Nonsingular solutions to the Einstein equations on piecewise-Lorentzian manifolds

We consider 4-dimensional spacetime manifolds that are piecewise Lorentzian, where the Lorentzian components of the manifold are separated by codimension-one planes (spacelike or timelike) on which the metric is degenerate. Such manifolds are of interest because they enlarge the smooth and nonsingular solution space of the Einstein equations. Planes of degeneracy that are perpendicular to each other can exist simultaneously. We describe various solutions of this type to the vacuum equations $G_{\mu\nu}=0$ and $G_{\mu\nu}+\Lambda g_{\mu\nu}=0$, and to $G_{\mu\nu}= 8\pi G T_{\mu\nu}$ for a perfect fluid. Novel examples include static gravitational lumps of finite curvature and a spacetime that responds to a cosmological constant via oscillations in time and/or space. A spacelike degeneracy plane can be used to avoid the big bang singularity, as we have further described elsewhere.