Feuilletages de Killing. (Killing foliations)
The main result of the paper is the following: Let \((M,F,g_ T)\) be a Killing foliation of codimension q in a closed connected manifold of dimension \(n=p+q\). If the minimal dimension of the closure of the leaves is \(p+r\), then: (i) There are r global transverse fields \(X_ 1,X_ 2,...,X_ r\) which commute and are linearly independent at each point. (ii) The orbits of the Lie algebra of the transverse fields spanned by \(X_ 1,...,X_ r\) give a new Riemannian foliation F' of codimension q-r where the closure of the leaves is the same as in F. Moreover, F' has at least one compact leaf. Here \(g_ T\) is a transverse metric defining a Riemannian foliation F. A Killing foliation is defined to be a Riemannian foliation with a trivial central transversal sheaf. An interesting consequence of this theorem is: If the span of M is k (maximal number of linearly independent vector fields) then every Killing foliation of codimension q in M admits closures of the leaves of codimension at least q-r. If \(k=0\) (or \(\chi\) (M)\(\neq 0)\) the foliation has a compact leaf, and this is a result proved by \textit{E. Ghys} [Ann. Inst. Fourier 34, No.4, 203-223 (1984; Zbl 0525.57024)].
scientific article; zbMATH DE number 3994458