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Classification des feuilletages totalement géodésiques de codimension un. (French) Zbl 0534.57015
The aim of this paper is to construct the totally geodesic differentiable foliations of codimension 1. The author constructs some model foliations \((M_ D,F_ D)\), which are torus bundles, and he proves that if m is a compact orientable manifold, and F is a \(C^{\infty}\) foliation of codimension 1 and transversally orientable on M, then M has a Riemannian metric such that the leaves of F are totally geodesic hypersurfaces, iff either F is transversal to a locally free action of \(S^ 1\) on M or M is diffeomorphic to \(M_ D\) and F is conjugate to the corresponding model \(F_ D\). Similarly, if M is noncompact but either F is \(C^{\infty}\) and M has a fundamental group of finite type or F is \(C^{\omega}\), one has a complete metric making F totally geodesic in the same cases as for compact M, and in the case where F is transversal to a trivial fibration of M with fibre \({\mathbb{R}}\), and whose restriction to every leaf is a covering map.
Reviewer: I.Vaisman

MSC:
57R30 Foliations in differential topology; geometric theory
53C12 Foliations (differential geometric aspects)
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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