REAL ANALYTICITY OF THE HYPERKAHLER ALMOST KÄHLER MANIFOLDS

Abstract

1. Preliminaries LetM be a C∞-smooth paracompact 4n-dimensional manifold and J , K be antiinvolutive automorphisms of the tangent bundle on M which anticommute, i.e. J = K = −E and JK = −KJ . Let h be a riemannian metric on M satisfying the equalities h(JX, Y ) = h(X, JY ), h(KX,Y ) = h(X,KY ) for each two vector fields X , Y defined on an open set inM . Then h is called an almost hermitian metric with respect to the almost complex structures J andK and (M,J, h), (M,K, h) are called almost hermitian manifolds and (M ; J,K, h) is called a hyperkahler manifold . The two-form Ω on an almost hermitian manifold (M,J, h) defined by the equality Ω(X,Y ) = h(X, JY ) where X , Y are vector fields defined on an open set in M is called a fundamental form of the almost hermitian manifold. An almost hermitian manifold is called an almost Kahler manifold , if the fundamental form Ω is a closed two-form, i.e. if dΩ = 0.