In this paper, we study non-integrable distributions in a Riemannian manifold with a semi-symmetric metric connection, a kind of semi-symmetric non-metric connection and a statistical connection. We obtain the Gauss, Codazzi, and Ricci equations for non-integrable distributions with respect to the semi-symmetric metric connection, the semi-symmetric non-metric connection and the statistical connection. As applications, we obtain Chen’s inequalities for non-integrable distributions of real space forms endowed with a semi-symmetric metric connection and a kind of semi-symmetric non-metric connection. We give some examples of non-integrable distributions in a Riemannian manifold with affine connections. We find some new examples of Einstein distributions and distributions with constant scalar curvature.

中文翻译：

---

不可积分布的仿射连接

在本文中，我们研究了具有半对称度量连接、一种半对称非度量连接和统计连接的黎曼流形中的不可积分布。我们获得了关于半对称度量连接、半对称非度量连接和统计连接的不可积分布的 Gauss、Codazzi 和 Ricci 方程。作为应用，我们得到了具有半对称度量连接和一种半对称非度量连接的实空间形式的不可积分布的陈不等式。我们给出了一些具有仿射连接的黎曼流形中不可积分布的例子。我们发现了一些新的爱因斯坦分布和具有恒定标量曲率的分布的例子。