Negative sectional curvature and the product complex structure
  We prove that a product complex manifold cannot admit a complete K\"ahler metric with sectional curvature $K<c<0$ and Ricci curvature $Ric > d$, where $c$ and $d$ are constants.   In particular, a product domain in $\C$ cannot cover a compact K\"ahler manifold with negative sectional curvature.   On the other hand, we observe that there are complete K\"ahler metrics with negative sectional curvature on $\C$. Hence the upper sectional curvature bound is necessary. 
Harish Seshadri