On the geometry of locally conformal almost Kähler manifolds.
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Date
2020
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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.
Description
Masters Degree. University of KwaZulu-Natal, Pietermaritzburg.