Gradings appear in geometry and physics in a plethora of scenarios, such as vector bundles, higher-order tangent bundles or particles with spin. A natural and simple way to encode a grading on a manifold is by endowing it with a weight vector field, a generalisation of the Euler vector field on a vector bundle. If the grading is by non-negative integers and the weight vector field is complete, the latter generates a smooth action of the multiplicative monoid of real numbers on the manifold, known as a homogeneity structure. Using either the weight vector field or the homogeneity structure, one can easily extend several results to the graded setting with the tools and language of classical differential geometry, without resorting to intricate algebro-geometric artefacts.

In the first part of my talk, I will present Darboux theorems for homogeneous one-forms and presymplectic forms on graded (super)manifolds. In the second part, I will sketch how the Frobenius theorem for N-graded manifolds proven by Bursztyn, Cueca and Mehta can be recovered in a straightforward manner by applying some results on weight vector fields and replicating the proof of the standard Frobenius theorem. This is joint work with Prof. Janusz Grabowski.