Let L be a holomorphic line bundle over a compact locally conformally Kähler manifold \((X,\omega ,\theta )\) with negative sectional curvature \(sec\le -K<0\) . Let \(\pi :(\tilde{X},\tilde{\omega },\tilde{\theta })\rightarrow (X,\omega ,\theta )\) be the universal covering map for X. In this article, we first observe that the fundamental form \(\tilde{\omega }\) of \(\tilde{X}\) is \(d_{-\tilde{\theta }}\) exact. Specifically, there exists a 1-form \(\eta \) such that \(\tilde{\omega }=d\eta -\tilde{\theta }\wedge \eta \) . Moreover, under the assumption \(\Vert \theta \Vert ^{2}_{L^{\infty }(X)}<K\) , we can prove that \(\eta \) is bounded. Finally, building upon the boundedness of \(\eta \) , we derive a lower bound for the Euler characteristic \(\chi (X)\) of X under the non-vanishing condition \(\int _{X}c^{n}_{1}(L)\ne 0\) . Specifically, we show that \((-1)^{n}\chi (X)\ge (n+1)+\lfloor \frac{C(n)K}{nC} \rfloor \) , where \(C:=|[\Theta (L),\Lambda ]|\) is a constant determined by the curvature of L.
