The Cotangent Bundle

Abstract

In this chapter we introduce a construction that is not typically seen in elementary calculus: tangent covectors, which are linear functionals on a tangent space to a smooth manifold M. The space of all covectors at p∈M is a vector space called the cotangent space at p; in linear-algebraic terms, it is the dual space to T _p M. The union of all cotangent spaces at all points of M is a vector bundle called the cotangent bundle. Whereas tangent vectors give us a coordinate-free interpretation of derivatives of curves, it turns out that derivatives of real-valued functions on a manifold are most naturally interpreted as tangent covectors. Thus we define the differential of a real-valued function as a covector field (a smooth section of the cotangent bundle); it is a coordinate-independent analogue of the gradient. In the second half of the chapter we introduce line integrals of covector fields, which satisfy a far-reaching generalization of the fundamental theorem of calculus.