Two Conditions Of The Integrability For Almost Kaehler Manifolds

The study concerning the integrability of almost Kaehler manifolds is stemed from a well-known conjecture referred firstly by S.I.Goldberg in 1969[4].Until to now,there are abundant results concerning the issue.In this paper,by making use of K.-D.Kirchberg's methods to study the integrability of almost Kaehler manifolds,we attain some new conditions for almost Kaehler manifolds.The main results consists of three parts.At first,we introduce some major results and recent development about the conjecture.In the second part,we introduce some basic conceptions and notations which will be used in proving lemmas and theorems.Most of notations are introduced in[1].At last,we prove firstly a lemma,and then apply the lemma to prove our results.Theorem 1.Let M be a compact almost Kaehler *-Einstein manifold whose Ricci tensor is J-invariant and nonnegative,then M is a Kaehler manifold.Theorem 2.Let M be a 4-dimensional almost Kaehler manifold whose Ricci tensor is J-invariant.Suppose there exists a real numberλ≥0 such thatλg(X,X)≤p^*sym(X,X)≤2λg(X,X), (?)X∈TM.Then M is a Kaehler manifold.