# On Lie algebras of vector fields related to Riemannian foliations

Annales Polonici Mathematici (1993)

- Volume: 58, Issue: 2, page 111-122
- ISSN: 0066-2216

## Access Full Article

top## Abstract

top## How to cite

topTomasz Rybicki. "On Lie algebras of vector fields related to Riemannian foliations." Annales Polonici Mathematici 58.2 (1993): 111-122. <http://eudml.org/doc/262263>.

@article{TomaszRybicki1993,

abstract = {Riemannian foliations constitute an important type of foliated structures. In this note we prove two theorems connecting the algebraic structure of Lie algebras of foliated vector fields with the smooth structure of a Riemannian foliation.},

author = {Tomasz Rybicki},

journal = {Annales Polonici Mathematici},

keywords = {Riemannian foliation; Lie algebra; ideal; isomorphism; vector field; generalized manifold; stratification; space of leaves; singular foliation; Riemannian foliations of compact, connected manifolds; Lie algebra of vector fields; Lie algebra of foliated vector fields; Lie algebra isomorphism; Satake diffeomorphism},

language = {eng},

number = {2},

pages = {111-122},

title = {On Lie algebras of vector fields related to Riemannian foliations},

url = {http://eudml.org/doc/262263},

volume = {58},

year = {1993},

}

TY - JOUR

AU - Tomasz Rybicki

TI - On Lie algebras of vector fields related to Riemannian foliations

JO - Annales Polonici Mathematici

PY - 1993

VL - 58

IS - 2

SP - 111

EP - 122

AB - Riemannian foliations constitute an important type of foliated structures. In this note we prove two theorems connecting the algebraic structure of Lie algebras of foliated vector fields with the smooth structure of a Riemannian foliation.

LA - eng

KW - Riemannian foliation; Lie algebra; ideal; isomorphism; vector field; generalized manifold; stratification; space of leaves; singular foliation; Riemannian foliations of compact, connected manifolds; Lie algebra of vector fields; Lie algebra of foliated vector fields; Lie algebra isomorphism; Satake diffeomorphism

UR - http://eudml.org/doc/262263

ER -

## References

top- [1] K. Abe, Pursell-Shanks type theorem for orbit spaces of G-manifolds, Publ. RIMS Kyoto Univ. 18 (1982), 685-702. Zbl0519.57033
- [2] I. Amemiya, Lie algebra of vector fields and complex structure, J. Math. Soc. Japan 27 (1975), 545-549. Zbl0311.57012
- [3] M. Davis, Smooth G-manifolds as collections of fiber bundles, Pacific J. Math. 77 (1978), 315-363. Zbl0403.57002
- [4] R. P. Filipkiewicz, Isomorphisms between diffeomorphism groups, Ergodic Theory Dynamical Systems 2 (1982), 159-171. Zbl0521.58016
- [5] K. Fukui, Pursell-Shanks type theorem for free G-manifolds, Publ. RIMS Kyoto Univ. 17 (1981), 249-265. Zbl0464.57018
- [6] K. Fukui and N. Tomita, Lie algebra of foliation preserving vector fields, J. Math. Kyoto Univ. 22 (1983), 685-699. Zbl0511.58030
- [7] F. Guedira et A. Lichnerowicz, Géométrie des algèbres de Lie locales de Kirillov, J. Math. Pures Appl. 63 (1984), 407-484. Zbl0562.53029
- [8] P. Molino, Géométrie globale des feuilletages riemanniens, Nederl. Akad. Wetensch. Proc. 85 (1982), 45-76. Zbl0516.57016
- [9] P. Molino, Riemannian Foliations, Progr. Math. 73, Birkhäuser, 1988.
- [10] M. Pierrot, Orbites des champs feuilletés pour un feuilletage riemannien sur une variété compacte, C. R. Acad. Sci. Paris 301 (1985), 443-445. Zbl0593.58003
- [11] L. E. Pursell and M. E. Shanks, The Lie algebra of a smooth manifold, Proc. Amer. Math. Soc. 5 (1954), 468-472. Zbl0055.42105
- [12] T. Rybicki, On the Lie algebra of a transversally complete foliation, Publ. Sec. Mat. Univ. Autònoma Barcelona 31 (1987), 5-16. Zbl0637.57018
- [13] T. Rybicki, Lie algebras of vector fields and codimension one foliations, Publ. Mat. 34 (1990), 311-321. Zbl0721.57016
- [14] G. W. Schwarz, Lifting smooth homotopies of orbit spaces, Publ. IHES 51 (1980), 37-135. Zbl0449.57009
- [15] R. A. Wolak, Maximal subalgebras in the algebra of foliated vector fields of a Riemannian foliation, Comment. Math. Helv. 64 (1989), 536-541. Zbl0695.53023

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.