Negative sectional curvature and the product complex structure

Details Negative sectional curvature and the product complex structure Negative sectional curvature and the product complex structure

2006-02-14

arxiv.org/abs/math/0602289v1

Mathematics

Differential Geometry

6 Pages. To appear in Mathematical Research Letters

Scientific paper

We prove that a product complex manifold cannot admit a complete K\"ahler metric with sectional curvature $K d$, 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.

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