Geometric characterization of level set families of harmonic functions on Riemannian manifolds

Abstract 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 $$\vert {\text {grad}}\;u \vert ^{-1}\varDelta u=\psi$$ | grad u | - 1 Δ u = ψ , where $$\psi$$ ψ is a smooth function on the manifold.