The Covariant Surface Derivative

Abstract

In the preceding chapter, we gave an overview of embedded surfaces. We now turn to the important question of covariant differentiation on the surface. We will divide the construction of the covariant derivative into two parts. We will first define this operator for objects with surface indices. The definition will be completely analogous to that of the covariant derivative in the ambient space. While the definition will be identical, some of the important characteristics of the surface covariant derivative will be quite different. In particular, surface covariant derivatives do not commute. Our proof of commutativity for the ambient derivative was based on the existence of affine coordinates in Euclidean spaces. Since affine coordinates may not be possible on a curved surface, that argument is no longer available. We will also discover that the surface covariant derivative is not metrinilic with respect to the covariant basis S α. This will prove fundamental and will give rise to the curvature tensor, which will be further developed in Chap. 12