Affine Connections

Abstract

In this central chapter we introduce an important additional structure on differentiable manifolds, thus making it possible to define a “covariant derivative” which transforms tensor fields into other tensor fields. This allows us to introduce many important notions, such as the parallel transport of tensor fields along a curve, geodesics, exponential mappings in the neighborhood of points, normal coordinates, and—most importantly—the concepts of curvature and torsion of an affine connection. For a pseudo-Riemannian manifold there are distinguished affine connections, called metric. Of prime importance is the fact that for every pseudo-Riemannian manifold, there exists a unique affine connection with vanishing torsion (symmetric connection) that is metric. Its Bianchi identities play a crucial role in Einstein’s field equations. Cartan’s structure equations lead to an alternative, compact formulation of these identities, by making use of the absolute exterior differential of tensor valued differential forms. Additional subsections are devoted to a characterisation of locally flat manifolds, the Weyl tensor and conformally flat manifolds. At the end we extend the covariant derivative to tensor densities.