  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.
