# Higher dimensional holonomy map for ruled submanifolds in graded manifolds

## TL;DR

This paper introduces a higher dimensional holonomy map for ruled submanifolds in graded manifolds, simplifying the deformability PDE system to ODEs and providing criteria for singularities and deformability.

## Contribution

It develops a novel higher dimensional holonomy map and simplifies the deformability analysis for ruled submanifolds in graded manifolds.

## Key findings

- Reduction of PDE system to ODEs along characteristic directions
- Characterization of singularities in the holonomy map
- Deformability criterion for ruled submanifolds

## Abstract

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, we introduce a natural choice of coordinates, which allows to deeply simplify the formal expression of the system, and reduce it to a system of ODEs along a characteristic direction. We introduce a notion of higher dimensional holonomy map in analogy with the one-dimensional case [J. Differential Geom., 36(3):551-589, 1992], and we provide a characterization for singularities as well as a deformability criterion.