On the $h$-principle for Horizontal Immersions in Certain Corank $2$ Fat Distributions

Abstract

In this article we consider a class of fat corank $2$ distribution on a manifold, which includes the holomorphic contact structures. We prove the h-principle for regular horizontal immersion $\Sigma\to (M,\mathcal{D})$ for such a distribution $\mathcal{D}$ on $M$ if $\dim M \ge 4\dim\Sigma + 6$. In particular, we show that $\mathcal{D}$-horizontal maps always exist provided $\dim M \ge \max \{4\dim\Sigma + 6, 5\dim\Sigma-1\}$.Comment: 16 page