Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A note on Euler number of locally conformally K\"{a}hler manifolds   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. $>$) $0$.

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The categories of this paper are: dg-ga
27
The categories of this paper are: comp-gas
22
The categories of this paper are: chao-dyn
1
The categories of this paper are: cs
32
The categories of this paper are: atom-ph
30
The categories of this paper are: q-alg
7
The categories of this paper are: hep-th
23
The categories of this paper are: solv-int
28
The categories of this paper are: alg-geom
8
The categories of this paper are: hep-ex
18
The categories of this paper are: econ
5
The categories of this paper are: gr-qc
9
The categories of this paper are: nlin
0
The categories of this paper are: hep-ph
11
The categories of this paper are: hep-lat
25
The categories of this paper are: patt-sol