具有非负数量曲率的局部对称嵌入空间及其特征
On some locally symmetric embedded spaces with non-negative scalar curvature and their characterization
摘要：
在这项工作中，我们进行了一般性的研究的性质的一类局部对称嵌入超曲面在时空中承认一个$1+1+2$时空分解。超曲面通过指定关于诱导度量的Ricci张量的形式给出。这些是时空中恒定时间的片段。首先给出了一般超曲面的Ricci张量的形式，并给出了一般超曲面退化为常数时间片的条件。我们提供了这些超曲面的表征，在时空中的关键物理量发挥作用，在指定这些超曲面的局部几何。此外，我们研究的情况下，这些超曲面承认一个里奇孤子结构。分析了与孤子相关的矢量场为Killing矢量场或共形Killing矢量场的特殊情况。最后，在具有局部旋转对称性的时空的背景下，证明了只有具有零旋转和空间扭曲的这类时空才能容纳所考虑的超曲面类型，并且这些超曲面必然是平坦的。如果这种超曲面确实存在Ricci孤子结构，则孤子是稳定的，孤子场的分量是常数。
英文摘要：
In this work we perform a general study of properties of a class of locally symmetric embedded hypersurfaces in spacetimes admitting a $1+1+2$ spacetime decomposition. The hypersurfaces are given by specifying the form of the Ricci tensor with respect to the induced metric. These are slices of constant time in the spacetime. Firstly, the form of the Ricci tensor for general hypersurfaces is obtained and the conditions under which the general case reduces to those of constant time slices are specified. We provide a characterization of these hypersurfaces, with key physical quantities in the spacetime playing a role in specifying the local geometry of these hypersurfaces. Furthermore, we investigate the case where these hypersurfaces admit a Ricci soliton structure. The particular cases where the vector fields associated to the solitons are Killing or conformal Killing vector fields are analyzed. Finally, in the context of spacetimes with local rotational symmetry it is shown that, only spacetimes in this class with vanishing rotation and spatial twist can admit the hypersurface types considered, and that the hypersurfaces are necessarily flat. And if such hypersurface do admit a Ricci soliton structure, the soliton is steady, with the components of the soliton field being constants.
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