Translation foliations of codimension one on compact affine manifolds

Francisco Turiel

Banach Center Publications (1997)

  • Volume: 39, Issue: 1, page 171-182
  • ISSN: 0137-6934

Abstract

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Consider two foliations 1 and 2 , of dimension one and codimension one respectively, on a compact connected affine manifold ( M , ) . Suppose that T 1 T 2 T 2 ; T 2 T 1 T 1 and T M = T 1 T 2 . In this paper we show that either 2 is given by a fibration over S 1 , and then 1 has a great degree of freedom, or the trace of 1 is given by a few number of types of curves which are completely described. Moreover we prove that 2 has a transverse affine structure.

How to cite

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Turiel, Francisco. "Translation foliations of codimension one on compact affine manifolds." Banach Center Publications 39.1 (1997): 171-182. <http://eudml.org/doc/208660>.

@article{Turiel1997,
abstract = {Consider two foliations $\{\mathcal \{F\}\}_1$ and $\{\mathcal \{F\}\}_2$, of dimension one and codimension one respectively, on a compact connected affine manifold $(M,\nabla )$. Suppose that $\nabla _\{T\{\mathcal \{F\}\}_1\} T\{\mathcal \{F\}\}_2\subset T\{\mathcal \{F\}\}_2$; $\nabla _\{T\{\mathcal \{F\}\}_2\} T\{\mathcal \{F\}\}_1\subset T\{\mathcal \{F\}\}_1$ and $TM = T\{\mathcal \{F\}\}_1\oplus T\{\mathcal \{F\}\}_2$. In this paper we show that either $\{\mathcal \{F\}\}_2$ is given by a fibration over $S^1$, and then $\{\mathcal \{F\}\}_1$ has a great degree of freedom, or the trace of $\{\mathcal \{F\}\}_1$ is given by a few number of types of curves which are completely described. Moreover we prove that $\{\mathcal \{F\}\}_2$ has a transverse affine structure.},
author = {Turiel, Francisco},
journal = {Banach Center Publications},
keywords = {translation foliations; transverse affine structure},
language = {eng},
number = {1},
pages = {171-182},
title = {Translation foliations of codimension one on compact affine manifolds},
url = {http://eudml.org/doc/208660},
volume = {39},
year = {1997},
}

TY - JOUR
AU - Turiel, Francisco
TI - Translation foliations of codimension one on compact affine manifolds
JO - Banach Center Publications
PY - 1997
VL - 39
IS - 1
SP - 171
EP - 182
AB - Consider two foliations ${\mathcal {F}}_1$ and ${\mathcal {F}}_2$, of dimension one and codimension one respectively, on a compact connected affine manifold $(M,\nabla )$. Suppose that $\nabla _{T{\mathcal {F}}_1} T{\mathcal {F}}_2\subset T{\mathcal {F}}_2$; $\nabla _{T{\mathcal {F}}_2} T{\mathcal {F}}_1\subset T{\mathcal {F}}_1$ and $TM = T{\mathcal {F}}_1\oplus T{\mathcal {F}}_2$. In this paper we show that either ${\mathcal {F}}_2$ is given by a fibration over $S^1$, and then ${\mathcal {F}}_1$ has a great degree of freedom, or the trace of ${\mathcal {F}}_1$ is given by a few number of types of curves which are completely described. Moreover we prove that ${\mathcal {F}}_2$ has a transverse affine structure.
LA - eng
KW - translation foliations; transverse affine structure
UR - http://eudml.org/doc/208660
ER -

References

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  1. [1] R. Brouzet, P. Molino and F. J. Turiel, Géométrie des systèmes bihamiltoniens, Indag. Math. (N.S.) 4(3) (1993), 269-296. 
  2. [2] G. Châtelet and H. Rosenberg, Manifolds which admit 𝐑 𝐧 actions, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 245-260. Zbl0305.57031
  3. [3] G. Darboux, Leçons sur la Théorie générale de Surfaces, Gauthier-Villars, Paris. 
  4. [4] C. Godbillon, Feuilletages: études géométriques, Progr. Math. 98, Birkhäuser, 1991. 
  5. [5] I. M. Gelfand and I. Zakharevich, Webs, Veronese curves, and Bihamiltonian systems, J. Funct. Anal. 99 (1991), 150-178. Zbl0739.58021
  6. [6] G. Hector, Quelques exemples de feuilletages-Espèces rares, Ann. Inst. Fourier (Grenoble) 26(1) (1976), 239-264. Zbl0313.57015
  7. [7] G. Hector and U. Hirsch, Introduction to the Geometry of Foliations. Part B, Aspects Math. E3, Friedr. Vieweg & Sohn, Braunschweig/Wiesbaden, 1987. Zbl0704.57001
  8. [8] R. Sacksteder, Foliations and pseudo-groups, Amer. J. Math. 87 (1965), 79-102. Zbl0136.20903
  9. [9] B. Seke, Sur les structures transversalement affines des feuilletages de codimension un, Ann. Inst. Fourier (Grenoble) 30(1) (1980), 1-29. Zbl0417.57011

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