Generalized de Rham-Hodge complexes, the related characteristic Chern classes, and some applications to integrable multidimensional differential systems on Riemannian manifolds

Abstract

We study the differential-geometric aspects of generalized de Rham-Hodge complexes naturally related to integrable multidimensional differential systems of the M. Gromov type, as well as the geometric structure of the Chern characteristic classes. Special differential invariants of the Chern type are constructed, their importance for the integrability of multidimensional nonlinear differential systems on Riemannian manifolds is discussed. An example of the three-dimensional Davey-Stewartson-type nonlinear integrable differential system is considered, its Cartan type connection mapping, and related Chern-type differential invariants are analyzed.