of the procedure. The Cartan test generalizes the Frobenius theorem on the solubility of first-order linear systems of partial differential equations. If the coframes on M and N (obtained by a thorough application of the first three steps of the algorithm) agree and satisfy the Cartan test, then the two G-structures are equivalent. (Actually, to the best of the author's knowledge, the coframes must be real analytic in order for this to hold, because the Cartan-Kähler theorem requires analyticity.) Prolongation. This is the most intricate case. In fact there are two sub-cases. In the first sub-case, all of the torsion can be uniquely absorbed into the connection form. (Riemannian manifolds are an example, since the Levi-Civita connection absorbs all of the torsion). The connection coefficients and their invariant derivatives form a complete set of invariants of the structure, and the equivalence problem is solved. In the second subcase, however, it is either impossible to absorb all of the torsion, or there is some ambiguity (as is often the case in Gaussian elimination, for example). Here, just as in Gaussian elimination, there are additional parameters which appear in attempting to absorb the torsion. These parameters themselves turn out to be additional invariants of the problem, so the structure group G must be prolonged into a subgroup of a jet group. Once this is done, one obtains a new coframe on the prolonged space and has to return to the first step of the equivalence method. (See also prolongation of G-structures.) Degeneracy. Because of a non-uniformity of some rank condition, the equivalence method is unsuccessful in handling this particular equivalence problem. For example, consider the equivalence problem of mapping a manifold M with a single one-form θ to another manifold with a single one-form γ such that φ*γ=θ. The zeros of these one forms, as well as the rank of their exterior derivatives at each point need to be taken into account. The equivalence method can handle such problems if all of the ranks are uniform, but it is not always suitable if the rank changes. Of course, depending on the particular application, a great deal of information can still be obtained with the equivalence method. References Differential geometry Diffeomorphisms