On the rigidity of webs

Michel Belliart

Bulletin de la Société Mathématique de France (2007)

  • Volume: 135, Issue: 1, page 1-24
  • ISSN: 0037-9484

Abstract

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Plane d -webs have been studied a lot since their appearance at the turn of the 20th century. A rather recent and striking result for them is the theorem of Dufour, stating that the measurable conjugacies between 3-webs have to be analytic. Here, we show that even the set-theoretic conjugacies between two d -webs, d ≥ 3 are analytic unless both webs are analytically parallelizable. Between two set-theoretically conjugate parallelizable d -webs, however, there always exists a nonmeasurable conjugacy; still, every pair of set-theoretically conjugate 3 -webs (parallelizable or not) also are analytically conjugate, while if d ≥ 4 there exist pairs of d -webs which are set-theoretically conjugate but not even measurably so.

How to cite

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Belliart, Michel. "On the rigidity of webs." Bulletin de la Société Mathématique de France 135.1 (2007): 1-24. <http://eudml.org/doc/272367>.

@article{Belliart2007,
abstract = {Plane $d$-webs have been studied a lot since their appearance at the turn of the 20th century. A rather recent and striking result for them is the theorem of Dufour, stating that the measurable conjugacies between 3-webs have to be analytic. Here, we show that even the set-theoretic conjugacies between two $d$-webs, $d\ge 3$ are analytic unless both webs are analytically parallelizable. Between two set-theoretically conjugate parallelizable $d$-webs, however, there always exists a nonmeasurable conjugacy; still, every pair of set-theoretically conjugate $3$-webs (parallelizable or not) also are analytically conjugate, while if $d\ge 4$ there exist pairs of $d$-webs which are set-theoretically conjugate but not even measurably so.},
author = {Belliart, Michel},
journal = {Bulletin de la Société Mathématique de France},
keywords = {foliations; 3-webs; conjugacy; rigidity},
language = {eng},
number = {1},
pages = {1-24},
publisher = {Société mathématique de France},
title = {On the rigidity of webs},
url = {http://eudml.org/doc/272367},
volume = {135},
year = {2007},
}

TY - JOUR
AU - Belliart, Michel
TI - On the rigidity of webs
JO - Bulletin de la Société Mathématique de France
PY - 2007
PB - Société mathématique de France
VL - 135
IS - 1
SP - 1
EP - 24
AB - Plane $d$-webs have been studied a lot since their appearance at the turn of the 20th century. A rather recent and striking result for them is the theorem of Dufour, stating that the measurable conjugacies between 3-webs have to be analytic. Here, we show that even the set-theoretic conjugacies between two $d$-webs, $d\ge 3$ are analytic unless both webs are analytically parallelizable. Between two set-theoretically conjugate parallelizable $d$-webs, however, there always exists a nonmeasurable conjugacy; still, every pair of set-theoretically conjugate $3$-webs (parallelizable or not) also are analytically conjugate, while if $d\ge 4$ there exist pairs of $d$-webs which are set-theoretically conjugate but not even measurably so.
LA - eng
KW - foliations; 3-webs; conjugacy; rigidity
UR - http://eudml.org/doc/272367
ER -

References

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  1. [1] W. Blaschke – Einführung in die Geometrie der Waben, Birkhaüser-Verlag, 1955. Zbl0068.36501MR75630
  2. [2] W. Blaschke & G. Bol – Geometrie der Gewebe, Springer, 1938. Zbl0020.06701JFM64.0727.03
  3. [3] É. Cartan – « Les sous-groupes des groupes continus de transformations », Ann. École Norm. Sup.25 (1908), p. 57–124. Zbl39.0206.04JFM39.0206.04
  4. [4] S. S. Chern – « Web geometry », Bull. Amer. Math. Soc.6 (1982), p. 1–8. Zbl0483.53013MR634430
  5. [5] J.-P. Dufour – « Rigidity of webs », in Web Theory and Related Topics, World Scientific, 2001. Zbl1048.53010MR1837885
  6. [6] J.-P. Dufour & P. Jean – « Rigidity of webs and families of hypersurfaces, Singularities and dynamical systems (Iráklion, 1983) », North-Holland Math. Stud., vol. 103, 1985, p. 271–283. Zbl0583.57015MR806194
  7. [7] A. Haefliger & G. Reeb – « Variétés (non séparées) à une dimension et structures feuilletées du plan », Enseign. Math.3 (1957), p. 107–125. Zbl0079.17101MR89412
  8. [8] M. Herman – « Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations », Publ. Math. Inst. Hautes Études Sci.49 (1979), p. 5–233. Zbl0448.58019
  9. [9] W. Kaplan – « Regular curve-families filling the plane, I », Duke Math. J.7 (1940), p. 154–185. MR4116JFM66.0966.05
  10. [10] —, « Regular curve-families filling the plane, II », Duke Math. J.8 (1941), p. 11–46. Zbl0025.09301MR4117JFM67.0744.02
  11. [11] R. Mañé – Ergodic theory and differentiable dynamics, Ergeb. Math. Grenzgeb., vol. 8, Springer, 1987. Zbl0616.28007MR889254
  12. [12] I. Nakai – « Topology of complex webs of codimension one and geometry of projective space curves », Topology26 (1987), p. 475–504. Zbl0647.57018MR919731
  13. [13] —, « A naive guide to web geometry », in Web Theory and Related Topics, World Scientific, 2001. Zbl1011.53011MR1837881

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