Regular embeddings of Cartesian product graphs

It is proved in this paper that if a Cartesian power XnXn of a prime graph XX (with respect to the Cartesian multiplication) has an orientable regular embedding, then XX has an orientable regular embedding too; and if a graph with some extra conditions has orientable regular embeddings, then its Cartesian power also has. As an application of our main theorems, the regular embeddings of grid-like graphs are studied.