Chapter 2 Curvature and Differential Forms

Abstract

In Chapter 1 we developed, in terms of vector fields, the classical theory of curvature for curves and surfaces in \(\mathbb{R}^{3}\). There is a dual approach using differential forms. Differential forms arise naturally even if one is interested only in vector fields. For example, the coefficients of tangent vectors relative to a frame on an open set are differential 1-forms on the open set. Differential forms are more supple than vector fields: they can be differentiated and multiplied, and they behave functorially under the pullback by a smooth map. In the 1920s and 30s Elie Cartan pioneered the use of differential forms in differential geometry [4], and these have proven to be tools of great power and versatility. In this chapter we redevelop the theory of connections and curvature in terms of differential forms.