Visual Differential Geometry and Forms

An inviting, intuitive, and visual exploration of differential geometry and forms

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 new 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 provides an intuitive, geometrical introduction to Differential Forms, elucidating 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, providing a self-contained introduction to Differential Forms.

Requiring only basic calculus and geometry, Visual Differential Geometry and Forms provocatively rethinks the way this important area of mathematics should be understood and taught.

Table of Contents

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 lsometries and Complex Numbers

7 Exercises for Act 11

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 111

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

About the Author

Tristan Needham is professor of mathematics at the University of San Francisco. He is the author of Visual Complex Analysis.