Given a foliation F on a smooth manifold M one may ask if any foliation G close to F in some suitable topology is actually conjugate to F with respect to some diffeomorphism on M. This property is called stability. Infinitesimally, this problem is controlled by H1(F,NF), the first foliated cohomology with coefficients in the normal bundle. It is natural to ask whether infinitesimal stability (H1(F,NF)=0) implies stability for F. This question was first answered to the positive by Hamilton for Hausdorff foliations on closed manifolds. Around the same time Epstein & Rosenberg gave a postive answer for compact Hausdorff Cr -foliations on manifolds without boundary. In this talk we extend Hamiltons result and give a positive answer for Riemannian foliations on closed manifolds. This is based on joint work with Stephane Geudens.
