Timelike Killing Vector Fields on a Spacelike Complete Lorentzian Manifold

In this article, we show that there is no nonvanishing timelike Killing vector field on a spacelike complete Lorentzian manifold M of negative sectional curvature. In addition, we show that if E is a nonvanishing timelike Killing vector field on a manifold M and the square of the norm of the integrability tensor of the orthogonal complement of E is not less than the sectional curvature of M, then the local flow of E is geodesic.