A foliated metric rigidity theorem for higher rank irreducible symmetric spaces

Let F be a foliation of a compact manifold with a transverse invariant measure of finite total mass. We prove that if F admits a leafwise metric such that every leaf is an irreducible symmetric space of noncompact type and higher rank, then any other leafwise metric of nonpositive curvature is also symmetric along any leaf in the support of the transverse measure. A rank one version of this result is also exposed.