Differentiable mappings of affine spaces into manifolds of $m$-planes in a multidimensional Euclidean space

In this paper we consider a differentiable mapping of a $p$-dimensional affine space into the differentiable manifold $\mathfrak M_N$ of all centered $m$-planes in the $n$-dimensional Euclidean space. We pay the special attention to describing geometric images defined by the fundamental geometric object of a certain mapping.