Sadeghian, S (reprint author), Inst Res Fundamental Sci IPM, Sch Phys, POB 19395-5531, Tehran, Iran.
; Sheikh-Jabbari, M. M.] Inst Res Fundamental Sci IPM, Sch Phys, Tehran, Iran
; [Sadeghian, S.] Alzahra Univ, Dept Phys, Tehran 91167, Iran
; [Yavartanoo, H.] Chinese Acad Sci, Inst Theoret Phys, State Key Lab Theoret Phys, Beijing 100190, Peoples R China
Motivated by the Extremal Vanishing Horzon (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-A 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 AdS(3) with an interval or a circle. In five dimensions the geometries are of the form of warped product of AdS(3) and a 2d surface Sigma(2). For the Einsten-Maxwell-Lambda theory we prove that 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) x U(1)(d-4) isometry.