Tensor Fields and Differential Forms

Abstract

This chapter deals with general vector bundles, including the ‘cocycle approach’; other topics are: Tensors and tensor fields, exterior forms, the Lie derivative and the interior product; the calculus of differential forms and distributions.

Some examples related to manifolds studied in the previous chapter are considered, as the infinite Möbius strip, considered as a vector bundle and the tautological bundle over the real Grassmannian.

A number of problems intend to familiarise the reader with computations of vector fields, differential forms, the Lie derivative, the interior product, the exterior differential, and their relationships. Other problems are also intended to get certain practical capability with integral distributions and differential ideals.

The last section is devoted to almost symplectic manifolds, Hamilton’s equations, and the relation with principal U(1)-bundles, also considered in Chapter 5.