Nonholonomic $(n+1)$-webs

M. I. Kabanova^a, A. M. Shelekhov<sup>b

a</sup> Chair of Geometry, Moscow State Pedagogical University, 1 Malaya Pirogovskaya str., Bld. 1, Moscow, 119991 Russia
^b Chair of Functional Analysis and Geometry, Tver State University, 33 Zhelyabov str., Tver, 170000 Russia

Abstract: On an $n$-manifold $M$ we consider nonholonomic $(n+1)$-web $NW$, which consists of $n+1$ distributions of codimension 1. We prove that the web $NW$ is equivalent to $G$-structure with structure group $\lambda E$, the group of scalar matrices. We obtain structure equations of the nonholonomic web $NW$ and find the integrability conditions of all its distributions. We show that a connection $\Gamma$ arises on the manifold $M$ carrying the web $NW$. Distributions of the web $NW$ are totally geodesic with respect to this connection. We consider the special case when the curvature of $\Gamma$ equals zero and in particular when the $(n+1)$-web $NW$ is formed by invariant distributions on the Lie group. We find the equations of the group when all distributions of $NW$ are integrable.

Keywords: nonholonomic $(n+1)$-web, $(n+1)$-web, $G$-structure, $\lambda E$-structure.

UDC: 514.763

Received: 21.05.2013

 English version: Russian Mathematics (Izvestiya VUZ. Matematika), 2014, 58:12, 23–31

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