Geometry of Triples of Vector Fields in R4

Abstract

This chapter describes the geometry of triples of vector fields in R 4 . Interest for k-uples of vector field comes from control theory. The chapter focuses on the singularites of the objects. A general theory for singularities of k-uples was initiated in the articles of B. Jakubczyk and F. Przytycki. The chapter highlights how some geometrical considerations may help the study. Some results for the triples of vector fields in R 4 are established in the chapter. The existence of local explicit models implies that the germs of generic triple of vector fields are stable, in an evident sense, for points of codimensions ≤ 2. The chapter describes that the first model in the theorem presented in the chapter corresponds to the regular case (codimension 0 – stratum of singularity) and the second one to a singularity stratum of codimension 2. This chapter also presents some definitions coming from the general theory of Jakubczyk and Przytycki.