Translation foliations of codimension one on compact affine manifolds

@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 -