Deux remarques sur les flots Riemanniens. (Two remarks on Riemannian flows) (Q1070546)

Let M be a compact, connected, oriented Riemannian n-manifold and let \({\mathcal F}\) be a 1-dimensional oriented foliation such that the metric is bundle-like for \({\mathcal F}\). Theorem A. The following conditions are equivalent: (a) \({\mathcal F}\) generates an isometric flow. (b) The structure sheaf of \({\mathcal F}\) is globally trivial. (c) The basic cohomology is nonzero in dimension n-1. Theorem B. The flow of \({\mathcal F}\) has a section if and only if the natural map from the basic cohomology in dimension n-1 into the cohomology of M is nonzero. The Molino structure theory for Riemannian foliations is used to reduce Theorem A to the case where the flow is transversally parallelizable by infinitesimal automorphisms of \({\mathcal F}\). In this case, the symmetry makes most of the proof rather straightforward. The proof of Theorem B depends on the de Rham duality theorem for Riemannian foliations, which is proved in the necessary generality in a paper of \textit{V. Sergiescu} [Ann. Inst. Fourier 35, No.3, 137-158 (1985; Zbl 0563.57012)].