On the Schouten and Wagner curvature tensors

We recall and prove some basic properties of both the Schouten and Wagner curvature tensors. We consider-in some detail-the construction of the Wagner tensor, and then discuss how the vanishing of this tensor characterizes the flat structures, i.e., those for which the parallel translation is path-independent. (As a special case, we revisit the flatness of three-dimensional nonholonomic Riemannian manifolds.) In order to facilitate our presentation, we employ an extension of the notion of a connection to a "restricted connection." Such connections are equivalently viewed as horizontal lifts (or horizontal distributions); hence we revisit the Schouten and Wagner tensors from this perspective. In particular, it turns out that the construction of the Wagner tensor may be equivalently formulated as a flag of horizontal distributions.