Visual differential geometry and forms : a mathematical drama in five acts

Includes bibliographical references (pages 485-489) and index.

The Nature of Space. Euclidean and non-Euclidean geometry -- Gaussian curvature -- Exercises for prologue and act I -- The Metric. Mapping surfaces : the metric -- The pseudosphere and hyperbolic plane -- Isometries and complex numbers -- Exercises for act II -- Curvature. Curvature of plane curves -- Curves in 3-space -- The principal curvatures of a surface -- Geodesics and geodesic curvature -- The extrinsic curvature of a surface -- Gauss's Theorema Egregium -- The curvature of a spike -- The shape operator -- Introduction to the Global Gauss-Bonnet theorem -- First (heuristic) proof of the Global Gauss-Bonnet theorem -- Second (angular excess) proof of the Global Gauss-Bonnet theorem -- Expercises for act III -- Parallel Transport. An historical puzzle -- Extrinsic constructions -- Intrinsic constructions -- Holonomy -- An intuitive geometric proof of the Theorem Egregium -- Fourth (holonomy) proof of the global Gauss-Bonnet theorem -- Geometric proof of the metric curvature formula -- Curvature as a force between neighboring geodesics -- Riemann's curvature -- Einstein's curved spacetime -- Exercises for act IV -- Forms. 1-forms -- Tensors -- 2-forms -- 3-forms -- Differentiation -- Integration -- Differential geometry via forms -- Exercises for act V.

"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"--