By definition, a coframe on a manifold is a “complete” collection of one-forms in the sense that, at each point, it provides a basis for the cotangent space. Two coframes are said to be equivalent if they are mapped to each other by a diffeomorphism. The equivalence problem for coframes is, in fact, the most important of the equivalence problems that we are to treat, because it ultimately includes all the others as special cases. Indeed, the remarkable and powerful Cartan equivalence method, [39], [78], [230], provides an explicit, practical algorithm for reducing most other equivalence problems to an equivalence problem for a suitable coframe. (The exceptions are those problems admitting an infinite dimensional symmetry group, which will be handled by similar, but more sophisticated methods to be discussed later on.) Therefore, it is crucial that we learn how to deal properly with this apparently special, but, in reality, quite general equivalence problem. In this chapter, we describe the basic constructions required to solve the fundamental equivalence problem for coframes. The solution will include a complete list of fundamental invariants associated with the coframe, and consequential necessary and sufficient conditions for equivalence.

Frames and Coframes

We begin by presenting the basic definitions. Let M be a smooth manifold of dimension m. If x = (x1, …, xm) are local coordinates on M, then the translational vector fields ∂/∂x1, …, ∂/∂xm provide a basis for the tangent space TM|x at each point x in the coordinate chart; these vector fields are known as the “coordinate frame” associated with the local coordinate system.