论文标题
爱因斯坦 - 希尔伯特（Einstein-Hilbert
The Einstein-Hilbert type action on metric-affine almost-product manifolds
论文作者
论文摘要
我们继续研究混合爱因斯坦 - 希尔伯特（Einstein-Hilbert）的作用，作为伪 - 利曼式公制和线性连接的功能。它的几何部分是在具有分布或叶片的光滑歧管上的总混合标态曲率。我们开发了分布在公制植物空间上分布的外部几何形状量的变异公式，并使用它们来得出Euler-Lagrange方程（在时空的情况下类似于Einstein-Cartan理论中的时空），并表征了该动作对真空时空的关键点。加上指标和连接的任意变化，我们还考虑了部分保留度量的变化，例如沿分布，以及杰出连接类别之间的变化（例如，统计和度量兼容，这是根据contorsion tensor的限制来表示的）。爱因斯坦 - 希尔伯特（Einstein-Hilbert）作用的Euler-Lagrange方程之一是cartan旋转连接方程的类似物，另一个可以以类似于爱因斯坦方程的形式表示，而RICCI曲率被新的RICCI型Tensor代替。该张量通常具有复杂的形式，但在纸上给出了半对称性连接之间的变化。
We continue our study of the mixed Einstein-Hilbert action as a functional of a pseudo-Riemannian metric and a linear connection. Its geometrical part is the total mixed scalar curvature on a smooth manifold endowed with a distribution or a foliation. We develop variational formulas for quantities of extrinsic geometry of a distribution on a metric-affine space and use them to derive Euler-Lagrange equations (which in the case of space-time are analogous to those in Einstein-Cartan theory) and to characterize critical points of this action on vacuum space-time. Together with arbitrary variations of metric and connection, we consider also variations that partially preserve the metric, e.g., along the distribution, and also variations among distinguished classes of connections (e.g., statistical and metric compatible, and this is expressed in terms of restrictions on contorsion tensor). One of Euler-Lagrange equations of the mixed Einstein-Hilbert action is an analog of the Cartan spin connection equation, and the other can be presented in the form similar to the Einstein equation, with Ricci curvature replaced by the new Ricci type tensor. This tensor generally has a complicated form, but is given in the paper explicitly for variations among semi-symmetric~connections.
