The Riemann tensor for nonholonomic manifolds

Abstract

For every nonholonomic manifold, i.e., manifold with nonintegrable distribution, the analog of the Riemann tensor is introduced. It is calculated here for the contact and Engel structures: for the contact structure it vanishes (another proof of Darboux's canonical form); for the Engel distribution the target space of the tensor is of dimension 2. In particular, the Lie algebra preserving the Engel distribution is described. The tensors introduced are interpreted as modifications of the Spencer cohomology and, as such, provide with a new way to solve partial differential equations. Goldschmidt's criteria for formal integrability (vanishing of certain Spencer cohomology) are only applicable to ``one half'' of all differential equations, the ones whose symmetries are induced by point transformations. Lie's theorem says that the ``other half'' consists of differential equations whose symmetries are induced by contact transformations. Therefore, we can now extend Goldschmidt's criteria for formal integrability to all differential equations.