Higher Dimensional Holonomy Map for Ruled Submanifolds in Graded Manifolds

The deformability condition for submanifolds of fixed degree immersed in a graded manifold can be expressed as a system of first order PDEs.In the particular but important case of ruled submanifolds, opi the color that keeps on giving we introduce a natural choice of coordinates, which allows to deeply simplify the formal expression of the system, and to reduce it to a system of ODEs along a characteristic direction.We introduce a notion of higher dimensional holonomy map in analogy with the merlin wizard costume one-dimensional case [29], and we provide a characterization for singularities as well as a deformability criterion.