Gravitation

The covariant derivative defines the rate of change of a tensor field in the direction of a vector. Covariant derivatives in two different directions do not in general commute; their commutator determines the curvature of space.

In field theory one has to deal with derivatives of vector fields and in modern theories there appear quantities which are vectors not in space-time but in an internal space. In both cases one deals with vector bundles where vectors at different points are not canonically oriented towards each other. A chart independent notion of a derivative requires an additional structure, the so-called connection. It will be the subject of this chapter. As one hopes that eventually space, time and internal space will turn out to be only different directions in a unifying entity we start with some definitions which allow us to treat both cases in the same way.