2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/108956We prove that a product complex manifold cannot admit a complete Kähler 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 Kähler manifold with negative sectional curvature. On the other hand, we observe that there are complete Kähler metrics with negative sectional curvature on $\C$. Hence the upper sectional curvature bound is necessary.6 Pages. To appear in Mathematical Research LettersDifferential GeometryComplex Variables53C21text