Integrability Conditions For Almost Hermitian And Almost Kaehler 4-Manifolds

Abstract

If $W_{+}$ denotes the self dual part of the Weyl tensor of any Kähler 4-manifold and $S$ its scalar curvature, then the relation $|W_{+}{|}^2=S^2/6$ is well-known. For any almost Kähler 4-manifold with $S\ge 0$, this condition forces the Kähler property. A compact almost Kähler 4-manifold 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 4-manifold $(M,g,J)$ 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 $\delta W_{+}+\nabla \log |W_{+}|┘W_{+}=0$, respectively. The almost complex structure $J$ enters here explicitely via the star scalar curvature $S_{\star }$ only.

19 pages, Latex

References in corpus (2)

• A splitting theorem for Kahler manifolds whose Ricci tensors have constant eigenvalues
• Some integrability conditions for almost Kähler manifolds

Cited by in corpus (1)

• Geometry of neutral metrics in dimesnion four