Structure of a leaf of some codimension one riemannian foliation

Bugajska, Krystyna

Annales de l'Institut Fourier, Tome 38 (1988), p. 169-174 / Harvested from Numdam

Résumé

Quelques propriétés sont démontrées de la “portée” sur une feuille ouverte ${\cal L}$ d’un feuilletage arbitraire de co-dimension 1; celles-ci diffèrent des propriétés connues de la distance de feuilles. Elles comprennent que la feuille ${\cal L}$ est d’un type fibré sur une variété riemannienne complète avec marge, ainsi que l’existence d’un champ vectoriel $v$ sur ${\cal L}$. Si $v$ est parallèle, ${\cal L}$ est difféomorphe de ${\cal L}^{\prime }\times {\bf R}$ et d’une courbure non-positive.

Abstract

Some properties of the range on an open leaf ${\cal L}$ of some codimension-one foliation are shown. They are different from the known properties of the distance of leaves. They imply that leaf ${\cal L}$ is of fibred type over a complete Riemannian manifold with boundary, as well that there exists some vector field $v$ on ${\cal L}$. If $v$ is parallel then ${\cal L}$ is diffeomorphic to ${\cal L}^{\prime }\times {\bf R}$ and has non-positive curvature.

Détail

Publié le : 1988-01-01

MR 89f:53052 | Zbl 0652.53024

DOI : https://doi.org/10.5802/aif.1128

Bibliographie

[1] R.A. Blumenthal, J.J. Hebda. — Complementary distributions which preserve the leaf geometry and applications to totally geodesic foliation, Quart. J. Math. Oxford, (2) 35 (1984), 383-392. | MR 86e:53021 | Zbl 0572.57016

[2] D. Gromoll, W. Meyer. — On complete open manifolds of positive curvature, Ann. Math., 90 (1969), 75-90. | MR 40 #854 | Zbl 0191.19904

[3] D.I. Jonson, L.B. Whitt. — Totally geodesic foliations, J. Diff. Geom., 15 (1980), 225-235. | MR 83h:57037 | Zbl 0444.57017

[4] S. Kashiwabara. — The structure of a Riemannian manifold admitting a parallel field of one dimensional tangent vector subspaces, Tohoku Math. J., 11 (1959), 119-132. | MR 22 #4076 | Zbl 0106.15103

[5] S. Kobayashi, K. Nomizu. — Foundations of differential geometry, vol. I, Interscience, New York, 1967.

[6] B.L. Reinhart. — Foliated manifolds with bundle like metrics, Annals of Math., 69 (1959), 119-132. | MR 21 #6004 | Zbl 0122.16604

[7] D.J. Welsh. — On the existence of complete parallel vector fields, Proc. Am. Math. Soc., 97 (1986), 311-314. | MR 87g:53065 | Zbl 0603.53024

[8] S.T. Yau. — Remarks on the group of isometries of a Riemannian manifold, Topology, 16 (1977), 239-247. | MR 56 #6686 | Zbl 0372.53020