Iso-contact embeddings of manifolds in co-dimension $2$

Abstract

The purpose of this article is to study co-dimension $2$ iso-contact embeddings of closed contact manifolds. We first show that a closed contact manifold $(M^{2n-1}, \xi_M)$ iso-contact embeds in a contact manifold $(N^{2n+1}, \xi_N),$ provided $M$ contact embeds in $(N, \xi_N)$ with a trivial normal bundle and the contact structure induced on $M$ via this embedding is homotopic as an almost-contact structure to $\xi_M.$ We apply this result to first establish that a closed contact $3$--manifold having no $2$--torsion in its second integral cohomology iso-contact embeds in the standard contact $5$--sphere if and only if the first Chern class of the contact structure is zero. Finally, we discuss iso-contact embeddings of closed simply connected contact $5$--manifolds.