О неголономном комплексе пространства $P_4$
С. И. Григелионис
Аннотация:
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].
Библ. 7.