On immersion of the space of projective connection with different dimensions of base and fibre into the projective space (Q1333651)

Let \(M\) be an \(n\)-dimensional manifold and \(P_{n,n}\) be the projective tangent bundle of \(M\) whose fibre is the \(n\)-dimensional projective space. Given a section \(s: M\to P_{n,n}\) one can consider a subbundle \(\Delta\) in \(P_{n,n}\) whose fibre over \(p\in M\) is an \(m\)-plane \((m\leq n)\) passing through the point \(s(p)\) (\(\Delta\) is called a distribution in \(P_{n,n}\), see \textit{G. F. Laptev} and \textit{N. M. Ostianu} [Tr. Geom. Semin. 3, 49-94 (1971)]). The paper deals with the problem of immersion of \(\Delta\) endowed with a projective connection into a projective space \(P_ N\). More precisely, the problem is ``to find an \(n\)-dimensional surface \(\Sigma \subset P_ N\); to connect with each point of \(\Sigma\) an \(n\)-dimensional plane \(S_ n\) which has to be non-tangent to a manifold where an \(m\)-dimensional subspace \(S_ m\) has been chosen; to set a connection on the manifold of the planes \(S_ m\) so that it coincides with the given connection.'' With the use of E. Cartan's method of specialization of frames the author proves the following theorem: if \(N\geq n(m+4)/2 -1\), then such an immersion exists.