论文标题
复发的洛伦兹韦尔空间
Recurrent Lorentzian Weyl spaces
论文作者
论文摘要
我们发现所有未关闭的Lorentzian Weyl歧管$（m，c，\ nabla）的局部形式，带有复发曲率张量。如果歧管的尺寸大于3，则保形结构是平坦的，并且通过单个功能局部确定复发的Weyl结构。当且仅当相应函数通过$ \ Mathrm {Saff} _1（\ MathBb {r}）\ Times \ Times \ MathRM {PSL} _2（\ Mathbb {r}）\ times \ times \ mathbb {z z} _2 _2 $。我们发现了该谎言组动作的理性标量差异不变的领域的发电机。歧管$ m $的全球结构可以用具有横向投影结构的叶面来描述。结果表明，所有局部均匀的结构都是局部等效的，只有一个简单地连接的均匀的非相关复发的Lorentzian Weyl歧管。此外，共有5类同构性一个空间，所有其他空间都是共同体的两个。如果$ \ dim m = 3 $，则未关闭的lorentzian Weyl结构由两个变量的一个函数或一个变量的两个函数在局部确定，具体取决于其自由度代数是1-或二维。在这种情况下，当且仅当它们分别通过从无限维谎言pseudogroup或$ \ mathrm {aff}（aff}的4维亚组）转换时，具有相同载体代数的两个结构在本地相等。同样，我们为理性差异不变的领域提供了生成器。我们找到了局部均匀的非闭合复发洛伦兹韦尔歧管的局部表达，以及同一性一和二的局部表达。最后，我们对也是爱因斯坦·韦尔（Einstein-Weyl）的非封闭的洛伦兹（Lorentzian Weyl）歧管的局部描述。它们都是三维的，具有二维全能代数。
We find the local form of all non-closed Lorentzian Weyl manifolds $(M,c,\nabla)$ with recurrent curvature tensor.If the dimension of the manifold is greater than 3, then the conformal structure is flat, and the recurrent Weyl structure is locally determined by a single function. Two local structures are equivalent if and only if the corresponding functions are related by a transformation from $\mathrm{SAff}_1(\mathbb{R}) \times \mathrm{PSL}_2(\mathbb{R}) \times \mathbb{Z}_2$. We find generators for the field of rational scalar differential invariants of this Lie group action. The global structure of the manifold $M$ may be described in terms of a foliation with a transversal projective structure. It is shown that all locally homogeneous structures are locally equivalent, and there is only one simply connected homogeneous non-closed recurrent Lorentzian Weyl manifold. Moreover, there are 5 classes of cohomogeneity-one spaces, and all other spaces are of cohomogeneity-two. If $\dim M=3$, the non-closed recurrent Lorentzian Weyl structures are locally determined by one function of two variables or two functions of one variables, depending on whether its holonomy algebra is 1- or 2-dimensional. In this case, two structures with the same holonomy algebra are locally equivalent if and only if they are related, respectively, by a transformation from an infinite-dimensional Lie pseudogroup or a 4-dimensional subgroup of $\mathrm{Aff}(\mathbb R^3)$. Again we provide generators for the field of rational differential invariants. We find a local expression for the locally homogeneous non-closed recurrent Lorentzian Weyl manifolds of dimension 3, and also of those of cohomogeneity one and two. In the end we give a local description of the non-closed recurrent Lorentzian Weyl manifolds that are also Einstein-Weyl. All of them are 3-dimensional and have a 2-dimensional holonomy algebra.
