The geometry of a~seminonholonomic congruence of the first kind

I. V. Bliznikiene

Itogi Nauki i Tekhniki. Seriya Problemy Geometrii. Trudy Geometricheskogo Seminara, Trudy Geometricheskogo Seminara, Tome 3 (1971), pp. 125-148

Résumé

The Grassman manifold $\mathrm{Gr}(1,3)$ which is equipped by the tensor field $a_{pq}$ is called a semi-non-holonomic congruence. The structure of the first two fundamental geometrical objects of this congruence (in the elliptical case) is considered and the geometrical interpretation of subobjects $$ 1,\ h_\alpha^p,\ {\mathfrak U}_{\alpha\beta},\ H_{\alpha\beta}^{p,q},\ k_\alpha^p,\ H_{\alpha\beta},\ H_{\alpha\beta\gamma\epsilon} $$ and others is obtained. It is shown that with the semi-non-holonomic congruence we can associate principal fibre bundles $P$, $Q$ and $R$ with the linear differential-geometrical connection. The geometrical characteristics of the two points of the correlative non-holonomity and the four inflection centres of the straight line of this congruence are found.