A generalization of the connections of Schouten and Vranceanu on the \(f(2v+3,\epsilon)\)-varieties (Q2708417)

In this note the author generalizes the classic connection of J. A. Schouten and G. Vranceanu. The classic connections of Schouten and Vranceanu have been studied by Ianus (1971) and Ianus and Popovici (1980). The two types of connections have been generalized by Singh and Vohra (1975), Pripoaie (1986) and Demetropoulos-Psomopoulou (1998).NEWLINENEWLINENEWLINELet \(M\) be an \(m\)-dimensional differentiable variety. The author supposes that \(M\) is a \(f(2\nu+3,\varepsilon)\)-variety, i.e., there exists \(f\subset J^I_I(M)\) so that \(f^{2\nu+3}+\varepsilon f=0\) \((\varepsilon=\pm 1)\). Two complementary global distributions denoted by \(D\) and \(D'\), are defined: The linear connection \(\overline \nabla\) defined by \(\overline\nabla_X Y=\nabla_X^s Y+mB(X,mY)+ IB(X,IY)\) is called generalized Schouten connection. The generalized Vranceanu connection \(\widetilde\nabla\) is defined by \(\widetilde\nabla_X Y=\nabla_x^\nu Y+m[IX,mY]+ I[mX,IY]+mD(mX,mY)+ ID(IX,IY)\).NEWLINENEWLINENEWLINEIn the section entitled ``Parallelism of the distributions \(D\) and \(D'\)'', two theorems are proved. NEWLINENEWLINENEWLINETheorem 2.1. On a \(f(2\nu+3, \varepsilon)\)-variety \(M\), the distributions \(D\) and \(D'\) are parallel in relation to \(\overline\nabla\) and \(\widetilde\nabla\).NEWLINENEWLINENEWLINEIn the last paragraph, entitled ``The integrability of the \(f(2\nu+3, \varepsilon)\)-structure'', are proved three theorems. NEWLINENEWLINENEWLINETheorem 3.1. If the connection \(\nabla\) is symmetric, the tensorial field \(D\) is symmetric, and the distributions \(D\) and \(D'\) are integrable, then generalized Vranceanu connection is symmetric.