Functions with (non)timelike gradient on a space-time

We consider some generalization of a problem proposed by V. A. Toponogov for functions with nontimelike gradient on a globally hyperbolic space-time and a specific application of the positive solution of this problem to the cases of a Minkowski space-time and a de Sitter space-time of the first kind. Examples of smooth functions with timelike gradient on Lorentz manifolds are given. The authors obtain some sufficient conditions for level surfaces of functions with timelike gradient on a Lorentz manifold which guarantee that the manifold is globally hyperbolic. A description of the past and the future event horizons for timelike geodesics in a de Sitter space-time of the first kind is given. Some unsolved problems are formulated.