Invariants of smooth layerings

Background. Geometry of smooth layerings is one of the main objects of research in differential geometry, having multiple applications, particularly in theoretical physics. Differential invariants of layerings have been studied by one of the authors of the present article by the methods developed in work by A. Vinogradov, D. Alekseevsky and V. Lychagin. However, these methods do not represent invariant notation of differential equations of the studied objects, and that causes certain difficulties in research of complex differential-geometric structures. The work is aimed at the development of a universal approach to studying the layerings of various codimensionality. Materials and methods. The authors use the method of external forms and moving frames, developed by Elie Cartan and modified by G.F. Laptev and other geometers. In particular, G.F. Laptev built the invariant theory of differentiable mapping of the smooth manifold into the manifold of greater dimensionality. In the present work the authors show the ways to research the geometry of smooth submersions and smooth layerings determined by them using the method of Cartan - Laptev. Results. The authors found a canonical form of structural equations of smooth submersions, discovered the geometrical sense of canonization. It is shown that canonical submersions are connected with G -structures of the first and second order and a certain trivalent tensor. Conclusions. The method of Cartan - Laptev allows effective rsearching of the geometry of smooth layerings of various codimansionality both on random smooth manifolds and on manifolds, supplied by an additional structure.