On 4‐dimensional Lorentzian affine hypersurfaces with an almost symplectic form

In this paper we study 4‐dimensional 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 ifR k · ω = 0 or∇ k ω = 0 for some positive integer k . This result is the final step in a classification of Lorentzian affine hypersurfaces with higher order parallel almost symplectic forms.