Nonlinear $m^p$-connectivities in principal fibrations

The paper deals with the multidimensional analog of the concept of nonlinear connectivity of order $p$, the content of the analog being that, for each $m$-submanifold of the basis of the principal fibration, there is defined a linear connectivity with horizontal $m$-distribution in the submanifolds passing through the common $m^p$-element of tangency. It has been possible to reduce consideration to the case of a linear connectivity in the appropriately chosen principal fibration.