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Keywords:
three-web; torsion tensor of a web; distribution; projector; manifold; connection; web
three-web; torsion tensor of a web; distribution; projector; manifold; connection; web
Summary:
A 3-web on a smooth $2n$-dimensional manifold can be regarded locally as a triple of integrable $n$-distributions which are pairwise complementary, [5]; that is, we can work on the tangent bundle only. This approach enables us to describe a $3$-web and its properties by invariant $(1,1)$-tensor fields $P$ and $B$ where $P$ is a projector and $B^2=$ id. The canonical Chern connection of a web-manifold can be introduced using this tensor fields, [1]. Our aim is to express the torsion tensor $T$ of the Chern connection through the Nijenhuis $(1,2)$-tensor field $[P,B]$, and to verify that $[P,B]=0$ is a necessary and sufficient conditions for vanishing of the torsion $T$.
References:
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[2] P. Nagy: Invariant tensor fields and the canonical connection of a 3-web. Aeq. Math. 35 (1988), 31-44. University of Waterloo, Birkhäuser Verlag, Basel. MR 0939620
[3] P. Nagy: On complete group 3-webs and 3-nets. Arch. Math. 53 (1989), 411-413. Birkhäuser Veгlag, Basel. DOI 10.1007/BF01195223 | MR 1016007 | Zbl 0696.53008
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[5] A. Vanžurová: On (3,2, n)-webs. Acta Sci. Math. 59 (1994), 3-4. Szeged. MR 1317181 | Zbl 0828.53017
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[7] Webs & quasigroups. (1993). Tver State University, Russia. Zbl 0776.00019
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