高维情形下的复Goldberg-Sachs定理
The complex Goldberg-Sachs theorem in higher dimensions
摘要：
研究了$(2m+ε)$维复黎曼流形$(\mathcal{M}, \bm{g})$上全零$m$ -平面的全纯分布的几何性质，其中$ε\in {0,1}$和$m \geq 2$ .特别地，给定这样一个分布$\mathcal{N}$，我们得到了保证$\mathcal{N}$及其正交补在奇数维上可积的Weyl张量和Cotton-York张量的代数条件.这些结果概括了彼得罗夫分类的（反）自对偶部分的复杂外尔张量，和复杂的戈德堡萨克斯定理从四维到更高的维度。高维类似物的彼得罗夫型D条件的定义，我们表明，这些导致的可积性高达$2^m$全纯分布的完全空$m$ -平面。最后，我们适应这些发现的真正光滑的伪黎曼流形的类别，特别是评论的应用，埃尔米特几何和罗宾逊（或光学）几何。
英文摘要：
We study the geometric properties of holomorphic distributions of totally null $m$-planes on a $(2m+ε)$-dimensional complex Riemannian manifold $(\mathcal{M}, \bm{g})$, where $ε\in {0,1}$ and $m \geq 2$. In particular, given such a distribution $\mathcal{N}$, say, we obtain algebraic conditions on the Weyl tensor and the Cotton-York tensor which guarrantee the integrability of $\mathcal{N}$, and in odd dimensions, of its orthogonal complement. These results generalise the Petrov classification of the (anti-)self-dual part of the complex Weyl tensor, and the complex Goldberg-Sachs theorem from four to higher dimensions. Higher-dimensional analogues of the Petrov type D condition are defined, and we show that these lead to the integrability of up to $2^m$ holomorphic distributions of totally null $m$-planes. Finally, we adapt these findings to the category of real smooth pseudo-Riemannian manifolds, commenting notably on the applications to Hermitian geometry and Robinson (or optical) geometry.
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