Regular projectively Anosov flows with compact leaves

Noda, Takeo

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

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.

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.

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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