Higher Dimensional Holonomy Map for Ruled Submanifolds in Graded Manifolds

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 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 one-dimensional case [29], and we provide a characterization for singularities as well as a deformability criterion.

Keywords

• sub-riemannian manifolds
• graded manifolds
• regular and singular ruled submanifolds
• higher-dimensional holonomy map
• admissible variations
• 58h99
• 49q99
• 58a17