On torsion of a 3 -web

Alena Vanžurová

Mathematica Bohemica (1995)

  • Volume: 120, Issue: 4, page 387-392
  • ISSN: 0862-7959

Abstract

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A 3-web on a smooth 2 n -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 .

How to cite

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Vanžurová, Alena. "On torsion of a $3$-web." Mathematica Bohemica 120.4 (1995): 387-392. <http://eudml.org/doc/247808>.

@article{Vanžurová1995,
abstract = {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 $T$.},
author = {Vanžurová, Alena},
journal = {Mathematica Bohemica},
keywords = {three-web; torsion tensor of a web; distribution; projector; manifold; connection; web; three-web; torsion tensor of a web},
language = {eng},
number = {4},
pages = {387-392},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On torsion of a $3$-web},
url = {http://eudml.org/doc/247808},
volume = {120},
year = {1995},
}

TY - JOUR
AU - Vanžurová, Alena
TI - On torsion of a $3$-web
JO - Mathematica Bohemica
PY - 1995
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 120
IS - 4
SP - 387
EP - 392
AB - 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 $T$.
LA - eng
KW - three-web; torsion tensor of a web; distribution; projector; manifold; connection; web; three-web; torsion tensor of a web
UR - http://eudml.org/doc/247808
ER -

References

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  1. P. Nagy, On the canonical connection of a three-web, Publ. Math. Debrecen 32 (1985), 93-99. (1985) MR0810595
  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. (1988) MR0939620
  3. P. Nagy, 10.1007/BF01195223, Arch. Math. 53 (1989), 411-413. Birkhäuser Veгlag, Basel. (1989) Zbl0696.53008MR1016007DOI10.1007/BF01195223
  4. J. Vanžura, 10.2996/kmj/1138847161, Kódai Math. Sem. Rep. 27 (1976), 42-60. (1976) MR0400106DOI10.2996/kmj/1138847161
  5. A. Vanžurová, On (3,2, n)-webs, Acta Sci. Math. 59 (1994), 3-4. Szeged. (1994) Zbl0828.53017MR1317181
  6. A. G. Walker, Almost-product stгuctures, Differential geometry, Proc. of Symp. in Pure Math.. vol. III, 1961, pp. 94-100. (1961) MR0123993
  7. Webs & quasigroups, (1993). Tver State University, Russia. (1993) Zbl0776.00019

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