Integrability Conditions For Almost Hermitian And Almost Kaehler 4-Manifolds

2006/05/23 by Kirchberg, Klaus-Dieter
#53B20 #53C25 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.math/0605611

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 = S²/6 is well-known. For any almost Kähler 4-manifold with S ≥ 0, this condition forces the Kähler property. A compact almost Kähler 4-manifold is already Kähler if it satisfies the conditions | W₊ |² = S²/6 and δW₊=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 supp (W₊)=M is already Kähler if it satisfies the condition | W₊ |² = 3 (S_⋆ - S/3)² /8 together with |∇ W₊ | = | ∇ |W₊|| or with δW₊ + ∇ log | W₊ | \lrcorner W₊ =0, respectively. The almost complex structure J enters here explicitely via the star scalar curvature S_⋆ only.

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