Abstract
Let M be a connected oriented closed n-manifold. A riemannian flow\(\mathfrak{F}\) on M is an oriented one dimensional foliation which admits a bundle-like metric.
We give a caracterization of isometric flows as riemannian flows whose basic cohomology H n−1b (M,\(\mathfrak{F}\)) is non trivial in degree (n−1). A second caracterization involves the triviality of the central sheaf.
We show also that\(\mathfrak{F}\) has a section if and only if H n−1b (M,\(\mathfrak{F}\)) has a non trivial image in Hn−1(M).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Bibliographie
P. Caron-Y. Carrière “Flots transversalement de Lie Rn, flots transversalement de Lie minimaux” C. R. Ac. Sc. Paris, 291 (1980), série A, pp. 477–478
Y. Carrière “Flots riemanniens”-Journées sur les structures transverses, Toulouse 1982-Asterisque (1984)
El Kacimi-G. Hector-V. Sergiescu “La cohomologie basique d'un feuilletage riemannien est de dimension finie”. A paraître dans Mathematishe Zeitschrift
E. Ghys “Classification des feuilletages totalement géodésiques de codimension un”. Comentari Math. Helv. 58, 4 (1983), pp. 543–572
E. Ghys “Feuilletages riemanniens sur les variétés simplement connexes”. Ann. Inst. Fourrier (à paraître)
H. Gluck “Dynamical behaviour of geodesic flows” In Lectures Notes no 819
A. Haefliger “Some remarks on foliations with minimal leaves”. Journal Diff. Geom., 15 (1980), 269–284
F. Kamber, P. Tondeur “Duality for riemannian foliations”. Proc. Symp. Pure Math 40 (1983)
P. Molino “Géométrie globale des feuilletages riemanniens”. Proc. Kon. Ned, Akad, Al, 85 (1982), pp. 45–76
P. Molino “Flots riemanniens et flots isométriques”. dans “Séminaire de Géométrie différentielle 1982–1983”, Montpellier
M. Pierrot “Groupe des automorphismes d'un flot linéaire sur le tore” dans “Séminaire de Géométrie Différentielle 1983–1984”. Montpellier
B. Reinhart “Foliated manifolds with bundle-like metrics”. Ann. of Maths, 69 (1959), pp. 119–132
V. Sergiescu “Cohomologie basique et dualité des feuilletages riemanniens” preprint Lille 1984
D. Sullivan “Cycles for the dynamical study of foliated manifolds” Inv. Math. 36 (1976) pp. 225–255
D. B. A. Epstein “Transversaly hyperbolic 1-dimensional foliations”. Journées sur les structures transverses, Toulouse 1982-Asterisque (1984)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Molino, P., Sergiescu, V. Deux remarques sur les flots riemanniens. Manuscripta Math 51, 145–161 (1985). https://doi.org/10.1007/BF01168350
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01168350