  Following Geroch, Traschen, Mars and Senovilla, we consider Lorentzian
manifolds with distributional curvature tensor. Such manifolds represent
spacetimes of general relativity that possibly contain gravitational waves,
shock waves, and other singular patterns. We aim here at providing a
comprehensive and geometric (i.e., coordinate-free) framework. First, we
determine the minimal assumptions required on the metric tensor in order to
give a rigorous meaning to the spacetime curvature within the framework of
distribution theory. This leads us to a direct derivation of the jump relations
associated with singular parts of connection and curvature operators. Second,
we investigate the induced geometry on a hypersurface with general signature,
and we determine the minimal assumptions required to define, in the sense of
distributions, the curvature tensors and the second fundamental form of the
hypersurface and to establish the Gauss-Codazzi equations.
