Abstract
We give a characterization of graded symplectic forms by studying the module of derivations of a graded sheaf. When the graded sheaf is the sheaf of differentiable forms on the underlying manifold M, we find canonical liftings from metrics on TM to odd symplectic forms, and from symplectic forms on M and metrics on TM to even symplectic forms. These graded symplectic forms give rise to canonical Poisson brackets on the graded manifold.
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