On riemannian foliations with minimal leaves
Lopez, Jesús A. Alvarez
Annales de l'Institut Fourier, Tome 40 (1990), p. 163-176 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

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.

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.

@article{AIF_1990__40_1_163_0,
     author = {Lopez, Jes\'us A. Alvarez},
     title = {On riemannian foliations with minimal leaves},
     journal = {Annales de l'Institut Fourier},
     volume = {40},
     year = {1990},
     pages = {163-176},
     doi = {10.5802/aif.1209},
     mrnumber = {92a:53038},
     zbl = {0688.57017},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1990__40_1_163_0}
}

Lopez, Jesús A. Alvarez. On riemannian foliations with minimal leaves. Annales de l'Institut Fourier, Tome 40 (1990) pp. 163-176. doi : 10.5802/aif.1209. http://gdmltest.u-ga.fr/item/AIF_1990__40_1_163_0/

[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 89m:53050 | Zbl 0644.57014

[2] J.A. Alvarez López, Duality in the spectral sequence of Riemannian foliations, American J. of Math., 111 (1989), 905-926. | MR 90k:58004 | Zbl 0685.57017

[3] J.A. Alvarez López, A decomposition theorem for the spectral sequence of Lie foliations, to appear. | Zbl 0756.57016

[4] R. Bott, L.W. Tu, Differential Forms in Algebraic Topology, GTM N° 82, Springer-Verlag, 1982. | MR 83i:57016 | Zbl 0496.55001

[5] Y. Carrière, Flots riemanniens, Astérisque, 116 (1984), 31-52. | MR 86m:58125a | Zbl 0548.58033

[6] Y. Carrière, Feuilletages riemanniens à croissance polynomiale, Comm. Math. Helv., 63 (1988), 1-20. | MR 89a:57033 | Zbl 0661.53022

[7] A. El Kacimi, G. Hector, Décomposition de Hodge basique pour un feuilletage riemannien, Ann. Inst. Fourier, 36-3 (1986), 207-227. | Numdam | MR 87m:57029 | Zbl 0586.57015

[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 0536.57013

[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 82j:57027 | Zbl 0444.57016

[12] A. Haefliger, Pseudogroups of local isometries, Res. Notes in Math., 131 (1985), 174-197. | MR 88i:58174 | Zbl 0656.58042

[13] G. Hector, Cohomologies transversales des feuilletages riemanniens, Sém. Sud-Rhod., VII/2, Travaux en Cours, Hermann, 1987. | Zbl 0666.57022

[14] F. Kamber, Ph. Tondeur, Duality for Riemannian foliations, Proc. Symp. Pure Math., 40/1 (1983), 609-618. | MR 85e:57030 | Zbl 0523.57019

[15] F. Kamber, Ph. Tondeur, Foliations and metrics, Progr. in Math., 32 (1983), 103-152. | MR 85a:57017 | Zbl 0542.53022

[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 88f:57045 | Zbl 0646.57016

[18] P. Molino, Géométrie globale des feuilletages riemanniens, Proc. Kon. Ned. Akad., A1, 85 (1982), 45-76. | MR 84j:53043 | Zbl 0516.57016

[19] P. Molino, V. Sergiescu, Deux remarques sur les flots riemanniens, Manuscripta Math., 51 (1985), 145-161. | MR 86h:53035 | Zbl 0585.53026

[20] B. Reinhart, Foliated manifolds with bundle-like metrics, Ann. Math., 69 (1959), 119-132. | MR 21 #6004 | Zbl 0122.16604

[21] H. Rummler, Quelques notions simples en géométrie riemannienne et leurs applications aux feuilletages compactes, Comm. Math. Helv., 54 (1979), 224-239. | MR 80m:57021 | Zbl 0409.57026

[22] K.S. Sarkaria, A finiteness theorem for foliated manifolds, J. Math. Soc. Japan, Vol. 30, N° 4 (1978), 687-696. | MR 80a:57014 | Zbl 0398.57012

[23] V. Sergiescu, Sur la suite spectrale d'un feuilletage riemannien, Lille, 1986. | Zbl 0673.53021

[24] D. Sullivan, A cohomological characterization of foliations consisting of minimal surfaces, Com. Math. Helv., 54 (1979), 218-223. | MR 80m:57022 | Zbl 0409.57025