ABSTRACT
This chapter focuses on finding a supergravity action such that the corresponding dynamical equations reduce to ordinary Einstein equations after switching off all the fields, except the vierbein, of the Einstein supergravity multiplet. It explores that the full covariance group of the commutator algebra also includes, on a level with the superspace general coordinate and superlocal Lorentz transformations, the super Weyl transformations. Clearly, the Einstein supergravity action is not super Weyl invariant. However, it is not difficult to find a supergravity action possessing such an invariance. The chapter considers linearized conformal supergravity. The action describing linearized conformal supergravity is given in terms of the linearized super Weyl tensor. Einstein supergravity action s suitable only for describing empty supersymmetric universes. The massless vector multiplet model is uniquely continued to a curved superspace when taking the superfield variable to be a real scalar superfield and requiring the invariance of the curved-superspace action under the gauge transformations.
