An obstruction for existence of codimension two contact embeddings in a Darboux chart

We prove the vanishing of the first Chern class of a codimension 2 closed contact submanifold of a cooriented contact manifold with trivial integral 2-dimensional cohomology group. Hence the first Chern class is an obstruction for the existence of codimension 2 contact embeddings in a Darboux chart. For the existence of such an embedding, we prove that a closed cooriented contact 3-manifold can be a contact submanifold of the 5-dimensional Euclidean space for a certain contact structure, if its first Chern class vanishes.