Tenseness of Riemannian flows
Annales de l'Institut Fourier, Volume 64 (2014) no. 4, p. 1419-1439

We show that any transversally complete Riemannian foliation of dimension one on any possibly non-compact manifold M is tense; namely, M 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 M, éventuellement non compacte, est étiré ; c’est à dire, il existe une métrique riemanniene sur M 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.

DOI : https://doi.org/10.5802/aif.2885
Classification:  53C12,  57R30,  37C10
Keywords: Riemannian foliation, taut foliation, mean curvature, basic cohomology
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {64},
     number = {4},
     year = {2014},
     pages = {1419-1439},
     doi = {10.5802/aif.2885},
     mrnumber = {3329668},
     zbl = {06387312},
     language = {en},
     url = {http://www.numdam.org/item/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://www.numdam.org/item/AIF_2014__64_4_1419_0/

[1] Álvarez López, Jesús A. A finiteness theorem for the spectral sequence of a Riemannian foliation, Illinois J. Math., Tome 33 (1989) no. 1, pp. 79-92 | MR 974012 | Zbl 0644.57014

[2] Álvarez López, Jesús A. The basic component of the mean curvature of Riemannian foliations, Ann. Global Anal. Geom., Tome 10 (1992) no. 2, pp. 179-194 | Article | MR 1175918 | Zbl 0759.57017

[3] Álvarez López, Jesús A. On the first secondary invariant of Molino’s central sheaf, Ann. Polon. Math., Tome 64 (1996) no. 3, pp. 253-265 | MR 1410344 | Zbl 0863.53021

[4] Cairns, Grant; Escobales, Richard H. Jr. 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, Tome 62 (1997) no. 1, pp. 46-63 | Article | MR 1427628 | Zbl 0884.57023

[5] Candel, Alberto; Conlon, Lawrence Foliations. I, American Mathematical Society, Providence, RI, Graduate Studies in Mathematics, Tome 23 (2000), pp. xiv+402 | MR 1732868 | Zbl 0936.57001

[6] Caron, Patrick; Carrière, Yves Flots transversalement de Lie R n , flots transversalement de Lie minimaux, C. R. Acad. Sci. Paris Sér. A-B, Tome 291 (1980) no. 7, p. A477-A478 | MR 595919 | Zbl 0453.57019

[7] Carrière, Yves Flots riemanniens, Astérisque (1984) no. 116, pp. 31-52 (Transversal structure of foliations (Toulouse, 1982)) | MR 755161 | Zbl 0548.58033

[8] Domínguez, Demetrio Finiteness and tenseness theorems for Riemannian foliations, Amer. J. Math., Tome 120 (1998) no. 6, pp. 1237-1276 | Article | MR 1657170 | Zbl 0964.53019

[9] El Kacimi-Alaoui, A.; Hector, G. Décomposition de Hodge basique pour un feuilletage riemannien, Ann. Inst. Fourier (Grenoble), Tome 36 (1986) no. 3, pp. 207-227 | Article | Numdam | MR 865667 | Zbl 0586.57015

[10] Ghys, Étienne Classification des feuilletages totalement géodésiques de codimension un, Comment. Math. Helv., Tome 58 (1983) no. 4, pp. 543-572 | Article | MR 728452 | Zbl 0534.57015

[11] Ghys, Étienne Feuilletages riemanniens sur les variétés simplement connexes, Ann. Inst. Fourier (Grenoble), Tome 34 (1984) no. 4, pp. 203-223 | Article | Numdam | MR 766280 | Zbl 0525.57024

[12] Haefliger, André Some remarks on foliations with minimal leaves, J. Differential Geom., Tome 15 (1980) no. 2, pp. 269-284 | MR 614370 | Zbl 0444.57016

[13] Haefliger, André Pseudogroups of local isometries, Differential geometry (Santiago de Compostela, 1984), Pitman, Boston, MA (Res. Notes in Math.) Tome 131 (1985), pp. 174-197 | MR 864868 | Zbl 0656.58042

[14] Haefliger, André Leaf closures in Riemannian foliations, A fête of topology, Academic Press, Boston, MA (1988), pp. 3-32 | MR 928394 | Zbl 0667.57012

[15] Kamber, Franz W.; Tondeur, Philippe Foliated bundles and characteristic classes, Springer-Verlag, Berlin-New York, Lecture Notes in Mathematics, Vol. 493 (1975), pp. xiv+208 | MR 402773 | Zbl 0308.57011

[16] Kamber, Franz W.; Tondeur, Philippe Duality for Riemannian foliations, Singularities, Part 1 (Arcata, Calif., 1981), Amer. Math. Soc., Providence, RI (Proc. Sympos. Pure Math.) Tome 40 (1983), pp. 609-618 | MR 713097 | Zbl 0523.57019

[17] Kamber, Franz W.; Tondeur, Philippe Foliations and metrics, Differential geometry (College Park, Md., 1981/1982), Birkhäuser Boston, Boston, MA (Progr. Math.) Tome 32 (1983), pp. 103-152 | MR 702530 | Zbl 0542.53022

[18] Kamber, Franz W.; Tondeur, Philippe Duality theorems for foliations, Astérisque (1984) no. 116, pp. 108-116 (Transversal structure of foliations (Toulouse, 1982)) | MR 755165 | Zbl 0559.58022

[19] Masa, Xosé Duality and minimality in Riemannian foliations, Comment. Math. Helv., Tome 67 (1992) no. 1, pp. 17-27 | Article | MR 1144611 | Zbl 0778.53029

[20] Moerdijk, I.; Mrčun, J. Introduction to foliations and Lie groupoids, Cambridge University Press, Cambridge, Cambridge Studies in Advanced Mathematics, Tome 91 (2003), pp. x+173 | MR 2012261 | Zbl 1029.58012

[21] Molino, Pierre Feuilletages de Lie à feuilles denses (1982-1983) (Séminaire de Géométrie Différentielle, Montpellier)

[22] Molino, Pierre Riemannian foliations, Birkhäuser Boston, Inc., Boston, MA, Progress in Mathematics, Tome 73 (1988) (Translated from the French by Grant Cairns, With appendices by Cairns, Y. Carrière, É. Ghys, E. Salem and V. Sergiescu) | MR 932463 | Zbl 0633.53001

[23] Molino, Pierre; Sergiescu, Vlad Deux remarques sur les flots riemanniens, Manuscripta Math., Tome 51 (1985) no. 1-3, pp. 145-161 | Article | MR 788676 | Zbl 0585.53026

[24] Nozawa, Hiraku Rigidity of the Álvarez class, Manuscripta Math., Tome 132 (2010) no. 1-2, pp. 257-271 | Article | MR 2609297 | Zbl 1193.53086

[25] Nozawa, Hiraku Haefliger cohomology of Riemannian foliations (2012) (arXiv:1209.3817, preprint)

[26] Reinhart, Bruce L. Foliated manifolds with bundle-like metrics, Ann. of Math. (2), Tome 69 (1959), pp. 119-132 | Article | MR 107279 | Zbl 0122.16604

[27] Royo Prieto, José Ignacio The Euler class for Riemannian flows, C. R. Acad. Sci. Paris Sér. I Math., Tome 332 (2001) no. 1, pp. 45-50 | Article | MR 1805626 | Zbl 0987.53009

[28] Royo Prieto, José Ignacio; Saralegi-Aranguren, Martintxo; Wolak, Robert Tautness for Riemannian foliations on non-compact manifolds, Manuscripta Math., Tome 126 (2008) no. 2, pp. 177-200 | Article | MR 2403185 | Zbl 1155.57026

[29] Royo Prieto, José Ignacio; Saralegi-Aranguren, Martintxo; Wolak, Robert Cohomological tautness for Riemannian foliations, Russ. J. Math. Phys., Tome 16 (2009) no. 3, pp. 450-466 | Article | MR 2551892 | Zbl 1178.57021

[30] Rummler, Hansklaus Quelques notions simples en géométrie riemannienne et leurs applications aux feuilletages compacts, Comment. Math. Helv., Tome 54 (1979) no. 2, pp. 224-239 | Article | MR 535057 | Zbl 0409.57026

[31] Sarkaria, K. S. A finiteness theorem for foliated manifolds, J. Math. Soc. Japan, Tome 30 (1978) no. 4, pp. 687-696 | Article | MR 513077 | Zbl 0398.57012

[32] Sergiescu, Vlad Cohomologie basique et dualité des feuilletages riemanniens, Ann. Inst. Fourier (Grenoble), Tome 35 (1985) no. 3, pp. 137-158 | Article | Numdam | MR 810671 | Zbl 0563.57012

[33] Sullivan, Dennis A homological characterization of foliations consisting of minimal surfaces, Comment. Math. Helv., Tome 54 (1979) no. 2, pp. 218-223 | Article | MR 535056 | Zbl 0409.57025

[34] Tondeur, Philippe A characterization of Riemannian flows, Proc. Amer. Math. Soc., Tome 125 (1997) no. 11, pp. 3403-3405 | Article | MR 1415373 | Zbl 0899.53023

[35] Žukova, N. On the stability of leaves of Riemannian foliations, Ann. Global Anal. Geom., Tome 5 (1987) no. 3, pp. 261-271 | Article | MR 962299 | Zbl 0658.53026