  The double tetrahedron is the triangulation of the three-sphere gotten by
gluing together two congruent tetrahedra along their boundaries. As a piecewise
flat manifold, its geometry is determined by its six edge lengths, giving a
notion of a metric on the double tetrahedron. We study notions of Einstein
metrics, constant scalar curvature metrics, and the Yamabe problem on the
double tetrahedron, with some reference to the possibilities on a general
piecewise flat manifold. The main tool is analysis of Regge's Einstein-Hilbert
functional, a piecewise flat analogue of the Einstein-Hilbert (or total scalar
curvature) functional on Riemannian manifolds. We study the
Einstein-Hilbert-Regge functional on the space of metrics and on discrete
conformal classes of metrics.
