The Wong-Rosay theorem is a theorem characterizing the strongly pseudoconvex domain in \(\mathbb {C}^n\) by their automorphism groups. It has a lot of generalizations to other domains (for example, the weakly pseudoconvex domains). However, most of them are those domains of \(\mathbb {C}^n\) . In this note, we generalize the Wong-Rosay theorem to the simply-connected complete Kähler manifold with negative sectional curvature. One aim of this note is to exhibit a Wong-Rosay type theorem in the manifold with a holomorphic non-invariant metric.
