Projective and correlative bending of the families \(L^m_{2n-1}\) (Q1599438)

In the real projective space \(\mathbb{R}\mathbb{P}_{2n-1}\) a smooth \(m\)-parametric family of \((n-1)\)-dimensional planes is denoted by \(L_{2n-1}^m\). This is a continuation of the author's study of the projective bendings (deformations) of \(L_{2n-1}^m\) [see Russ. Math. 41, No. 9, 11-14 (1997); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1997, No. 9(424), 13-16 (1997; Zbl 0914.53008)]. Necessary and sufficient conditions for the existence of a second-order projective deformation of \(L_{2n-1}^m\) to \({\overline L}_{2n-1}^m\) are considered. Also, conditions for the existence of second-order projective or correlative deformations of \(L_{2n-1}^m\) are expressed in terms of the corresponding linear projective elements. The study is based on the classical Fubini-Cartan approach to the projective deformations [see \textit{S. P. Finikov}, The method of exterior forms of Cartan in differential geometry. (Russian) (OGIZ, Moskva and Leningrad) (1948; Zbl 0033.06004)].