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hal-00371685, version 1

## Tautness for riemannian foliations on non-compact manifolds

M. Saralegi-Aranguren 1, J. I. Royo Prieto () 2, R. Wolak 3

manuscripta mathematica 126 (2008) 177-200

Abstract: For a riemannian foliation $\mathcal{F}$ on a closed manifold $M$, it is known that $\mathcal{F}$ is taut (i.e. the leaves are minimal submanifolds) if and only if the (tautness) class defined by the mean curvature form $\kappa_\mu$ (relatively to a suitable riemannian metric $\mu$) is zero. In the transversally orientable case, tautness is equivalent to the non-vanishing of the top basic cohomology group $H^{^{n}}(M/\mathcal{F})$, where $n = \codim \mathcal{F}$. By the Poincaré Duality, this last condition is equivalent to the non-vanishing of the basic twisted cohomology group $H^{^{0}}_{_{\kappa_\mu}}(M/\mathcal{F})$, when $M$ is oriented. When $M$ is not compact, the tautness class is not even defined in general. In this work, we recover the previous study and results for a particular case of riemannian foliations on non compact manifolds: the regular part of a singular riemannian foliation on a compact manifold (CERF).

• 1:  Laboratoire de Mathématiques de Lens (LML)
• Université d'Artois : EA2462
• 2:  Departamento de Matemática Aplicada
• Universidad del Pais Vasco- Euskal Herriko Unibertsitatea
• 3:  Instytut Matematyki
• Uniwersytet Jagiellonski
• Domain : Mathematics/Differential Geometry
Mathematics/Algebraic Topology
• Keywords : Feuilletages
• Comment : 18 pages

• hal-00371685, version 1
• oai:hal.archives-ouvertes.fr:hal-00371685
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• Submitted on: Monday, 30 March 2009 11:35:59
• Updated on: Thursday, 3 December 2009 16:54:29