  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.
