In first-order formulation of pure gravity, we find a new class of solutions to the equations of motion represented by degenerate 4-geometries. These configurations are described by noninvertible tetrads with two zero eigenvalues and admit nonvanishing torsion. The homogeneous ones among these infinitely many degenerate solutions admit a geometric classification provided by the three fundamental geometries that a closed 2-surface can accomodate, namely, $E^2$, $S^2$, and $H^2$.

• Received 8 September 2016

DOI:https://doi.org/10.1103/PhysRevD.94.104047