Regular projectively Anosov flows with compact leaves

Noda, Takeo

Annales de l'Institut Fourier, Volume 54 (2004) no. 2, pp. 481-497.

This paper concerns projectively Anosov flows ${\phi }^t$ with smooth stable and unstable foliations ${ℱ}^s$ and ${ℱ}^u$ on a Seifert manifold $M$. We show that if the foliation ${ℱ}^s$ or ${ℱ}^u$ contains a compact leaf, then the flow ${\phi }^t$ is decomposed into a finite union of models which are defined on $T^2\times I$ and bounded by compact leaves, and therefore the manifold $M$ is homeomorphic to the 3-torus. In the proof, we also obtain a theorem which classifies codimension one foliations on Seifert manifolds with compact leaves which are incompressible tori.

Cet article concerne les flots projectivement Anosov, dont les feuilletages stable et instable ${ℱ}^s$ et ${ℱ}^u$ sont lisses, sur une variété de Seifert $M$. Nous prouvons que si l’un des feuilletages ${ℱ}^s$ ou ${ℱ}^u$ contient une feuille compacte, alors le flot ${\phi }^t$ se décompose en union finie de modèles définis sur $T^2\times I$ et ayant pour bord les feuilles compactes. La variété $M$ est donc homeomorphe au tore $T^3$. Dans la preuve, nous obtenons également un théorème qui classifie les feuilletages de codimension un sur les variétés de Seifert ayant des feuilles compactes qui sont des tores incompressibles.

Classification: 57R30, 37D30, 53C12, 53C15
Keywords: projectively Anosov flows, stable foliations, bi-contact structures
Mot clés : flots projectivement Anosov, feuilletages stables, structures de bi-contact

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