Even and odd geometries on supermanifolds

We analyze from a general perspective all possible supersymmetric generalizations of symplectic and metric structures on smooth manifolds. There are two different types of structures according to the even/odd character of the corresponding quadratic tensors. In general we can have even/odd symplectic supermanifolds, Fedosov supermanifolds and Riemannian supermanifolds. The geometry of even Fedosov supermanifolds is strongly constrained and has to be flat. In the odd case, the scalar curvature is only constrained by Bianchi identities. However, we show that odd Riemannian supermanifolds can only have constant scalar curvature. We also point out that the supersymmetric generalizations of AdS space do not exist in the odd case.