Projective connections in canonical bundles of manifolds of planes

Ü. G. Lumiste

Sbornik. Mathematics, Tome 20 (1973) no. 2, pp. 223-248

Résumé

Given a submanifold $B$ of the Grassmann manifold $\Omega(m,n)$ of $m$-dimensional planes in $n$-dimensional projective space $P_n$, there is defined a fiber bundle with base space $B$ and with the planes of $B$ as fibers. The projective connections in this fiber bundle are studied. The cases condidered are when either 1) $B=\Omega(m,n)$, or 2) $m=n-1$, or 3) $m=1$ and $\operatorname{codim}B=1$. It is proved that in these cases the fiber bundle admits only a perspective projective connection, apart from the following two possibilities: a) $m=n-1$ and $\dim B=1$; b) $m=1$ and $B$ consists of the tangent lines to a hypersurface of maximum rank. Under assumptions a) and b) there exist nonperspective connections, and a complete geometric description is given of them. Bibliography: 13 titles.