SYMPLECTIC 4-MANIFOLDS VIA LORENTZIAN GEOMETRY

We observe that, in dimension four, symplectic forms may be obtained via Lorentzian geometry; in particular, null vector fields can give rise to exact symplectic forms. That a null vector field is nowhere vanishing yet orthogonal to itself is essential to this construction. Specifically, we show that on a Lorentzian 4-manifold (M, g), if k is a complete null vector field with geodesic flow along which Ric(k, k) > 0, and if f is any smooth function on M with k(f) nowhere vanishing, then dg(e(f)k, .) is a symplectic form and k/k(f) is a Liouville vector field; any null surface to which k is tangent is then a Lagrangian submanifold. Even if the Ricci curvature condition is not satisfied, one can still construct such symplectic forms with additional information from k. We give an example of this, with k a complete Liouville vector field, on the maximally extended "rapidly rotating" Kerr spacetime.