Abstract
In this note, we study the relation between the existence of a negatively curved complete hermitian metric on a complex manifold M and the product structure of (or contained in) M. We introduce the concept of geometric ranks and give a curvature characterization of the rank one manifolds, which generalizes the previous results of P. Yang and N. Mok (see below). In the proof, we used the old techniques of Yau's Schwartz lemma and Cheng-Yau's result on the existence of Kahler-Einstein metrics.
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