The contact number of an affine immersion and its upper bounds (Q2655725)

Following \textit{B.~Y.~Chen} and \textit{S.-J.~Li} [Proc. Edinb. Math. Soc., II. Ser. 47, No.~1, 69--100 (2004; Zbl 1065.53044)], the contact number of a Euclidean submanifold \(M\) is defined as the maximal number \(k\) such that the unit speed geodesic and the normal section to any pair \((p,u)\) consisting of a point and tangent vector of \(M\) are in contact of order \(k\). The author generalizes this concept to an affine immersion with transversal bundle \(\sigma\). The affine contact number is defined via the contact order of geodesic and \(\sigma\)-section. The main results concern affine immersions into a projectively flat space and generalize findings in the Euclidean case: {\parindent5mm \begin{itemize}\item[1)] The affine immersion is isotropic in the sense of \textit{L.~Vrancken} [Tohoku Math. J., II. Ser. 53, No.~4, 511--531 (2001; Zbl 1008.53014)] if and only if \(k \geq 3\). \item[2)] If the contact number \(k\) is finite, it lies between \(2\) and \(q+1\) where \(q\) is the immersion's codimension. \item[3)] The contact number \(k\) is infinite if and only if the affine immersion has geodesic \(\sigma\)-sections. In particular, this is the case for planar and cubic geodesic immersions. \end{itemize}}