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

The Riemannian geometry of the space of compact spacelike Cauchy hypersurfaces