Lorentzian affine hypersurfaces with an almost symplectic form

In this paper, we study affine hypersurfaces with a Lorentzian second fundamental form additionally equipped with an almost symplectic structure ω. We prove that the rank of the shape operator is at most one if the hypersurface is of dimension at least 6 and Rk⋅ω=0 or ∇kω=0 for some positive integer k.