Visual Differential Geometry and Forms

Synopsis

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 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.

Summary

Chapter 1: The Differential Geometry of Surfaces

* Introduces the concept of a surface as a two-dimensional manifold embedded in three-dimensional space.
* Defines the first and second fundamental forms of a surface and their geometric interpretation.
* Examines the Gaussian and mean curvatures of a surface and their relationship to the surface's shape.

Example: The sphere is a surface with positive Gaussian curvature everywhere, meaning it has a positive curvature at every point. This curvature gives the sphere its characteristic round shape.

Chapter 2: Differential Forms

* Introduces the concept of a differential form, which is a tensor field with values in the cotangent space of a manifold.
* Covers exact and closed forms, and the Poincaré lemma.
* Explores the de Rham cohomology of a manifold and its topological significance.

Example: The magnetic field around a current-carrying wire can be represented as a differential 1-form. The closedness of this form implies that the magnetic field is conservative, meaning it can be written as the gradient of a scalar potential.

Chapter 3: Integration of Differential Forms

* Defines the integral of a differential form over a submanifold of a manifold.
* Introduces Stokes' theorem, which relates the integral of a differential form over a boundary to its integral over the interior.
* Examines the applications of Stokes' theorem to the computation of line integrals, surface integrals, and volumes.

Example: The surface area of a surface can be computed using the integral of the differential 2-form representing the surface area element.

Chapter 4: Lie Groups and Lie Algebras

* Introduces the concept of a Lie group, which is a group where the group operations are smooth.
* Defines the Lie algebra of a Lie group and its relation to the group's infinitesimal generators.
* Examines the exponential map and its role in connecting Lie groups and their Lie algebras.

Example: The rotation group in three dimensions is a Lie group. Its Lie algebra is the vector space of all skew-symmetric matrices, which represent the infinitesimal rotations.

Chapter 5: Vector Bundles and Connections

* Defines a vector bundle as a fiber space over a manifold whose fibers are vector spaces.
* Introduces the concept of a connection on a vector bundle and its role in defining parallel transport and curvature.
* Examines the curvature form of a connection and its relation to the Gauss-Bonnet theorem.

Example: The tangent bundle of a manifold is a vector bundle. A connection on the tangent bundle defines the concept of parallel transport, which is essential for understanding how vectors vary along curves on the manifold.