Geometry of isotypic Kronecker webs
type: Item
instance of: scholarly article
title: Geometry of isotypic Kronecker webs
zbMATH Open document ID: 1275.53016
author: Wojciech Kryński
published in: Central European Journal of Mathematics
publication date: 2013-02-18T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 53A60
Mathematics Subject Classification ID: 34A26
zbMATH DE Number: 6137089
zbMATH Keywords: Veronese web
zbMATH Keywords: Kronecker web
zbMATH Keywords: ordinary differential equation
zbMATH Keywords: three-web
reviewed by: A. M. Shelekhov
MaRDI profile type: Publication
full work available at URL: s11533-012-0081-z
OpenAlex ID: W1989356183
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DOI: 10.2478/S11533-012-0081-Z
description: scientific article; zbMATH DE number 6137089