Negative sectional curvature and the product complex structure
Abstract
Description
We 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 Letters
6 Pages. To appear in Mathematical Research Letters