  Let $M^{2n}$ be a compact Riemannian manifold of non-positive (resp.
negative) sectional curvature. We call $(M,J,\theta)$ a $d$(bounded) locally
conformally K\"{a}hler manifold if the lifted Lee form $\tilde{\theta}$ on the
universal covering space of $M$ is $d$(bounded). We shown that if $M^{2n}$ is
homeomorphic to a $d$(bounded) LCK manifold, then its Euler number satisfies
the inequality $(-1)^{n}\chi(M^{2n})\geq$ (resp. $>$) $0$.
