We prove that the first Chern class of a codimension two closed contact submanifold of the odd dimensional Euclidean space is trivial. For any closed co-oriented contact 3-manifold with trivial Chern class, we prove that there is a contact structure on the 5-dimensional Euclidean space which admits a contact embedding of it.
Naohiko KASUYA. "An obstruction for codimension two contact embeddings in the odd dimensional Euclidean spaces." J. Math. Soc. Japan 68 (2) 737 - 743, April, 2016. https://doi.org/10.2969/jmsj/06820737