- 6635 South Hall
- and Physics Seminar
We investigate timelike and null vector flows on closed Lorentzian manifolds and their relationship to Ricci curvature. The guiding observation, first observed for closed Riemannian 3-manifolds by Harris & Paternain '13, is that positive Ricci curvature tends to yield contact forms, namely, 1-forms metrically equivalent to unit vector fields with geodesic flow. We carry this line of thought over to the Lorentzian setting. First, we observe that the same is true on a closed Lorentzian 3-manifold: if X is a global timelike unit vector field with geodesic flow satisfying Ric(X,X) > 0, then g(X,•) is a contact form with Reeb vector field X, at least one of whose integral curves is closed. Second, we show that on a closed Lorentzian 4-manifold, if X is a global null vector field satisfying nabla_XX = X and Ric(X) > divX - 1, then dg(X,•) is a symplectic form and X is a Liouville vector field.