The Riemannian geometry of the space of compact spacelike Cauchy hypersurfaces

We study the geometry of a weak Riemannian metric on the infinite dimensional manifold of compact spacelike Cauchy hypersurfaces in a globally hyperbolic spacetime. We show that the geodesic distance (i.e. the infimum of lengths of paths between two given points) is positive, and that the sectional curvature is well defined and non-positive.