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

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

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

@article{AIF_1988__38_1_169_0,
     author = {Bugajska, Krystyna},
     title = {Structure of a leaf of some codimension one riemannian foliation},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {38},
     number = {1},
     year = {1988},
     pages = {169-174},
     doi = {10.5802/aif.1128},
     zbl = {0652.53024},
     mrnumber = {89f:53052},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1988__38_1_169_0}
}
Bugajska, Krystyna. Structure of a leaf of some codimension one riemannian foliation. Annales de l'Institut Fourier, Volume 38 (1988) no. 1, pp. 169-174. doi : 10.5802/aif.1128. http://www.numdam.org/item/AIF_1988__38_1_169_0/

[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