Communications of the
Korean Mathematical Society CKMS

We show that on a compact locally conformal K\"ahler manifold $M^{2n}$ (dim $M^{2n}=2n\geq 4$), $M^{2n}$ is K\"ahler if and only if its conformal scalar curvature $k$ is not smaller than the scalar curvature $s$ of $M^{2n}$ everywhere. As a consequence, if a compact locally conformal K\"ahler manifold $M^{2n}$ is both conformally flat and scalar flat, then $M^{2n}$ is K\"ahler. In contrast with the compact case, we show that there exists a locally conformal K\"ahler manifold with $k$ equal to $s$, which is not K\"ahler.

Keywords: compact locally conformal K\"ahler manifold, conformal scalar curvature, K\"ahler, conformally flat and scalar flat, a locally conformal K\"ahler manifold with $k$ equal to $s$

MSC numbers: 53A30, 53B35, 53C25, 53C55, 53C56