On pseudo Z-symmetric Lorentzian manifolds admitting a type of semi-symmetric metric connection

The paper aims to investigate the general properties of pseudo Z-symmetric Lorentzian manifolds with semi-symmetric metric ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \rho $$\end{document}-connection ∇¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\bar{\nabla }} $$\end{document} and examine compatibility conditions. Moreover, such a manifold is applied to general relativity and its physical consequences are given.