Connections in the bundles associated with the manifold of pairs of tangent and contact planes of a surface

Surface $X_m$ of a projective space $P_n$ is considered as a manifold of triples $(A,T_m,T_s)$, where $A$ is a point of surface, $T_m$ is a tangent plane, $T_s$ ($s\equiv\frac12m(m+3)<n$) is a contact plane. Principle bundle $G(X_m)$ over a base surface $X_m$, a typical fiber of which is subgroup of stationarity $G$, $(A,T_m,T_s)$ associates with the surface $X_m$. This bundle contains eight principle subbundles with the same base $X_m$ and with typical fibers — subgroups of the group $G$, acting in fibers of bundles adjoined to the given ones. In the associated bundle $G(X_m)$ fundamental-group connection is given by G. F. Laptev's way, eight subobjects of which defines connections in corresponding subbundles. Composition equipment of the surface $X_m$ is made, that is next planes are joined to the point $A$: a) Norden's normal of the second kind $N_{m-1}$: $A\notin N_{m-1}\subset T_m$, b) plane $P_{s-m-1}$: $T_m\oplus P_{s-m-1}=T_s$, c) plane $P_{n-s-1}$: $T_s\oplus P_{n-s-1}=P_n$. Two type of scopes of connection object $\Gamma$ by fundamental object of surface, equipping quasitensor and their prolongations are found. Three subobjects of linear connections are equipped equally. Coincidence conditions of different scopes of the rest of components of connection object $\Gamma$ are obtained. They fix a hyperplane drawn on all equipping planes.
Parallel displacement of equipping planes along lines on the surface are considered. They characterize six subconnections defined the subobjects $\Gamma$ of both types. Three directions are considered, i.e. straight lines passing through the point $A$ and intersecting one of equipping planes respectively. A notion of a projective-covariant differential is introduced, it permits to interpret linear connection by parallel displacements of those directions. In particular, a parallel displacement of a straight line $AM$ in the tangent linear connection $\Gamma_{jk}^i$ is investigated, when $M$ is removed in the plane drawn on the straight line $AM$ and the plane $P_{s-m-1}$. Under our view at the surface this displacement is analogue of Norden's tangent direction displacement. More strong parallel displacements and connections characterized by them are studied by means of covariant differentials of equipping quasi tensors.
Connections pencils of dimension $m^2$ and ms are obtained, parameters of which are components $\Gamma_{ij}$, $\Gamma_{ai}$ of connection object $\Gamma$. Parallel displacements of normal (belonged and unbelonged to the contact plane $T_s$) directions in the subconnections defined by the pencils are described.