论文标题
平滑度量度量空间中的真空爱因斯坦场方程：各向同性情况
Vacuum Einstein field equations in smooth metric measure spaces: the isotropic case
论文作者
论文摘要
在平滑的度量度量时空$（m，g，e^{ - f} dvol_g）$上，我们定义了加权的爱因斯坦张量。它是根据bakry-émeryricci张量作为对称，无差异，伴随度量和密度函数的张量的。我们考虑了相关的真空加权爱因斯坦场方程，并表明各向同性溶液具有Nilpotent Ricci操作员。此外，如果它是$ 2 $ -Step nilpotent，则基础歧管是Brinkmann浪潮，如果它是$ 3 $ - 步骤Nilpotent，则是kundt Spacetime。在尺寸$ 3 $中获得更具体的结果，其中所有各性解决方案均以局部坐标为平面波或昆特空间。
On a smooth metric measure spacetime $(M,g,e^{-f} dvol_g)$, we define a weighted Einstein tensor. It is given in terms of the Bakry-Émery Ricci tensor as a tensor which is symmetric, divergence-free, concomitant of the metric and the density function. We consider the associated vacuum weighted Einstein field equations and show that isotropic solutions have nilpotent Ricci operator. Moreover, the underlying manifold is a Brinkmann wave if it is $2$-step nilpotent and a Kundt spacetime if it is $3$-step nilpotent. More specific results are obtained in dimension $3$, where all isotropic solutions are given in local coordinates as plane waves or Kundt spacetimes.
