On the geometry of locally conformal almost Kähler manifolds.

Abstract

In this work, we study a class of almost Hermitian manifolds called locally conformal almost Kähler manifolds. These are almost Hermitian manifolds which contains an open cover fUtgt2I and a family of C1 functions ft : Ut ! R such that each conformal metric gt on Ut is an almost Kähler metric. Locally conformal almost Kähler manifolds also falls under a class of locally conformal symplectic manifolds. More precisely, locally conformal almost Kähler manifolds are manifolds whose fundamental 2-form is locally conformal symplectic. We first recall some of the existing geometric properties of almost Hermitian manifolds. Then further use these properties to derive those of locally conformal almost Kähler manifolds. A new example of a locally conformal almost Kähler manifold is given. We further investigate the relationship between the covariant derivative and the Nijenhuis tensor on a locally conformal almost Kähler manifold. The equivalence of the Nijenhuis tensor de ned on each Ut and the one de ned globally is also proven. The relationship between the curvature tensors induced by the two conformal metrics on a locally conformal Kähler manifolds are considered. In particular, we show that a locally conformal almost Kähler manifold is an almost Kähler manifold under some curvature conditions. To achieve our goal, we first prove the relation between scalar curvatures and together with the corresponding scalar-curvatures and of a locally conformal almost Kähler manifold. Moreover, among other results, we also investigate the canonical foliations of locally conformal almost Kähler manifolds. Accurately, we give necessary and sufficient conditions for the metric on a locally conformal almost Kähler manifolds to be a bundle-like for foliations F.