Lorentzian affine hypersurfaces with an almost symplectic form

Abstract

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 R k ⋅ ω = 0 or ∇ k ω = 0 for some positive integer k .