III型Ricci平坦self-dual中立型Walker四维流形的非紧性和最大可动性
Noncompactness and maximum mobility of type III Ricci-flat self-dual neutral Walker four-manifolds
摘要：
证明了一个自对偶中立型Einstein四流形的Petrov III型，承认一个二维零平行分布兼容的方向，不能是紧凑的或局部齐次的，它的最大可能的程度的流动性是3。Diaz-Ramos、Garcia-Rio和Vazquez-Lorenzo发现了这种流形的一般坐标形式。本文件还提供了一个修改后的版本，坐标形式，有效的一个适当定义的一般情况下，在某种意义上说，“更规范”比通常的配方。此外，还对迁移度等于3的局部等距流形进行了分类。进一步的结果包括在明确的描述，第一，内核和图像的Killing运营商的任何torsionfree表面连接无处不在的非零，斜对称的Ricci张量，其次，一个模曲线的表面连接的属性刚才提到的，此外，局部均匀。最后，双曲平面几何是用来展示的例子余维2叶理的维数大于2的紧致流形承认一个横向torsionfree联系，其中的Ricci张量是反对称的和非零无处不在。在任何闭曲面上都不存在这样的连接，因此在维度2中没有类似的例子。
英文摘要：
It is shown that a self-dual neutral Einstein four-manifold of Petrov type III, admitting a two-dimensional null parallel distribution compatible with the orientation, cannot be compact or locally homogeneous, and its maximum possible degree of mobility is 3. Diaz-Ramos, Garcia-Rio and Vazquez-Lorenzo found a general coordinate form of such manifolds. The present paper also provides a modified version of that coordinate form, valid in a suitably defined generic case and, in a sense, "more canonical" than the usual formulation. Moreover, the local-isometry types of manifolds as above having the degree of mobility equal to 3 are classified. Further results consist in explicit descriptions, first, of the kernel and image of the Killing operator for any torsionfree surface connection with everywhere-nonzero, skew-symmetric Ricci tensor, and, secondly, of a moduli curve for surface connections with the properties just mentioned that are, in addition, locally homogeneous. Finally, hyperbolic plane geometry is used to exhibit examples of codimension-two foliations on compact manifolds of dimensions greater than 2 admitting a transversal torsionfree connection, the Ricci tensor of which is skew-symmetric and nonzero everywhere. No such connection exists on any closed surface, so that there are no analogous examples in dimension 2.
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