Nous étudions les feuilletages lisses totalement géodésiques de codimension des variétés lorentziennes. Nous nous intéressons notamment aux relations entre les flots riemanniens et les feuilletages géodésiques. Nous prouvons que, quitte à prendre un revêtement d’ordre , tout fibré de Seifert possède un tel feuilletage.
We study totally geodesic codimension smooth foliations on Lorentzian manifolds. We are in particular interested in the relations between riemannian flows and geodesic foliations. We prove that, up to a -cover, any Seifert bundle admits such a foliation.
Classification : 57R30, 53C50
Mots clés : feuilletages totalement géodésiques, flots riemanniens
@article{AIF_2005__55_4_1411_0, author = {Mounoud, Pierre}, title = {Feuilletages totalement g\'eod\'esiques, flots riemanniens et vari\'et\'es de Seifert}, journal = {Annales de l'Institut Fourier}, pages = {1411--1438}, publisher = {Association des Annales de l'institut Fourier}, volume = {55}, number = {4}, year = {2005}, doi = {10.5802/aif.2128}, zbl = {1080.53024}, mrnumber = {2157171}, language = {fr}, url = {http://www.numdam.org/item/AIF_2005__55_4_1411_0/} }
Mounoud, Pierre. Feuilletages totalement géodésiques, flots riemanniens et variétés de Seifert. Annales de l'Institut Fourier, Tome 55 (2005) no. 4, pp. 1411-1438. doi : 10.5802/aif.2128. http://www.numdam.org/item/AIF_2005__55_4_1411_0/
[A-S] A note on actions of the cylinder , Topology and its applications, Volume 123 (2002), pp. 533-535 | Article | MR 1924050 | Zbl 1039.57024
[B-M-T] Foliations admitting a transverse connection; applications in dimension (à paraître dans Ergodic Theory Dynam. Systems.)
[C-C1] Foliations I, Graduate Studies in Math., Volume 23 | Zbl 0936.57001
[C-C2] Endsets of exceptional leaves; a theorem of G. Duminy (preprint) | Zbl 1011.57009
[C-G] Feuilletages totalement géodésiques, An. Acad. Brasil Cienc., Volume 53 (1981) no. 3, pp. 427-432 | MR 663239 | Zbl 0486.57013
[Ca] Flots riemanniens, Structure transverse des feuilletages, Toulouse 1982 (Asterisque), Volume 116 (1984), pp. 31-52 | Zbl 0548.58033
[E-H-N] Transverse foliations of Seifert bundles and self homeomorphism of the circle, Comment. Math. Helv., Volume 56 (1981), pp. 638-660 | Article | MR 656217 | Zbl 0516.57015
[G-K] The fundamental group of a compact flat Lorentz space form is virtually polycyclic., J. Differential Geom., Volume 19 (1984) no. 1, pp. 233-240 | MR 739789 | Zbl 0546.53039
[G-N] A Hochschild homology Euler characteristic for circle actions, K-Theory, Volume 18 (1999), pp. 99-135 | Article | MR 1711720 | Zbl 0947.55004
[Gh] Rigidité différentiable des groupes fuchsiens, Inst. Hautes Études Sci. Publ. Math. (1993) no. 78, pp. 163-185 | Numdam | MR 1259430 | Zbl 0812.58066
[He] Feuilletages en cylindres (Lecture Notes in Math.), Volume 597 (1977), pp. 252-270 | Zbl 0361.57020
[Kl] Complétude des variétés lorentziennes à courbure constante, Math. Ann., Volume 306 (1996) no. 2, pp. 353-370 | MR 1411352 | Zbl 0862.53048
[Le] Feuilletages des variétés de dimension 3 qui sont des fibrés en cercles, Comment. Math. Helv., Volume 32 (1957-58), pp. 215-223 | MR 511848 | Zbl 0393.57004
[M-R] Relations de conjugaison et de cobordisme entre certains feuilletages, Publ. math. IHES, Volume 43 (1973), pp. 143-168 | Numdam | MR 358810 | Zbl 0356.57018
[Mo1] Dynamical properties of the space of Lorentzian metrics, Comment. Math. Helv., Volume 78 (2003), pp. 463-485 | Article | MR 1998389 | Zbl 1033.58012
[Mo2] Complétude et flots nul-géodésiques en géométrie lorentzienne, Bulletin de la SMF, Volume 132 (2004), pp. 463-475 | Numdam | MR 2081222 | Zbl 1066.53087
[Mol] Riemannian foliations, Progress in mathematics (1988) | MR 932463 | Zbl 0633.53001
[R-R] Reeb foliations, Annals of Math., Volume 91 (1970), pp. 1-24 | Article | MR 258057 | Zbl 0198.28402
[Sa] Variétés anti-de Sitter de dimension 3 exotiques, Ann. Inst. Fourier, Volume 50 (2000) no. 1, pp. 257-284 | Article | Numdam | MR 1762345 | Zbl 0951.53047
[Wo] Bundle with totally disconnected structure group, Comment. Math. Helv., Volume 46 (1971), pp. 257-273 | Article | MR 293655 | Zbl 0217.49202
[Yo] Examples of Lorentzian geodesible foliations of closed three manifolds having Heegard splitting of genus one, Tohoku Math. J., Volume 56 (2004), pp. 423-443 | Article | MR 2075776 | Zbl 1065.57029
[Ze1] Geodesic foliations in Lorentz 3-manifolds, Comment. Math. Helv., Volume 74 (1999), pp. 1-21 | Article | MR 1677118 | Zbl 0919.53011
[Ze2] Isometry group and geodesic foliations of Lorentz manifolds. Part II: geometry of analytic Lorentz manifold with large isometry groups, Geom. func. anal., Volume 9 (1999), pp. 823-854 | Article | MR 1719610 | Zbl 0946.53036