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.

我们继续研究混合爱因斯坦-希尔伯特作用作为伪黎曼度量和线性连接的函数。它的几何部分是具有分布或叶理的光滑流形上的总混合标量曲率。我们开发了度量仿射空间上分布的外在几何量的变分公式，并使用它们来推导出欧拉-拉格朗日方程（在时空的情况下类似于爱因斯坦-嘉当理论中的方程）并表征关键的这个动作点在真空时空上。连同度量和连接的任意变化，我们还考虑部分保留度量的变化，例如，沿着分布，以及不同连接类别之间的变化（例如，统计和度量兼容，这以对扭曲张量的限制表示）。混合爱因斯坦-希尔伯特作用的欧拉-拉格朗日方程之一是嘉当自旋连接方程的类比，另一个可以用类似于爱因斯坦方程的形式表示，用新的Ricci型张量代替Ricci曲率。该张量通常具有复杂的形式，但在论文中明确给出了半对称连接之间的变化。