Distributions of $m$-dimensional line elements in a space with projective connection. I

In this article is traced a theory of distribution of $m$-dimensional linear elements in the space of projective connection with $n$-dimensional base and $n$-dimensional fibers. The authors consider only those properties which are preserved for flat projective space. It is supposed that $n-m<m(m+1)/2$ and later the hyperplane distribution is excluded; this case will be studied in a separate paper. The results of this paper were highly influenced by the classical results of the projective differential geometry of surfaces, such as contained in well known monographies [9], [2], [15], [12]. All constructions are carried out in an almost arbitrary moving frame (the zero order frame), so that all results are stated in an invariant form. The distribution is considered actually as a manifold immersed in the space of all $m$-dimensional elements and the study of the differential geometry of the distribution is reduced to construction of geometrical objects subordinated to the fundamental objects of the manifold. Thus the objects intrinsically defined by the distribution are studied without using any additional structures (excluding the structure of the space itself).
In § 1 the differential equation of the distribution with respect to the zero order frame is obtained. In the process of construction of the sequence of fundamental objects another sequence of so called fundamental subobjects appears which clarifies the similarity of the geometry of distributions with the geometry of surfaces. Those facts of the geometry of distributions which are based only on the fundamental subobjects are considered in a separate article [8]. In § 4 the distribution is considered for which the tensor of non-holonomity vanishes. In § 5 and § 7 fields of planes invariantly associated with the distribution are constructed. Projective connections in the distribution are regarded in § 6. In § 8 invariant conditions for the curves belonging to the distribution are obtained. In § 9 the focal places defined by the distribution are considered. § 10 is devoted to a generalization of the Bompiani–Pantazi projectivity for the case of distribution of $m$-dimensional elements. In § 11 the linear systems of osculating hyperquadrics are considered which are intrinsically defined by the distribution.