We investigate a necessary condition for a compact complex manifold X of dimension n in order that its universal cover be the Cartesian product C^n of a curve C = P1or H: the existence of a semispecial tensor ω. A semispecial tensor is a non zero section ω ∈ H0(X, SnΩ1 (−K)⊗η)), where η is an invertible sheaf of 2-torsion (i.e., η^2 = OX). We show that this condition works out nicely, as a sufficient condition, when coupled with some other simple hypothesis, in the case of dimension n = 2 or n = 3; but it is not sufficient alone, even in dimension 2. In the case of K¨ahler surfaces we use the above results in order to give a characterization of the surfaces whose universal cover is a product of two curves, distinguishing the 6 possible cases.