Topology and bottom spectrum of transversally negatively curved foliations

We show that for any Riemannian foliation with a simply connected and negatively curved leaf space the normal exponential map of a leaf is a diffeomorphism. As an application, if the leaves are furthermore minimal submanifolds, we give a sharp estimate for the bottom of the spectrum of such a Riemannian manifold. Our proof of the spectral estimate also yields an estimate for the bottom of the spectrum of the horizontal Laplacian.