×

A rigidity phenomenon for germs of actions of \(\mathbb{R}^2\). (English) Zbl 1214.22001

The authors deal with the maximality of Lie algebras generated by commuting vector fields without linear part. The work is restricted to those commuting vector fields whose Taylor development starts with a given pair of polynomial homogeneous commuting vector fields. The authors restrict their interest to three special maximal Lie algebras generated by commuting quadratic homogeneous vector fields in \(\mathbb R^2\), \(\mathbb R^3\) and \(\mathbb R^4\) respectively. In the first case it is proved that the quadratic algebra is a smooth normal form. In the second and third ones, the authors prove that the orbit structure is, from a topological viewpoint, the one of the quadratic part.

MSC:

22E05 Local Lie groups
17B66 Lie algebras of vector fields and related (super) algebras
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Dumortier (F.) and Roussarie (R.).— Smooth linearization of germs of \({\bf R}^2\)-actions and holomorphic vector fields. Ann. Inst. Fourier (Grenoble), Vol. 30 no. 1, p. 31-64 (1980). · Zbl 0418.58015
[2] Ghys (E.) and Rebelo (J.-C.).— Singularités des flots holomorphes. II. Ann. Inst. Fourier (Grenoble), Vol. 47 no. 4, p. 1117-1174 (1997). · Zbl 0938.32019
[3] Guillot (A.).— Sur les exemples de Lins Neto de feuilletages algébriques. C. R. Math. Acad. Sci. Paris, Vol. 334 no. 9, p. 747-750 (2002). · Zbl 1004.37029
[4] Palais (R. S.).— A global formulation of the Lie theory of transformation groups. Mem. Amer. Math. Soc., 22 (1957). · Zbl 0178.26502
[5] Palis (J.) and de Melo (W.).— Introdução aos sistemas dinâmicos, volume 6 of Projeto Euclides. Instituto de Matemática Pura e Aplicada, Rio de Janeiro (1978). · Zbl 0507.58001
[6] Rebelo (J. C.).— Singularités des flots holomorphes. Ann. Inst. Fourier (Grenoble), Vol. 46 no. 2, p. 411-428 (1996). · Zbl 0853.34002
[7] Rebelo (J. C.).— Réalisation de germes de feuilletages holomorphes par des champs semi-complets en dimension 2. Ann. Fac. Sci. Toulouse Math. (6), Vol. 9 no. 4, p. 735-763 (2000). · Zbl 1002.32025
[8] Reis (H.).— Equivalence and semi-completude of foliations. Nonlinear Anal., Vol. 64 no. 8, p. 1654-1665 (2006). · Zbl 1134.37345
[9] Salmon (G.).— A treatise on the higher plane curves. Hodges and Smith, Dublin (1852).
[10] Spivak (M.).— A comprehensive introduction to differential geometry. Vol. IV. Publish or Perish Inc., Wilmington, Del., second edition (1979). · Zbl 0439.53004
[11] Sternberg (S.).— Local contractions and a theorem of Poincaré. Amer. J. Math., Vol. 79: p. 809-824 (1957). · Zbl 0080.29902
[12] Thurston (W. P.).— Three-dimensional geometry and topology. Vol. 1. Princeton University Press, Princeton, NJ (1997). · Zbl 0873.57001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.