Closed geodesics on compact semi-Riemannian manifolds

We study the existence of closed geodesics on compact quotients of Lie groups endowed with left-invariant pseudo-Riemannian metrics. We construct a compact, geodesically complete pseudo-Riemannian manifold that admits no closed geodesics. We also establish the existence of closed geodesics in several settings, with particular focus on the Lorentzian case.