We show that any transversally complete Riemannian foliation of dimension one on any possibly non-compact manifold is tense; namely, admits a Riemannian metric such that the mean curvature form of is basic. This is a partial generalization of a result of Domínguez, which says that any Riemannian foliation on any compact manifold is tense. Our proof is based on some results of Molino and Sergiescu, and it is simpler than the original proof by Domínguez. As an application, we generalize some well known results including Masa’s characterization of tautness.
On montre que tout feuilletage riemannien de dimension un transversalement complet sur une variété , éventuellement non compacte, est étiré ; c’est à dire, il existe une métrique riemanniene sur pour laquelle la forme de courbure moyenne de est basique. Ceci est une généralisation partielle d’un résultat de Domínguez, qui dit que tout feuilletage riemannien sur une variété compacte est étiré. La preuve s’appuie sur certains résultats de Molino et Sergiescu, et elle est plus simple que la première démonstration de Domínguez. Comme application, on généralise certains résultats bien connus, comme la caractérisation des feuilletages tendus par Masa.
Keywords: Riemannian foliation, taut foliation, mean curvature, basic cohomology
Mot clés : feuilletage riemannien, feuilletage tendu, courbure moyenne, cohomologie basiqueidéal multiplicateur, métrique à singularités minimales
@article{AIF_2014__64_4_1419_0, author = {Nozawa, Hiraku and Royo Prieto, Jos\'e Ignacio}, title = {Tenseness of {Riemannian} flows}, journal = {Annales de l'Institut Fourier}, pages = {1419--1439}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {64}, number = {4}, year = {2014}, doi = {10.5802/aif.2885}, zbl = {06387312}, mrnumber = {3329668}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.2885/} }
TY - JOUR AU - Nozawa, Hiraku AU - Royo Prieto, José Ignacio TI - Tenseness of Riemannian flows JO - Annales de l'Institut Fourier PY - 2014 SP - 1419 EP - 1439 VL - 64 IS - 4 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2885/ DO - 10.5802/aif.2885 LA - en ID - AIF_2014__64_4_1419_0 ER -
%0 Journal Article %A Nozawa, Hiraku %A Royo Prieto, José Ignacio %T Tenseness of Riemannian flows %J Annales de l'Institut Fourier %D 2014 %P 1419-1439 %V 64 %N 4 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.2885/ %R 10.5802/aif.2885 %G en %F AIF_2014__64_4_1419_0
Nozawa, Hiraku; Royo Prieto, José Ignacio. Tenseness of Riemannian flows. Annales de l'Institut Fourier, Volume 64 (2014) no. 4, pp. 1419-1439. doi : 10.5802/aif.2885. http://archive.numdam.org/articles/10.5802/aif.2885/
[1] A finiteness theorem for the spectral sequence of a Riemannian foliation, Illinois J. Math., Volume 33 (1989) no. 1, pp. 79-92 | MR | Zbl
[2] The basic component of the mean curvature of Riemannian foliations, Ann. Global Anal. Geom., Volume 10 (1992) no. 2, pp. 179-194 | DOI | MR | Zbl
[3] On the first secondary invariant of Molino’s central sheaf, Ann. Polon. Math., Volume 64 (1996) no. 3, pp. 253-265 | MR | Zbl
[4] Further geometry of the mean curvature one-form and the normal plane field one-form on a foliated Riemannian manifold, J. Austral. Math. Soc. Ser. A, Volume 62 (1997) no. 1, pp. 46-63 | DOI | MR | Zbl
[5] Foliations. I, Graduate Studies in Mathematics, 23, American Mathematical Society, Providence, RI, 2000, pp. xiv+402 | MR | Zbl
[6] Flots transversalement de Lie , flots transversalement de Lie minimaux, C. R. Acad. Sci. Paris Sér. A-B, Volume 291 (1980) no. 7, p. A477-A478 | MR | Zbl
[7] Flots riemanniens, Astérisque (1984) no. 116, pp. 31-52 Transversal structure of foliations (Toulouse, 1982) | Numdam | MR | Zbl
[8] Finiteness and tenseness theorems for Riemannian foliations, Amer. J. Math., Volume 120 (1998) no. 6, pp. 1237-1276 | DOI | MR | Zbl
[9] Décomposition de Hodge basique pour un feuilletage riemannien, Ann. Inst. Fourier (Grenoble), Volume 36 (1986) no. 3, pp. 207-227 | DOI | Numdam | MR | Zbl
[10] Classification des feuilletages totalement géodésiques de codimension un, Comment. Math. Helv., Volume 58 (1983) no. 4, pp. 543-572 | DOI | MR | Zbl
[11] Feuilletages riemanniens sur les variétés simplement connexes, Ann. Inst. Fourier (Grenoble), Volume 34 (1984) no. 4, pp. 203-223 | DOI | Numdam | MR | Zbl
[12] Some remarks on foliations with minimal leaves, J. Differential Geom., Volume 15 (1980) no. 2, pp. 269-284 | MR | Zbl
[13] Pseudogroups of local isometries, Differential geometry (Santiago de Compostela, 1984) (Res. Notes in Math.), Volume 131, Pitman, Boston, MA, 1985, pp. 174-197 | MR | Zbl
[14] Leaf closures in Riemannian foliations, A fête of topology, Academic Press, Boston, MA, 1988, pp. 3-32 | MR | Zbl
[15] Foliated bundles and characteristic classes, Lecture Notes in Mathematics, Vol. 493, Springer-Verlag, Berlin-New York, 1975, pp. xiv+208 | MR | Zbl
[16] Duality for Riemannian foliations, Singularities, Part 1 (Arcata, Calif., 1981) (Proc. Sympos. Pure Math.), Volume 40, Amer. Math. Soc., Providence, RI, 1983, pp. 609-618 | MR | Zbl
[17] Foliations and metrics, Differential geometry (College Park, Md., 1981/1982) (Progr. Math.), Volume 32, Birkhäuser Boston, Boston, MA, 1983, pp. 103-152 | MR | Zbl
[18] Duality theorems for foliations, Astérisque (1984) no. 116, pp. 108-116 Transversal structure of foliations (Toulouse, 1982) | Numdam | MR | Zbl
[19] Duality and minimality in Riemannian foliations, Comment. Math. Helv., Volume 67 (1992) no. 1, pp. 17-27 | DOI | MR | Zbl
[20] Introduction to foliations and Lie groupoids, Cambridge Studies in Advanced Mathematics, 91, Cambridge University Press, Cambridge, 2003, pp. x+173 | MR | Zbl
[21] Feuilletages de Lie à feuilles denses (1982-1983) (Séminaire de Géométrie Différentielle, Montpellier)
[22] Riemannian foliations, Progress in Mathematics, 73, Birkhäuser Boston, Inc., Boston, MA, 1988 (Translated from the French by Grant Cairns, With appendices by Cairns, Y. Carrière, É. Ghys, E. Salem and V. Sergiescu) | MR | Zbl
[23] Deux remarques sur les flots riemanniens, Manuscripta Math., Volume 51 (1985) no. 1-3, pp. 145-161 | DOI | MR | Zbl
[24] Rigidity of the Álvarez class, Manuscripta Math., Volume 132 (2010) no. 1-2, pp. 257-271 | DOI | MR | Zbl
[25] Haefliger cohomology of Riemannian foliations (2012) (arXiv:1209.3817, preprint)
[26] Foliated manifolds with bundle-like metrics, Ann. of Math. (2), Volume 69 (1959), pp. 119-132 | DOI | MR | Zbl
[27] The Euler class for Riemannian flows, C. R. Acad. Sci. Paris Sér. I Math., Volume 332 (2001) no. 1, pp. 45-50 | DOI | MR | Zbl
[28] Tautness for Riemannian foliations on non-compact manifolds, Manuscripta Math., Volume 126 (2008) no. 2, pp. 177-200 | DOI | MR | Zbl
[29] Cohomological tautness for Riemannian foliations, Russ. J. Math. Phys., Volume 16 (2009) no. 3, pp. 450-466 | DOI | MR | Zbl
[30] Quelques notions simples en géométrie riemannienne et leurs applications aux feuilletages compacts, Comment. Math. Helv., Volume 54 (1979) no. 2, pp. 224-239 | DOI | MR | Zbl
[31] A finiteness theorem for foliated manifolds, J. Math. Soc. Japan, Volume 30 (1978) no. 4, pp. 687-696 | DOI | MR | Zbl
[32] Cohomologie basique et dualité des feuilletages riemanniens, Ann. Inst. Fourier (Grenoble), Volume 35 (1985) no. 3, pp. 137-158 | DOI | Numdam | MR | Zbl
[33] A homological characterization of foliations consisting of minimal surfaces, Comment. Math. Helv., Volume 54 (1979) no. 2, pp. 218-223 | DOI | MR | Zbl
[34] A characterization of Riemannian flows, Proc. Amer. Math. Soc., Volume 125 (1997) no. 11, pp. 3403-3405 | DOI | MR | Zbl
[35] On the stability of leaves of Riemannian foliations, Ann. Global Anal. Geom., Volume 5 (1987) no. 3, pp. 261-271 | DOI | MR | Zbl
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