On riemannian foliations with minimal leaves
Annales de l'Institut Fourier, Volume 40 (1990) no. 1, pp. 163-176.

For a Riemannian foliation, the topology of the corresponding spectral sequence is used to characterize the existence of a bundle-like metric such that the leaves are minimal submanifolds. When the codimension is 2, a simple characterization of this geometrical property is proved.

Pour un feuilletage riemannien, on utilise la topologie de la suite spectrale correspondante pour caractériser l’existence d’une métrique “bundle-like” telle que les feuilles sont des sous-variétés minimales. Quand la codimension est 2, on prouve une caractérisation cohomologique simple de cette propriété géométrique.

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     title = {On riemannian foliations with minimal leaves},
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Lopez, Jesús A. Alvarez. On riemannian foliations with minimal leaves. Annales de l'Institut Fourier, Volume 40 (1990) no. 1, pp. 163-176. doi : 10.5802/aif.1209. https://aif.centre-mersenne.org/articles/10.5802/aif.1209/

[1] J.A. Alvarez López, A finiteness theorem for the spectral sequence of a Riemannian foliation, Illinois J. of Math., 33 (1989), 79-92. | MR | Zbl

[2] J.A. Alvarez López, Duality in the spectral sequence of Riemannian foliations, American J. of Math., 111 (1989), 905-926. | MR | Zbl

[3] J.A. Alvarez López, A decomposition theorem for the spectral sequence of Lie foliations, to appear. | Zbl

[4] R. Bott, L.W. Tu, Differential Forms in Algebraic Topology, GTM N° 82, Springer-Verlag, 1982. | MR | Zbl

[5] Y. Carrière, Flots riemanniens, Astérisque, 116 (1984), 31-52. | MR | Zbl

[6] Y. Carrière, Feuilletages riemanniens à croissance polynomiale, Comm. Math. Helv., 63 (1988), 1-20. | MR | Zbl

[7] A. El Kacimi, G. Hector, Décomposition de Hodge basique pour un feuilletage riemannien, Ann. Inst. Fourier, 36-3 (1986), 207-227. | Numdam | MR | Zbl

[8] A. El Kacimi, G. Hector, V. Sergiescu, La cohomologie basique d'un feuilletage riemannien est de dimension finie, Math. Z., 188 (1985), 593-599. | Zbl

[9] E. Ghys, Riemannian Foliations : Examples and Problems, Appendix E of Riemannian Foliations (by P. Molino), Birkhäuser, 1988, 297-314.

[10] W. Greub, S. Halperin, R. Vanstone, Connections, curvature and cohomology, Academic Press, 1973-1975.

[11] A. Haefliger, Some remarks on foliations with minimal leaves, J. Diff. Geom., 15 (1980), 269-284. | MR | Zbl

[12] A. Haefliger, Pseudogroups of local isometries, Res. Notes in Math., 131 (1985), 174-197. | MR | Zbl

[13] G. Hector, Cohomologies transversales des feuilletages riemanniens, Sém. Sud-Rhod., VII/2, Travaux en Cours, Hermann, 1987. | Zbl

[14] F. Kamber, Ph. Tondeur, Duality for Riemannian foliations, Proc. Symp. Pure Math., 40/1 (1983), 609-618. | MR | Zbl

[15] F. Kamber, Ph. Tondeur, Foliations and metrics, Progr. in Math., 32 (1983), 103-152. | MR | Zbl

[16] E. Macías, Las cohomologías diferenciable, continua y discreta de una variedad foliada, Publ. do Dpto. de Xeometría e Topoloxía n° 60, Santiago de Compostela, 1983.

[17] X. Masa, Cohomology of Lie foliations, Res. Notes in Math., 131 (1985), 211-214. | MR | Zbl

[18] P. Molino, Géométrie globale des feuilletages riemanniens, Proc. Kon. Ned. Akad., A1, 85 (1982), 45-76. | MR | Zbl

[19] P. Molino, V. Sergiescu, Deux remarques sur les flots riemanniens, Manuscripta Math., 51 (1985), 145-161. | MR | Zbl

[20] B. Reinhart, Foliated manifolds with bundle-like metrics, Ann. Math., 69 (1959), 119-132. | MR | Zbl

[21] H. Rummler, Quelques notions simples en géométrie riemannienne et leurs applications aux feuilletages compactes, Comm. Math. Helv., 54 (1979), 224-239. | MR | Zbl

[22] K.S. Sarkaria, A finiteness theorem for foliated manifolds, J. Math. Soc. Japan, Vol. 30, N° 4 (1978), 687-696. | MR | Zbl

[23] V. Sergiescu, Sur la suite spectrale d'un feuilletage riemannien, Lille, 1986. | Zbl

[24] D. Sullivan, A cohomological characterization of foliations consisting of minimal surfaces, Com. Math. Helv., 54 (1979), 218-223. | MR | Zbl

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