Graded supermanifolds and homogeneity

We introduce the concept of a homogeneity supermanifold which is, roughly speaking, a supermanifold equipped with a privileged atlas whose coordinates have certain (real) homogeneity degrees. This defines a sheaf of graded algebras on the supermanifold, which is an additional structure on the supermanifold. The main principle of this approach is that grading is ultimately related to homogeneity. But assigning consistently homogeneity degrees to coordinates is the same as fixing a certain global vector field, the weight vector field. This approach is simple and more general than most of the approaches to graded manifolds present in the literature. In particular, the homogeneity degrees can be arbitrary reals, and the corresponding category includes compact supermanifolds. We systematically study homogeneity submanifolds, homogeneity Lie supergroups, tangent and cotangent lifts of homogeneity structures, homogeneous distributions and codistributions, and other related concepts like double homogeneity. The main achievements in this framework are proofs of the homogeneous Poincar´e Lemma and a homogeneous analog of the Darboux Theorem, which seem to be interesting even for purely even homogeneity manifolds.