Proofs of the Jacobi Equation and the Sectional Jacobi Equation
29.7 The Ricci Tensor
29.7.1 Acceleration of the Area Enclosed by a Bundle of Geodesics
29.7.2 Definition and Geometrical Meaning of the Ricci Tensor
29.8 Coda
30 Einstein’s Curved Spacetime
30.1 Introduction: “The Happiest Thought of My Life.”
30.2 Gravitational Tidal Forces
30.3 Newton’s Gravitational Law in Geometrical Form
30.4 The Spacetime Metric
30.5 Spacetime Diagrams
30.6 Einstein’s Vacuum Field Equation in Geometrical Form
30.7 The Schwarzschild Solution and the First Tests of the Theory
30.8 Gravitational Waves
30.9 The Einstein Field Equation (with Matter) in Geometrical Form
30.10 Gravitational Collapse to a Black Hole
30.11 The Cosmological Constant: “The Greatest Blunder of My Life.”
30.12 The End
31 Exercises for Act IV
ACT V Forms
32 1-Forms
32.1 Introduction
32.2 Definition of a 1-Form
32.3 Examples of 1-Forms
32.3.1 Gravitational Work
32.3.2 Visualizing the Gravitational Work 1-Form
32.3.3 Topographic Maps and the Gradient 1-Form
32.3.4 Row Vectors
32.3.5 Dirac’s Bras
32.4 Basis 1-Forms
32.5 Components of a 1-Form
32.6 The Gradient as a 1-Form: df
32.6.1 Review of the Gradient as a Vector: ∇f
32.6.2 The Gradient as a 1-Form: df
32.6.3 The Cartesian 1-Form Basis: {dxj}
32.6.4 The 1-Form Interpretation of df=(Bxf)dx+(Byf)dy
32.7 Adding 1-Forms Geometrically
33 Tensors
33.1 Definition of a Tensor: Valence
33.2 Example: Linear Algebra
33.3 New Tensors from Old
33.3.1 Addition
33.3.2 Multiplication: The Tensor Product
33.4 Components
33.5 Relation of the Metric Tensor to the Classical Line Element
33.6 Example: Linear Algebra (Again)
33.7 Contraction
33.8 Changing Valence with the Metric Tensor
33.9 Symmetry and Antisymmetry
34 2-Forms
34.1 Definition of a 2-Form and of a p-Form
34.2 Example: The Area 2-Form
34.3 The Wedge Product of Two 1-Forms
34.4 The Area 2-Form in Polar Coordinates
34.5 Basis 2-Forms and Projections
34.6 Associating 2-Forms with Vectors in R3: Flux
34.7 Relation of the Vector and Wedge Products in R3
34.8 The Faraday and Maxwell Electromagnetic 2-Forms
35 3-Forms
35.1 A 3-Form Requires Three Dimensions
35.2 The Wedge Product of a 2-Form and 1-Form
35.3 The Volume 3-Form
35.4 The Volume 3-Form in Spherical Polar Coordinates
35.5 The Wedge Product of Three 1-Forms and of p 1-Forms
35.6 Basis 3-Forms
35.7 Is Ψ ∧ Ψ ̸= 0 Possible?
36 Differentiation
36.1 The Exterior Derivative of a 1-Form
36.2 The Exterior Derivative of a 2-Form and of a p-Form
36.3 The Leibniz Rule for Forms
36.4 Closed and Exact Forms
36.4.1 A Fundamental Result: d2 = 0
36.4.2 Closed and Exact Forms
36.4.3 Complex Analysis: Cauchy–Riemann Equations
36.5 Vector Calculus via Forms
36.6 Maxwell’s Equations
37 Integration
37.1 The Line Integral of a 1-Form
37.1.1 Circulation and Work
37.1.2 Path-Independence⇐⇒ Vanishing Loop Integrals
37.1.3 The Integral of an Exact Form: φ= df
37.2 The Exterior Derivative as an Integral
37.2.1 d(1-Form)
37.2.2 d(2-Form)
37.3 Fundamental Theorem of Exterior Calculus (Generalized Stokes’s Theorem)
37.3.1 Fundamental Theorem of Exterior Calculus
37.3.2 Historical Aside
37.3.3 Example: Area
37.4 The Boundary of a Boundary Is Zero!
37.5 The Classical Integral Theorems of Vector Calculus
37.5.1 Φ = 0-Form
37.5.2 Φ = 1-Form
37.5.3 Φ = 2-Form
37.6 Proof of the Fundamental Theorem of Exterior Calculus
37.7 Cauchy’s Theorem
37.8 The Poincaré Lemma for 1-Forms
37.9 A Primer on de Rham Cohomology
37.9.1 Introduction
37.9.2 A Special 2-Dimensional Vortex Vector Field
37.9.3 The Vortex 1-Form Is Closed
37.9.4 Geometrical Meaning of the Vortex 1-Form
37.9.5 The Topological Stability of the Circulation of a Closed 1-Form
37.9.6 The First de Rham Cohomology Group
37.9.7 The Inverse-Square Point Source in R3
37.9.8 The Second de Rham Cohomology Group
37.9.9 The First de Rham Cohomology Group of the Torus
38 Differential Geometry via Forms
38.1 Introduction: Cartan’s Method of Moving Frames
38.2 Connection 1-Forms
38.2.1 Notational Conventions and Two Definitions
38.2.2 Connection 1-Forms
38.2.3 WARNING: Notational Hazing Rituals Ahead!
38.3 The Attitude Matrix
38.3.1 The Connection Forms via the Attitude Matrix
38.3.2 Example: The Cylindrical Frame Field
38.4 Cartan’s Two Structural Equations
38.4.1 The Duals θi of mi in Terms of the Duals dxj of ej
38.4.2 Cartan’s First Structural Equation
38.4.3 Cartan’s Second Structural Equation
38.4.4 Example: The Spherical Frame Field
38.5 The Six Fundamental Form Equations of a Surface
38.5.1 Adapting Cartan’s Moving Frame to a Surface: The Shape Operator and the Extrinsic Curvature
38.5.2 Example: The Sphere
38.5.3 Uniqueness of Basis Decompositions
38.5.4 The Six Fundamental Form Equations of a Surface
38.6 Geometrical Meanings of the Symmetry Equation and the Peterson–Mainardi–Codazzi Equations
38.7 Geometrical Form of the Gauss Equation
38.8 Proof of the Metric Curvature Formula and the Theorema Egregium
38.8.1 Lemma: Uniqueness of ω12
38.8.2 Proof of the Metric Curvature Formula
38.9 A New Curvature Formula
38.10 Hilbert’s Lemma
38.11 Liebmann’s Rigid Sphere Theorem
38.12 The Curvature 2-Forms of an n-Manifold
38.12.1 Introduction and Summary
38.12.2 The Generalized Exterior Derivative
38.12.3 Extracting the Riemann Tensor from the Curvature 2-Forms
38.12.4 The Bianchi Identities Revisited
38.13 The Curvature of the Schwarzschild Black Hole
39 Exercises for Act V
Further Reading
Bibliography
Index