On Einstein Equations on Manifolds and Supermanifolds

Abstract

The Einstein equations (EE) are certain conditions on the Riemann tensor on the real Minkowski space M . In the twistor picture, after complexification and compactification M becomes the Grassmannian Gr 4 2 of 2-dimensional subspaces in the 4-dimensional complex one. Here we answer for which of the classical domains considered as manifolds with G-structure it is possible to impose conditions similar in some sense to EE. The above investigation has its counterpart on superdomains: an analog of the Riemann tensor is defined for any supermanifold with G-structure with any Lie supergroup G. We also derive similar analogues of EE on supermanifolds. Our analogs of EE are not what physicists consider as SUGRA (supergravity), for SUGRA see