A note on Euler number of locally conformally Kähler manifolds

Abstract

Let M 2n be a compact Riemannian manifold of non-positive (resp. negative) sectional curvature. We call (M, J, θ) a d(bounded) locally conformally Kähler manifold if the lifted Lee formθ 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 χ(M 2n ) ≥ (resp. >) 0.

Spheres and Circles With Respect to an Indefinite Metric on a Riemannian Manifold With a Skew-Circulant Structure

“…One subclass consists of the so-called locally conformal Kähler manifolds, determined by a special property of the covariant derivative of J. Some of the recent investigations of locally conformal Kähler manifolds are made in [2,3,10,14,15,16].…”

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“…In order to attack Hopf Conjecture in the Kählerian case when K ≤ 0 by extending Gromov's idea, Cao-Xavier [5] and Jost-Zuo [16] independently introduced the concept of Kähler parabolicity, which includes nonpositively curved closed Kähler manifolds, and showed that their Euler characteristics have the desired property. In [12], the author proved the Hopf conjecture in some locally conformally Kähler manifolds case.…”

“…12. ([19, Theorem 9.3]) The operator Dε has a finite projective L 2 index give byL 2 Index Gε ( Dε ) = X L X ∧ exp( ε2π[ω]).…”

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