A nonholonomic complex of the space~$P_4$

S. I. Grigelionis

Itogi Nauki i Tekhniki. Seriya Problemy Geometrii. Trudy Geometricheskogo Seminara, Trudy Geometricheskogo Seminara, Tome 3 (1971), pp. 149-172.

Résumé

Let $L$ be the six-dimensional manifold of all straight lines $l$ of the four-dimensional projective space $P_4$, and let $\Xi$ be the six-dimensional manifold of all two-dimensional planes $\xi$ of the same space. The twelve-parametric manifold $L\times\Xi$ will be denoted by $\tilde S$. We shall associate a three-dimensional manifold $К^\xi$ of all projective mappings $k_l^\xi$ of the points of $l$ onto the sheaf of hyperplanes the axis of which is $\xi$ to each element $(l,\xi)$ of $\tilde S$. The fifteen-dimensional manifold of all triplets $(l,\xi,k_l^\xi)$ may be regarded as a fibre bundle with the base $\tilde S$. The eight-dimensional submanifold formed by all those elements $(l,\xi)\in\tilde S$ for which $l$ and $\xi$ are incident will be denoted by $S$, and the restriction of the fibre space $\tilde T$ over the manifold $S$ will be denoted by $T$. In canonical way we define a mapping $\pi$ of $T$ onto $L:(l,\xi,k_l^\xi)$. Thus we have a fibre bundle $T$ with the base $L$ and the canonical projection $\pi$. Then a non-holonomic complex of the space $P_4$ is defined as a cross-section of the fibre bundle $T$. In the paper the first neighbourhood of an element $(l,\xi,k_l^\xi)$ of the non-holonomic complex of $P_4$ is considered applying the G. F. Laptev method [3].