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We prove the vanishing of the first Chern class of a codimension 2 closedcontact submanifold of a cooriented contact manifold with trivial integral2-dimensional cohomology group. Hence the first Chern class is an obstructionfor the existence of codimension 2 contact embeddings in a Darboux chart. Forthe existence of such an embedding, we prove that a closed cooriented contact3-manifold can be a contact submanifold of the 5-dimensional Euclidean spacefor a certain contact structure, if its first Chern class vanishes.

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