# A note on Euler number of locally conformally K\"{a}hler manifolds

## TL;DR

This paper investigates the Euler number of compact locally conformally K"{a}hler manifolds with non-positive curvature, establishing inequalities based on their topological and geometric properties.

## Contribution

It proves that the Euler number of such manifolds satisfies specific inequalities if they are homeomorphic to d-bounded LCK manifolds.

## Key findings

- Euler number satisfies $(-1)^n \, \chi(M^{2n}) \geq 0$ for non-positive curvature.
- Strict inequality holds if the manifold has negative curvature.
- Results connect geometric curvature conditions with topological invariants.

## Abstract

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. $>$) $0$.