局部齐性空间、诱导Killing向量场及其对Bianchi原型的应用
Locally Homogeneous Spaces, Induced Killing Vector Fields and Applications to Bianchi Prototypes
摘要：
这个问题的答案是：一般来说，采用一个给定的对称性是否会导致一个可能隐藏在第一个层次上的进一步的对称性？在局部齐性空间的微分几何的上下文中进行了尝试。基于E.利用Cartan的活动标架理论，给出了求n维局部齐性空间的所有对称的方法。分析应用到3维空间，从而嵌入到4维洛伦兹流形检查和爱因斯坦的场方程的特殊解决方案被恢复。分析主要是局部性的，因为兴趣集中在基于微分方程（及其对称性）的局部结构上，而不是在例如，它们的解的解析延拓和它们的动力学。
英文摘要：
An answer to the question: Can, in general, the adoption of a given symmetry induce a further symmetry, which might be hidden at a first level? has been attempted in the context of differential geometry of locally homogeneous spaces. Based on E. Cartan's theory of moving frames, a methodology for finding all symmetries for any n dimensional locally homogeneous space is provided. The analysis is applied to 3 dimensional spaces, whereby the embedding of them into a 4 dimensional Lorentzian manifold is examined and special solutions to Einstein's field equations are recovered. The analysis is mainly of local character, since the interest is focused on local structures based on differential equations (and their symmetries), rather than on the implications of, e.g., the analytic continuation of their solution(s) and their dynamics in the large.
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