On Classification of Geometries with SO(2,2) Symmetry

Motivated by the Extremal Vanishing Horizon (EVH) black holes, their near horizon geometry and the EVH/CFT proposal, we construct and classify solutions with (local) SO(2,2) symmetry to four and five dimensional Einstein-Maxwell-Dilaton (EMD) theory with positive, zero or negative cosmological constant Lambda, the EMD-$\mathrm{\Lambda }$ theory, and also $U(1{)}^4$ gauged supergravity in four dimensions and $U(1{)}^3$ gauged supergravity in five dimensions. In four dimensions the geometries are warped product of AdS3 with an interval or a circle. In five dimensions the geometries are of the form of warped product of AdS3 and a 2d surface ${\mathrm{\Sigma }}_2$. For the Einsten-Maxwell-$\mathrm{\Lambda }$ theory we prove that ${\mathrm{\Sigma }}_2$ should have a U(1) isometry, a rigidity theorem in this class of solutions. We also construct all d dimensional Einstein vacuum solutions with $SO(2,2)\times U(1{)}^{d-4}$ or $SO(2,2)\times SO(d-3)$ isometry.