Proyecciones Journal of Mathematics
Vol. 26, No 1, pp. 1-25, May 2007.
Universidad Católica del Norte
Antofagasta - Chile


ON SOME INFINITESIMAL AUTOMORPHISMS OF RIEMANNIAN FOLIATION


MOHAMED ALÍ CHAOUCH^1, NABILA TORKI - HAMZA²

¹UNIVERSITÉ DU 7 NOVEMBRE A CARTHAGE, TUNISIE
²UNIVERSITÉ DU 7 NOVEMBRE A CARTHAGE, TUNISIE

Correspondencia a:

Abstract

In Riemannian foliation, a transverse affine vector field preserves the curvature and its covariant derivatives. In this paper we solve the converse problem. Actually, we show that an infinitesimal automorphism of a Riemannian foliation which preserves the curvature and its covariant derivatives induces a transverse almost homothetic vector field. If in addition the manifold is closed and the foliation is irreducible harmonic , then a such infinitesimal automorphism induces a transverse killing vector field.

Keywords : Riemannian foliation. Harmonic foliation. Irreducible foliation. Transverse vector field.

REFERENCES

[1] R. A. Blumenthal, Transversaly homogeneous foliations, Ann. Institut Fourier 29, pp. 143-158, (1979).         [ Links ]

[2] R. A. Blumenthal, Foliated manifolds with flat basic connection, J. Diff.Geom. 16, pp. 401-406, (1981).         [ Links ]

[3] R. A. Blumenthal- J. J. Hebda, De Rham decomposition theorems for foliated manifolds, Ann. Inst Fourier. Grenoble. 33. 2, pp. 183-198, (1983).         [ Links ]

[4] Y. Carriére, Flots Riemanniens, Astérisque 116, pp. 31-52, (1984).         [ Links ]

[5] L. Conlon, Transversally parallelisable foliation of codimension two, Trans. Amer. Math. Soc. 194, pp. 79-102, (1974).         [ Links ]

[6] C. Godbillon, Feuilletages, etudes géométrique, Birkaüser Verlag, (1991).         [ Links ]

[7] K. Kobayashi - K. Nomizu, Fondations of differential geometry, Vol I .Interscience tracts in pure and applaied mathmatics. 15 Interscience New-York, (1963).         [ Links ]

[8] F. W. Kamber - Ph. Tondeur, Harmonic foliations, Lecture Notes in Math. 949, Springer -Verlag Berlin-Heidelberg-New York, pp 87-121, (1982).         [ Links ]

[9] F. W. Kamber- Ph. Tondeur, Infenitesimal automorphisms and second variation of the energy for harmonic foliations, Tóhoku. Math. J. 34, pp. 525-538, (1982). On some infinitesimal automorphisms of Riemannian Foliations 25         [ Links ]

[10] K. Nomizu- K. Yano, On infinitesimal transformations preserving the curvature tensor field and its covariant differentials, Ann. Ins. Four. 14-2 (1964).         [ Links ]

[11] Ph. Tondeur, Geometry of foliations, Birkaüser Verlag, (1997).         [ Links ]

Mohamed Ali Chaouch
Faculté des Sciences de Bizerte
Université du 7 Novembre A Carthage 7021-Zarzouna.
Bizerte -Tunisie.
e-mail : MohamedAli.Chaouch@fsb.rnu.tn

Nabila Torki-Hamza
Faculté des Sciences de Bizerte
Université du 7 Novembre A Carthage 7021-Zarzouna.
Bizerte -Tunisie.
e-mail : Nabila.Torki-Hamza@fsb.rnu.tn