Nous démontrons que si est une variété riemannienne complète simplement connexe et est un feuilletage totalement géodésique sur dont le fibré orthogonal est involutif, alors est topologiquement un produit et les deux feuilletages sont les feuilletages produits. Nous démontrons aussi un théorème de décomposition pour les feuilletages riemanniens et un théorème de structure pour les feuilletages riemanniens à courbure récurrente.
We prove that if is a complete simply connected Riemannian manifold and is a totally geodesic foliation of with integrable normal bundle, then is topologically a product and the two foliations are the product foliations. We also prove a decomposition theorem for Riemannian foliations and a structure theorem for Riemannian foliations with recurrent curvature.
@article{AIF_1983__33_2_183_0,
author = {Blumenthal, Robert A. and Hebda, James J.},
title = {De Rham decomposition theorems for foliated manifolds},
journal = {Annales de l'Institut Fourier},
volume = {33},
year = {1983},
pages = {183-198},
doi = {10.5802/aif.923},
mrnumber = {84j:53042},
zbl = {0487.57010},
language = {en},
url = {http://dml.mathdoc.fr/item/AIF_1983__33_2_183_0}
}
Blumenthal, Robert A.; Hebda, James J. De Rham decomposition theorems for foliated manifolds. Annales de l'Institut Fourier, Tome 33 (1983) pp. 183-198. doi : 10.5802/aif.923. http://gdmltest.u-ga.fr/item/AIF_1983__33_2_183_0/
[1] , Transversely homogeneous foliations, Annales Inst. Fourier, 29 (1979), 143-158. | Numdam | MR 81h:57011 | Zbl 0405.57016
[2] , Riemannian homogeneous foliations without holonomy, Nagoya Math. J., 83 (1981), 197-201. | MR 82m:57013 | Zbl 0427.57008
[3] , Riemannian foliations with parallel curvature, Nagoya Math. J. (to appear). | Zbl 0508.57020
[4] , Lectures on characteristic classes and foliations (notes by L. Conlon), Lecture Notes in Math., no. 279, Springer-Verlag, New York, 1972, 1-80. | MR 50 #14777 | Zbl 0241.57010
[5] and , Feuilletages totalement géodésiques, An. Acad. Brasil, Ciênc., 53 (1981), 427-432. | MR 83m:57019 | Zbl 0486.57013
[6] , Variétés feuilletées, Ann. Scuola Norm. Pisa, 16 (1962), 367-397. | Numdam | MR 32 #6487 | Zbl 0122.40702
[7] , On the differential geometry of foliations, Annals of Math., 72 (1960), 445-457. | MR 25 #5523 | Zbl 0196.54204
[8] , A sufficient condition that a mapping of Riemannian manifolds be a fibre bundle, Proc. Amer. Math. Soc., 11 (1960), 236-242. | MR 22 #3006 | Zbl 0112.13701
[9] and , Totally geodesic foliations on 3-manifolds, Proc. Amer. Math. Soc., 76 (1979), 355-357. | MR 81d:57020 | Zbl 0388.57015
[10] and , Totally geodesic foliations, Journal of Diff. Geom., 15 (1980), 225-235. | MR 83h:57037 | Zbl 0444.57017
[11] and , Foundations of differential geometry, vol. I, Interscience Tracts in Pure and Appl. Math., 15, Interscience, New York, 1963. | Zbl 0119.37502
[12] and , Secondary characteristic classes for Riemannian foliations, Journal of Diff. Geom., 11 (1976), 365-385. | MR 56 #3853 | Zbl 0356.57007
[13] , Etude des feuilletages transversalement complets et applications, Ann. Scient. Ec. Norm. Sup., 10 (1977), 289-307. | Numdam | MR 56 #16649 | Zbl 0368.57007
[14] , Foliations with measure preserving holonomy, Annals of Math., 102 (1975), 327-361. | MR 52 #11947 | Zbl 0314.57018
[15] , Measure preserving pseudogroups and a theorem of Sacksteder, Annales Inst. Fourier, 25, 1 (1975), 237-249. | Numdam | MR 51 #14100 | Zbl 0299.58007
[16] , Foliated manifolds with bundle-like metrics, Annals of Math., 69 (1959), 119-132. | MR 21 #6004 | Zbl 0122.16604
[17] , Sur la réductibilité d'un espace de Riemann, Comm. Math. Helv., 26 (1952), 328-344. | MR 14,584a | Zbl 0048.15701
[18] , Foliations and pseudogroups, Amer. J. of Math., 87 (1965), 79-102. | MR 30 #4268 | Zbl 0136.20903
[19] , On fibering certain foliated manifolds over S1, Topology, 9 (1970), 153-154. | MR 41 #1069 | Zbl 0177.52103