Connections in fiberings associated with the Grassman manifold and the space of centered planes

In the present paper we study connections in the fiberings associated with the Grassmann manifold and the space of the centered planes. The work is related to the studies in differential geometry. In the paper, we use the method of continuations and scopes of G. F. Laptev which generalizes the moving frame method and the exterior forms method of Cartan; the method depends on calculation of exterio differential forms. In the paper, we develop a new method of research in Grassman manifolds and some generalization of the method which includes the theory of the induced connections of the spaces of planes and centered planes in the $n$-dimensional projective space.