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. Dedication Contents Prologue Acknowledgements Act I: The Nature of Space 1 Euclidean and Non-Euclidean Geometry 2 Gaussian Curvature 3 Exercises for Prologue and Act I Act II: The Metric 4 Mapping Surfaces: The Metric 5 The Pseudosphere and the Hyperbolic Plane 6 Isometries and Complex Numbers 7 Exercises for Act II Act III: Curvature 8 Curvature of Plane Curves 9 Curves in 3-Space 10 The Principal Curvatures of a Surface 11 Geodesics and Geodesic Curvature 12 The Extrinsic Curvature of a Surface 13 Gauss’s Theorema Egregium 14 The Curvature of a Spike 15 The Shape Operator 16 Introduction to the Global Gauss–Bonnet Theorem 17 First (Heuristic) Proof of the Global Gauss–Bonnet Theorem 18 Second (Angular Excess) Proof of the Global Gauss–Bonnet Theorem 19 Third (Vector Field) Proof of the Global Gauss–Bonnet Theorem 20 Exercises for Act III Act IV: Parallel Transport 21 An Historical Puzzle 22 Extrinsic Constructions 23 Intrinsic Constructions 24 Holonomy 25 An Intuitive Geometric Proof of the Theorema Egregium 26 Fourth (Holonomy) Proof of the Global Gauss–Bonnet Theorem 27 Geometric Proof of the Metric Curvature Formula 28 Curvature as a Force between Neighbouring Geodesics 29 Riemann’s Curvature 30 Einstein’s Curved Spacetime 31 Exercises for Act IV Act V: Forms 32 1-Forms 33 Tensors 34 2-Forms 35 3-Forms 36 Differentiation 37 Integration 38 Differential Geometry via Forms 39 Exercises for Act V Further Reading Bibliography Index