Geometry of isotypic Kronecker webs
An isotypic Kronecker web of co-rank \(m\) and order \(k\) is a family of foliations given on a manifold \(M\) of dimension \((k+1)m\) by a system of Pfaffian forms \(\omega^j_t=\omega^j_0+t\omega^j_1+\dots+t^k\omega^j_k,\) where \(t\in I,\,j=0,1,2,\dots,m\), being a generalization of the so-called Veronese web introduced in [\textit{I. M. Gelfand} and \textit{I. Zakharevich}, J. Funct. Anal. 99, No. 1, 150--178 (1991; Zbl 0739.58021)]. In the present paper the problem of local classification of isotypic Kronecker webs is considered and a canonical connection for them is constructed. The following theorem is proved: There is a one-to-one correspondence between classes of germs of equivalent isotypic Kronecker webs and classes of time-preserving contact equivalent systems of ODE's such that the associated curvature operators vanish. If \(k=1\) every Kronecker web is equivalent to a three-web of special type (the so-called isoclinicly-geodesic three-web), formed by \(3m\)-dimensional foliations on a \(2r\)-dimensional manifold. For this web, the curvature tensor is found and investigated. Reviewer's remark: Isoclinicly-geodesic three-webs were introduced and investigated by \textit{M. A. Akivis} in [Tr. Geom. Semin. 2, 7--31 (1969; Zbl 0244.53014)]. In that paper the result of Theorem 7.1 can be found.
scientific article; zbMATH DE number 6137089