Théorèmes de slice et holonomie des feuilletages riemanniens singuliers
Annales de l'Institut Fourier, Volume 37 (1987) no. 4, p. 207-223

Let (M,) be a riemannian foliation on a closed manifold, ¯ the singular foliation defined by the closures of the leaves, (F ¯,) the induced foliation on a generic closure. We study the case where (F,) has no non trivial transverse vector field. Then the quotient space W=M/ ¯ has a natural structure of Sataké manifold, and the projection MW is a morphism (of Sataké manifolds) with folding along singular closures.

Soit (M,) un feuilletage riemannien sur une variété compacte; ¯ est le feuilletage singulier défini par les adhérences des feuilles (F ¯,) le feuilletage induit sur une adhérence générique. On étudie le cas où (F ¯,) n’a pas de champ transverse non trivial. Alors l’espace quotient W=M/ ¯ a une structure naturelle de variété de Sataké, de manière que la projection MW soit un morphisme (de variétés de Sataké) avec pliage autour des adhérences singulières.

@article{AIF_1987__37_4_207_0,
     author = {Molino, Pierre and Pierrot, M.},
     title = {Th\'eor\`emes de slice et holonomie des feuilletages riemanniens singuliers},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {37},
     number = {4},
     year = {1987},
     pages = {207-223},
     doi = {10.5802/aif.1118},
     zbl = {0625.57016},
     mrnumber = {89a:53040},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_1987__37_4_207_0}
}
Molino, Pierre; Pierrot, M. Théorèmes de slice et holonomie des feuilletages riemanniens singuliers. Annales de l'Institut Fourier, Volume 37 (1987) no. 4, pp. 207-223. doi : 10.5802/aif.1118. http://www.numdam.org/item/AIF_1987__37_4_207_0/

[1] R. Blumenthal, J. Hebda, De Rham decomposition theorems for foliated manifolds, Ann. Inst. Fourier, 33-2 (1983), 183-198. | Numdam | MR 84j:53042 | Zbl 0487.57010

[2] G. Cairns, A general description of totally geodesic foliations, Tohoku Math. Jour., 38 (1986), 37-55. | MR 87d:53062 | Zbl 0574.57012

[3] L. Conlon, Variational completeness and K-transversal domains, Jour. of Diff. geometry, 5 (1971), 135-147. | MR 45 #4320 | Zbl 0213.48602

[4] P. Dazord, Feuilletages à singularités, Proc. Kon. Akad. van Wet, 88 (1) (1985), 21-39. | MR 87a:57030 | Zbl 0584.57016

[5] A. Haefliger, Leaf closures in riemannian foliations, preprint, 1986. | Zbl 0667.57012

[6] J. L. Koszul, Sur certains groupes de transformations de Lie, Colloque de Géométrie différentielle, Strasbourg, 1953. | MR 15,600g | Zbl 0101.16201

[7] S. Kobayashi, K. Nomizu, Fundations of differential geometry, I-II, Wiley, New-York, 1963-1969.

[8] O. Loos, Symmetric spaces, I-II, Benjamin, New-York, 1969. | Zbl 0175.48601

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

[10] P. Molino, Feuilletages riemanniens réguliers et singuliers, preprint 1986. | Zbl 0627.53030

[11] M. Pierrot, Orbites des champs de vecteurs feuilletés pour un feuilletage riemannien sur une variété compacte, C.R. Ac. Sci., Paris, 301 (1985), 443-446. | MR 86k:53054 | Zbl 0593.58003

[12] J. Pradines, How to define the differentiable graph of a singular foliation, Cahiers de Top. et Geom. Diff., 26 (4) (1985), 339-380. | Numdam | MR 87g:57043 | Zbl 0576.57023

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

[14] I. Satake, The Gauss-Bonnet theorem for V-manifolds, Journal Math. Soc. of Japan, 9 (1957), 464-492. | MR 20 #2022 | Zbl 0080.37403

[15] P. Stefan, Accessible sets, orbits, and foliations with singularities, Proc. London Math. Soc., 29 (1974), 699-713. | Zbl 0342.57015

[16] H. Sussmann, Orbits of families of vector fields and integrability of distributions, Trans. Am. Math. Soc., 180 (1973), 171-188. | MR 47 #9666 | Zbl 0274.58002

[17] J. Szenthe, On the orbit structure of orthogonal actions with isotropy subgroups of maximal rank, Acta Sci. Math (Szeged), 43 (1981), 353-367. | MR 83j:57026 | Zbl 0493.57020

[18] J. Szenthe, Orthogonally transversal submanifolds and the generalization of the Weyl group, Periodica Math. Hungarica, 15 (4) (1984), 281-299. | MR 86m:53065 | Zbl 0583.53035

[19] D. Luna, Adhérences d'orbites et invariants, Inventions Math., 29 (1975), 231-238. | MR 51 #12879 | Zbl 0315.14018

[20] G.W. Schwartz, Lifting smooth homotopies of orbit spaces, Publ. Inst. Hautes Et. Scient., 51 (1980), 37-136. | Numdam | MR 81h:57024 | Zbl 0449.57009