ON 4-DIMENSIONAL LOCALLY CONFORMALLY FLAT ALMOST KÄHLER MANIFOLDS

Abstra t. Using the fundamental notions of the quaternionic analysis we show that there are no 4-dimensional almost Kähler manifolds which are locally confor-mally flat with a metric of a special form. I. Basic notions and the aim of the paper Let M2n be a real C∞-manifold of dimension 2n endowed with an almost com-plex structure J and a Riemannian metric g. If the metric g is invariant by the almost complex structure J, i.e. g(JX, JY) = g(X,Y) for any vector fields X and Y on M2n, then (M2n, J, g) is called almost Hermitian manifold. Define the fundamental 2-form Ω by Ω(X,Y): = g(X, JY). An almost Hermitian manifold (M2n, J, g,Ω) is said to be almost Kähler if Ω is a closed form, i.e. dΩ = 0. Suppose that n = 2. The aim of the paper is to prove the following