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Working in the geometric approach, we construct the lagrangians of N=1 andN=2 pure supergravity in four dimensions with negative cosmological constant,in the presence of a non trivial boundary of space-time. We find that thesupersymmetry invariance of the action requires the addition of topologicalterms which generalize at the supersymmetric level the Gauss-Bonnet term.Supersymmetry invariance is achieved without requiring Dirichlet boundaryconditions on the fields at the boundary, rather we find that the boundaryvalues of the fieldstrengths are dynamically fixed to constant values in termsof the cosmological constant \Lambda. From a group-theoretical point of viewthis means in particular the vanishing of the OSp(N|4)-supercurvatures at theboundary.

N=1 and N=2 pure supergravities on a manifold with boundary