We simultaneously consider two families of subspaces, which for some constant values of parameters give one family of subspaces. The transformation group here is restricted. Instead of usual transformation in $\Osc^kM$ here we use such transformation group, that $T(\Osc^kM)$ is the direct sum of $T(\Osc^kM_1)$ and $T(\Osc^kM_2)$, $\dim M_1+\dim M_2=\dim M$. The adapted bases of $T^*(\Osc^kM_1)$ and $T^*(\Osc^kM_2)$ are formed, and the relations between these spaces and $T^*(\Osc^kM)$ are given. The same is done for their dual spaces. We introduce generalized linear connection in the surrounding space and give transformation rule under the condition that covariant derivatives of the vector field are tensors. Using the decomposition of $T(\Osc^kM)$ in directions of two complementary subspaces, the induced connection on the subspaces are determined and examined. It is proved that almost all connection coefficients transform as tensor except some of them, which have second lower index $0a$, $0\alpha$ or $0\widehat\alpha$.