On transversal infinitesimal automorphisms for harmonic foliations (Q1094680)

Let F be a harmonic Riemannian foliation on a compact oriented Riemannian manifold M. The authors distinguish as \(\sigma\)-automorphisms those infinitesimal automorphisms of \({\mathcal F}\) the dual \(\omega\) of which verifies: \(\Delta_ B\omega -2\rho_{\nabla}(\omega)+\sigma d_ B\delta_ B\omega =0\), for some real \(\sigma\) ; it is proved that if \({\mathcal F}\) is stable and transversally oriented then every admissible \(\sigma\) is nonpositive. More restrictive bounds for \(\sigma\) are found under the assumption that either the transversal scalar curvature is constant or the foliation is (transversally) Einstein.