Narrow Results

It is shown how a system of differential forms can reproduce the complete set of differential equations generated by an SO(m) matrix Lax pair. By selecting the elements in the given matrices appropriately, examples of integrable nonlinear equations can be produced. The SO(3) case is discussed in detail, then extended to an m - 1 dimensional manifold immersed in Euclidean space.

An intrinsic version of the integrability theorem for the classical Bäcklund theorem is presented. It is characterized by a one-form which can be put in the form of a Riccati system. It is shown how this system can be linearized. Based on this result, a procedure for generating an infinite number of conservation laws is given.

We give an elegant formulation of the structure equations (of Cartan) and the Bianchi identities in terms of exterior calculus without reference to a particular basis and without the exterior covariant derivative. This approach allows both structure equations and the Bianchi identities to be expressed in terms of forms of arbitrary degree. We demonstrate the relationship with both the conventional vector version of the Bianchi identities and to the exterior covariant derivative approach. Contact manifolds, codimension one foliations and the Cartan form of classical mechanics are studied as examples of its flexibility and utility.