Structure of a leaf of some codimension one riemannian foliation
Annales de l'Institut Fourier, Tome 38 (1988) no. 1, pp. 169-174.

Quelques propriétés sont démontrées de la “portée” sur une feuille ouverte 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 ℒ est d’un type fibré sur une variété riemannienne complète avec marge, ainsi que l’existence d’un champ vectoriel v sur ℒ. Si v est parallèle, ℒ est difféomorphe de ℒ ′ ×R et d’une courbure non-positive.

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

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     title = {Structure of a leaf of some codimension one riemannian foliation},
     journal = {Annales de l'Institut Fourier},
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Bugajska, Krystyna. Structure of a leaf of some codimension one riemannian foliation. Annales de l'Institut Fourier, Tome 38 (1988) no. 1, pp. 169-174. doi : 10.5802/aif.1128. http://www.numdam.org/articles/10.5802/aif.1128/

[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 | Zbl

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

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

[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 | Zbl

[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 | Zbl

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

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

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