Integrability Conditions For Almost Hermitian And Almost Kaehler 4Manifolds
Abstract
If $W_+$ denotes the self dual part of the Weyl tensor of any Kähler 4manifold and $S$ its scalar curvature, then the relation $W_+^2 = S^2/6$ is wellknown. For any almost Kähler 4manifold with $S \ge 0$, this condition forces the Kähler property. A compact almost Kähler 4manifold is already Kähler if it satisfies the conditions $ W_+ ^2 = S^2/6$ and $\delta W_+=0$ and also if it is Einstein and $ W_+$ is constant. Some further results of this type are proved. An almost Hermitian 4manifold $(M,g,J)$ with $\mathrm{supp} (W_+)=M$ is already Kähler if it satisfies the condition $ W_+ ^2 = 3 (S_{\star}  S/3)^2 /8$ together with $\nabla W_+  =  \nabla W_+$ or with $\delta W_+ + \nabla \log  W_+  \lrcorner W_+ =0$, respectively. The almost complex structure $J$ enters here explicitely via the star scalar curvature $S_{\star}$ only.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 2006
 arXiv:
 arXiv:math/0605611
 Bibcode:
 2006math......5611K
 Keywords:

 Mathematics  Differential Geometry;
 53B20;
 53C25
 EPrint:
 19 pages, Latex