In (Bivens in Mathematics Magazine 65: 226–235, 1992), it is shown that the appearance of the curves completely determines whether a family of curves in the Euclidean plane is a family of level curves of some harmonic function free of critical points. In this paper, we extend the result of (Bivens in Mathematics Magazine 65: 226–235, 1992) to higher dimensional Riemannian manifolds and give a geometric characterization of the level set family of the solutions of the differential equation |gradu|-1Δu=ψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\vert {\ ext {grad}}\\;u \\vert ^{-1}\\varDelta u=\\psi$$\\end{document}, where ψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi$$\\end{document} is a smooth function on the manifold.