Partial immersions and partially free maps

In a recent paper DʼAmbra et al. (2011) [2] we studied basic properties of partial immersions and partially free maps, a generalization of free maps introduced first by Gromov (1970) in [4]. In this short note we show how to build partially free maps out of partial immersions and use this fact to prove that the partially free maps in critical dimension introduced in Theorems 1.1–1.3 of DʼAmbra et al. (2011) [2] for three important types of distributions can actually be built out of partial immersions. Finally, we show that the canonical contact structure on R2n+1 admits partial immersions in critical dimension for every n.