Tenseness of Riemannian flows

Nozawa, Hiraku; Royo Prieto, José Ignacio

Annales de l'Institut Fourier, Tome 64 (2014) no. 4, pp. 1419-1439.

On montre que tout feuilletage riemannien $ℱ$ 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 $ℱ$ 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.

We show that any transversally complete Riemannian foliation $ℱ$ 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 $ℱ$ 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.

Classification : 53C12, 57R30, 37C10
Keywords: Riemannian foliation, taut foliation, mean curvature, basic cohomology
Mot clés : feuilletage riemannien, feuilletage tendu, courbure moyenne, cohomologie basiqueidéal multiplicateur, métrique à singularités minimales

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