A rigidity phenomenon for germs of actions of ${\mathbf{R}}^2$

Arroyo, Aubin, and Guillot, Adolfo. "A rigidity phenomenon for germs of actions of ${\bf R}^2$." Annales de la faculté des sciences de Toulouse Mathématiques 19.3-4 (2010): 539-565. <http://eudml.org/doc/115860>.

@article{Arroyo2010,
abstract = {We study germs of Lie algebras generated by two commuting vector fields in manifolds that are maximal in the sense of Palais (those which do not present any evident obstruction to be the local model of an action of $\{\bf R\}^2$). We study three particular pairs of homogeneous quadratic commuting vector fields (in $\{\bf R\}^2$, $\{\bf R\}^3$ and $\{\bf R\}^4$) and study the maximal Lie algebras generated by commuting vector fields whose 2-jets at the origin are the given homogeneous ones. In the first case we prove that the quadratic algebra is a smooth normal form. In the second and third ones, we prove that the orbit structure is, from a topological viewpoint, the one of the quadratic part.},
affiliation = {Instituto de Matemáticas, Unidad Cuernavaca Universidad Nacional Autónoma de México, A.P. 273-3 Admon. 3, Cuernavaca, Morelos, 62251 México; Instituto de Matemáticas, Unidad Cuernavaca Universidad Nacional Autónoma de México, A.P. 273-3 Admon. 3, Cuernavaca, Morelos, 62251 México},
author = {Arroyo, Aubin, Guillot, Adolfo},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {Lie algebra; commuting vector fields},
language = {eng},
number = {3-4},
pages = {539-565},
publisher = {Université Paul Sabatier, Toulouse},
title = {A rigidity phenomenon for germs of actions of $\{\bf R\}^2$},
url = {http://eudml.org/doc/115860},
volume = {19},
year = {2010},
}

TY - JOUR
AU - Arroyo, Aubin
AU - Guillot, Adolfo
TI - A rigidity phenomenon for germs of actions of ${\bf R}^2$
JO - Annales de la faculté des sciences de Toulouse Mathématiques
PY - 2010
PB - Université Paul Sabatier, Toulouse
VL - 19
IS - 3-4
SP - 539
EP - 565
AB - We study germs of Lie algebras generated by two commuting vector fields in manifolds that are maximal in the sense of Palais (those which do not present any evident obstruction to be the local model of an action of ${\bf R}^2$). We study three particular pairs of homogeneous quadratic commuting vector fields (in ${\bf R}^2$, ${\bf R}^3$ and ${\bf R}^4$) and study the maximal Lie algebras generated by commuting vector fields whose 2-jets at the origin are the given homogeneous ones. In the first case we prove that the quadratic algebra is a smooth normal form. In the second and third ones, we prove that the orbit structure is, from a topological viewpoint, the one of the quadratic part.
LA - eng
KW - Lie algebra; commuting vector fields
UR - http://eudml.org/doc/115860
ER -