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

Description

Visual Differential Geometry and Forms fulfills two principal goals. In the first four acts, Tristan Needham puts the geometry back into differential geometry. Using 235 hand-drawn diagrams, Needham deploys Newton’s geometrical methods to provide geometrical explanations of the classical results. In the fifth act, he offers the first undergraduate introduction to differential forms that treats advanced topics in an intuitive and geometrical manner. Unique features of the first four acts include: four distinct geometrical proofs of the fundamentally important Global Gauss-Bonnet theorem, providing a stunning link between local geometry and global topology; a simple, geometrical proof of Gauss’s famous Theorema Egregium; a complete geometrical treatment of the Riemann curvature tensor of an n-manifold; and a detailed geometrical treatment of Einstein’s field equation, describing gravity as curved spacetime (General Relativity), together with its implications for gravitational waves, black holes, and cosmology. The final act elucidates such topics as the unification of all the integral theorems of vector calculus; the elegant reformulation of Maxwell’s equations of electromagnetism in terms of 2-forms; de Rham cohomology; differential geometry via Cartan’s method of moving frames; and the calculation of the Riemann tensor using curvature 2-forms. Six of the seven chapters of Act V can be read completely independently from the rest of the book. Requiring only basic calculus and geometry, Visual Differential Geometry and Forms provocatively rethinks the way this important area of mathematics should be considered and taught. 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