The commuting sheaf \({\mathcal C}\) of a G-foliation is defined and used to study \(\nabla\)-G-foliations. (A foliation \({\mathcal F}\) is a \(\nabla\)-G- foliation when it admits a transversal N with a holonomy invariant G- structure and a G-connection \(\nabla\) for which holonomy maps occur to be local affine transformations.) The main result says that if \({\mathcal C}\) is of compact type and \({\mathcal F}\) is a transversely complete \(\nabla\)-G- foliation, then \({\mathcal F}\) is Riemannian.