Gradient Einstein-type spacetimes and modified gravity

In this work, at first, we investigate gradient Einstein-type spacetimes. After that, we study the properties of a gradient Einstein-type spacetime admitting a parallel vector field. In the next parts, we present a systematic study about gradient Einstein-type spacetime obeying f(R, T) and f(R,Lm)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(R,L_m)$$\end{document} gravity theories. Finally, in the cosmological context we find some remarkable results.