Compacitification and positive mass theorem for fibered Euclidean end

In this note, we consider the positive mass theorem for Riemannian manifolds $(M^n,g)$ asymptotic to $({\mathbb{R}}^k\times X^{n-k},g_{{\mathbb{R}}^k}+g_X)$ for $k\ge 3$ by studying the corresponding compactification problem.