arXiv math/06056112006-05-23math.DG

几乎Hermitian和几乎Kaehler 4-流形的可积性条件

Integrability Conditions For Almost Hermitian And Almost Kaehler 4-Manifolds

Klaus-Dieter Kirchberg

备注：: 19 pages, Latex

摘要：

如果$W_+$表示任何Kähler 4-流形的Weyl张量的自对偶部分，而$S$表示其标量曲率，则关系式$|W_+|^2 = S^2/6$是众所周知的。对于任何具有$S \ge 0$的几乎Kähler 4-流形，这个条件强制Kähler性质。一个紧的几乎Kähler 4-流形已经是Kähler的，如果它满足条件$| W_+ |^2 = S^2/6$和$δW_+=0$，也如果它是爱因斯坦的，并且$| W_+|$是常数。进一步证明了这类结果。一个几乎厄米特4-流形$(M,g,J)$与$\mathrm{supp} (W_+)=M$已经是Kähler，如果它满足条件$| W_+ |^2 = 3 (S_{\star} - S/3)^2 /8$与$|\nabla W_+ | = | \nabla |W_+||$或与$δW_+ + \nabla \log | W_+ | \lrcorner W_+ =0$，分别。几乎复杂的结构$J$只通过星标量曲率$S_{\star}$进入这里。

英文摘要：

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 $δW_+=0$ 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 $\mathrm{supp} (W_+)=M$ 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 $δW_+ + \nabla \log | W_+ | \lrcorner W_+ =0$, respectively. The almost complex structure $J$ enters here explicitely via the star scalar curvature $S_{\star}$ only.

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英文摘要：

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