Some concave functions on Lorentzian manifolds

Let M be a Lorentzian manifold and $$\phi$$ be a future timelike isometry of M. We use $$\phi$$ to construct a concave function $$f_{\phi }$$ 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.