论文标题
在存在场上的巴赫和爱因斯坦方程
On the Bach and Einstein equations in presence of a field
论文作者
论文摘要
本文的目的是介绍和证明在field $φ$的情况下，在四维平滑的歧管$ m $上对经典的Bach场方程的概括是合理的，在此上下文中，这是由带有源$ M $的平滑地图给出的，而目标是另一种Riemannian歧管。这些方程式的特征是消失了两次协变，对称，无纹状体和共同不变的张量球场，称为$φ$ -BACH TENSOR，在没有田地的情况下，$φ$减少了经典的Bach Tensor。我们为$φ$ -BACH FLAT流形提供了各种表征，我们也为Harmonic-Einstein歧管（即Einstein Field方程的解决方案）做同样的表征，并具有保守的场$φ$。我们借此机会讨论了一些相关主题的概括：Yamabe问题，标量曲率图的图像，扭曲的产品解决方案和静态歧管。
The aim of this paper is to introduce and justify a possible generalization of the classic Bach field equations on a four dimensional smooth manifold $M$ in presence of field $φ$, that in this context is given by a smooth map with source $M$ and target another Riemannian manifold. Those equations are characterized by the vanishing of a two times covariant, symmetric, traceless and conformally invariant tensor field, called $φ$-Bach tensor, that in absence of the field $φ$ reduces to the classic Bach tensor. We provide a variational characterization for $φ$-Bach flat manifolds and we do the same also for harmonic-Einstein manifolds, i.e., solutions of the Einstein field equations with source the conservative field $φ$. We take the opportunity to discuss a generalization of some related topics: the Yamabe problem, the image of the scalar curvature map, warped product solutions and static manifolds.
