Negative sectional curvature and the product complex structure

Abstract

Let M = M1 × M2 be a product of complex manifolds. We prove that M cannot admit a complete Kähler metric with sectional curvature K < c < 0 and Ricci curvature Ric > d, where c and d are arbitrary constants. In particular, a product domain in ℂn 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 ℂn. Hence the upper sectional curvature bound is necessary. © International Press 2006.