Engel structures with trivial characteristic foliations.

Adachi, Jiro

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In this short note we find some conditions which ensure that a G foliation of finite type with all leaves compact is a Riemannian foliation of equivalently the space of leaves of such a foliation is a Satake manifold. A particular attention is paid to transversaly affine foliations. We present several conditions which ensure completeness of such foliations.

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Compact, $C^{2}$ -foliated manifolds of codimension one, having all leaves proper, are shown to be $C^{\infty}$ -smoothable. More precisely, such a foliated manifold is homeomorphic to one of class $C^{\infty}$ . The corresponding statement is false for foliations with nonproper leaves. In that case, there are topological distinctions between smoothness of class $C^{r}$ and of class $C^{r + 1}$ for every nonnegative integer $r$ .

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