OICC Press Mathematical Sciences 2251-7456 16 3 (September 2022) 2021 07 23 Some concave functions on Lorentzian manifolds 10.1007/s40096-021-00423-3 EN Reza Mirzaie Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University (IKIU), Qazvin, IR Omid Rezaei Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University (IKIU), Qazvin, IR Journal Article 2021 07 23 Abstract Let M be a Lorentzian manifold and ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi$$\end{document} be a future timelike isometry of M . We use ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi$$\end{document} to construct a concave function fϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_{\phi }$$\end{document} on M under some conditions on the curvature of timelike and spacelike planes. As a topological application, we characterize the fundamental group of M , when M has positive curvature along timelike planes.