Some questions on the geometry of seminonholonomic and nonholonomic congruences

I. V. Bliznikiene

Itogi Nauki i Tekhniki. Seriya Problemy Geometrii. Trudy Geometricheskogo Seminara, Trudy Geometricheskogo Seminara, Tome 5 (1974), pp. 97-121

Résumé

In this article the study of semi-non-holonomic and non-ho- Jonomic elliptic congruences in the three-dimensional projective space $P_3$ is continued (see the author [2]–[5]). Let $H^{(\nu)}(K_p)$ be the fundamental object of the order $\nu$ of the semi-nonholonomic elliptic congruence $K_p$ ($p=1,2$ denotes the kind of the congruence). In § 2 it is established that the $H^{(3)}(K_1)$ is the principal object and the $H^{(3)}(K_1)$ is the complete one (by G. F. Laptev's terminology [6]) of the semi-non-holonomis songruence $K_1$. In § 3 the algebraic structure of the first two fundamental objects $H^{(1)}(K_2)$ and $H^{(2)}(K_2)$ of the semi-non-holonomic elliptic congruence $K_2$ is considered. The geometrical interpretations of some comitants of $H^{(1)}(K_2)$ and $H^{(2)}(K_2)$ are given. In § 4 are shown some relations between the geometry of the semi-non-holonomic elliptic congruences and the geometry of the non-holonomic elliptic congruences.