GR with Riemannian (++++) metric? Neither the Einstein field equations nor the Einstein-Hilbert action requires pseudo-remannian metrc. Is there anything interesting about GR with Riemannian metric? Are there still black holes? Also, is it possible to find a solution to GR in which the metric is Riemannian in half of the manifold and pseudo-Riemannian in the other half? Joined up like how a black hole and a white hole are joined in Kruskal metric?