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Second order projective deformation of a family \(L_{2n-1}^m\). (English. Russian original) Zbl 0914.53008

Russ. Math. 41, No. 9, 11-14 (1997); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1997, No. 9(424), 13-16 (1997).
Let \(L_{2n-1}^m\) be a smooth \(m\)-parametric family of \((n-1)\)-dimensional planes in the real projective space \(P_{2n-1}\). The projective deformation (bending) in the sense of Fubini-Cartan of such an \(L_{2n-1}^m\) is considered, according to S. P. Finikov [‘Theory of pairs of convergences’ (Russian) (Gosudarstv. Izdat. Tehn.-Teor. Lit., Moskva) (1956; Zbl 0072.16801); French transl. (1976; Zbl 0342.53010)]. By means of the Cartan theory of involutive differential systems [see S. P. Finikov, ‘The method of exterior forms of Cartan in differential geometry. The theory of compatibility of systems of differential equations in total differentials and in partial derivatives’ (Russian) (OGIZ, Moskva and Leningrad) (1948; Zbl 0033.06004); R. L. Bryant, S. S. Chern, R. B. Gardner, H. L. Goldschmidt and P. A. Griffiths, ‘Exterior differential systems’ (Publ. MSRI 18, Springer, New York) (1991; Zbl 0726.58002)], the following three theorems are proved.
(1) A family \(L_{2n-1}^2\) which admits a second-order projective deformation exists with arbitrariness of \(2n-3\) holomorphic functions of two real arguments.
(2) A family \(L_{2n-1}^n\) which admits a second-order projective deformation exists with arbitrariness of \(3n^2-n\) holomorphic functions of one real argument.
(3) For \(m\) different from 2 and \(n\), a family \(L_{2n-1}^m\) which admits a second-order projective deformation exists with arbitrariness of \(n-m\) holomorphic functions of \(m\) real arguments. It is noted that the third-order projective deformation of \(L_{2n-1}^m\) is trivial.

MSC:

53A20 Projective differential geometry
53A25 Differential line geometry
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