On the geometry of static spacetimes

We review geometrical properties of a static spacetime (M,g), including geodesic completeness, causality, standard splittings, compact M, closed geodesics and geodesic connectedness. We pay special attention to the critical quadratic behaviour at infinity of the coefficients β, β-1 (β=-g(K,K), being K a timelike irrotational Killing vector field), which essentially control completeness, causality and geodesic connectedness. Recent references are specially discussed.