Distributions of tangent elements

G. F. Laptev

Itogi Nauki i Tekhniki. Seriya Problemy Geometrii. Trudy Geometricheskogo Seminara, Trudy Geometricheskogo Seminara, Tome 3 (1971), pp. 29-48.

Résumé

An $n$-dimensional differentiable manifold is considered, on which a Lie group operates. As an example, we can take point projective space or line projective space and the projective group operating on it etc. Any $m$-dimensional submanifold containing a fixed element of the manifold generates a geometric object (fundamental object of the first order), which we call an $m$-dlmensional tangent element. Thus a fibre bundle of $m$-dimensional tangent elements is defined; a cross section of this fibre bundle is called a non-holonomic manifold or a distribution. The system of differential equations of the distribution written in invariant form (3.3) generates a sequence of fundamental geometrical objects which are used to construct the differential geometry of distribution. A system (5.2) of differential equations (the associated system of the distribution) is introduced. An invariant condition of holonomity of the distribution is given. For a holonomic distribution the associated system is completely integrable and defines a $(n-m)$-parametric family of $m$-dimensional subrnanifolds envelopped by the elements of the distribution. In general case the class of curves (curves belonging to the distribution) is invariantly characterized; these curves are the 1-dimensional integral varieties of the associated system. In § 8 the distributions of tangent elements are considered in spaces with connection and arbitrary generating element.