Chapter 2 - Frame Fields

Abstract

This chapter focuses on the geometry of curves in R 3 because the basic method used to investigate curves has proved effective throughout the study of differential geometry. A curve in R 3 is studied by assigning at each point a certain frame—that is, set of three orthogonal unit vectors. The rate of change of these vectors along the curve is then expressed in terms of the vectors themselves by Frenet formulas. This method of moving frames is used to study a surface in R 3 . In the case of a curve, the chapter uses only the Frenet frame field T, N, B of the curve. Expressing the derivatives of these vector fields in terms of the vector fields themselves, this derives the curvature and torsion of the curve. The curvature and torsion reveal a lot about the geometry of a curve. The chapter highlights that the Cartan's generalization of the Frenet formulas follows the same pattern of expressing the (covariant) derivatives of these vector fields in terms of the vector fields themselves. Cartan's equations are not conspicuously more complicated than Frenet's, because the notion of 1-form is available for the coefficients ω ij , the connection forms.