supermanifold M is contained in its sheaf OM of "smooth functions". In the dual point of view, an injective map corresponds to a surjection of sheaves, and a surjective map corresponds to an injection of sheaves. An alternative approach to the dual point of view is to use the functor of points. If M is a supermanifold of dimension (p,q), then the underlying space M inherits the structure of a differentiable manifold whose sheaf of smooth functions is OM/I, where I is the ideal generated by all odd functions. Thus M is called the underlying space, or the body, of M. The quotient map OM → OM/I corresponds to an injective map M → M; thus M is a submanifold of M. Examples Let M be a manifold. The odd tangent bundle ΠTM is a supermanifold given by the sheaf Ω(M) of differential forms on M. More generally, let E → M be a vector bundle. Then ΠE is a supermanifold given by the sheaf Γ(ΛE*). In fact, Π is a functor from the category of vector bundles to the category of supermanifolds. Lie supergroups are examples of supermanifolds. Batchelor's theorem Batchelor's theorem states that every supermanifold is noncanonically isomorphic to a supermanifold of the form ΠE. The word "noncanonically" prevents one from concluding that supermanifolds are simply glorified vector bundles; although the functor Π maps surjectively onto the isomorphism classes of supermanifolds, it is not an equivalence of categories. It was published by Marjorie Batchelor in 1979. The proof of Batchelor's theorem relies in an essential way on the existence of a partition of unity, so it does not hold for complex or real-analytic supermanifolds. Odd symplectic structures Odd symplectic form In many physical and geometric applications, a supermanifold comes equipped with an Grassmann-odd symplectic structure. All natural geometric objects on a supermanifold are graded. In particular, the bundle of two-forms is equipped with a grading. An odd symplectic form ω on a supermanifold is a closed, odd form, inducing a non-degenerate pairing on TM. Such a supermanifold is called a P-manifold. Its graded dimension is necessarily (n,n), because the odd symplectic form induces a pairing of odd and even variables. There is a version of the Darboux theorem for P-manifolds, which allows one to equip a P-manifold locally with a set of coordinates where the odd symplectic form ω is written as where are even coordinates, and odd