All of the above constructs used to define a manifold with connection manipulate vectors, which means they can be naturally extended to operate on arbitrary tensor fields on \({M}\). This is the usual approach taken in general relativity; however, one can alternatively focus on \({k}\)-forms on \({M}\), an approach that generalizes more directly to gauge theories in physics. This viewpoint is sometimes called the Cartan formalism. We will cover both approaches.

Since many of the standard texts in this area only cover one of these viewpoints, and in addition often assume a coordinate frame, a metric, and/or zero torsion (to be defined here), we include a bit more calculational detail here than in other sections.