On differential-geometric structures on a manifold of nonholonomic $(n+1)$-web

M. I. Kabanova

Chair of Geometry, Moscow Pedagogical State University, 1 Malaya Pirogovskaya str., Bld. 1, Moscow, 119991 Russia

Abstract: We consider nonholonomic $(n+1)$-web $NW$ consisting of $n+1$ distributions of codimension $1$ on $n$-dimensional manifold $M$. We prove that an invariant pencil of projective connections exists on the manifold $M$. A unique curvilinear $(n+1)$-web corresponds to the ordered nonholonomic $(n+1)$-web and vice versa. The correspondence is defined by the polarity with respect to an invariant multilinear $n$-form or barycentric subdivision of an $(n+1)$-dimensional simplex. In conclusion we consider nonholonomic $(n+1)$-webs in affine space. The invariant pencil of affine connections is generated by every affine web. We also consider the case when the connections of the pencil are projective.

Keywords: nonholonomic $(n+1)$-web, curvilinear $(n+1)$-web, affine connection, affine nonholonomic $(n+1)$-web, projective connections, geodesic line.

UDC: 514.763

Received: 25.01.2014

 English version: Russian Mathematics (Izvestiya VUZ. Matematika), 2015, 59:7, 1–6

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