  Generalizations of symplectic and metric structures for supermanifolds are
analyzed. Two types of structures are possible according to the even/odd
character of the corresponding quadratic tensors. In the even case one has a
very rich set of geometric structures: even symplectic supermanifolds (or,
equivalently, supermanifolds with non-degenerate Poisson structures), even
Fedosov supermanifolds and even Riemannian supermanifolds. The existence of
relations among those structures is analyzed in some details. In the odd case,
we show that odd Riemannian and Fedosov supermanifolds are characterized by a
scalar curvature tensor. However, odd Riemannian supermanifolds can only have
constant curvature.
