Affine Hypersurfaces of Arbitrary Signature with an Almost Symplectic Form

Abstract

In this paper we study affine hypersurfaces with non-degenerate second fundamental form of arbitrary signature additionally equipped with an almost symplectic structure omega . We prove that if R^pomega =0 or nabla ^pomega =0 for some positive integer p then the rank of the shape operator is at most one. The results provide complete classification of affine hypersurfaces with higher order parallel almost symplectic forms and are generalization of recently obtained results for Lorentzian affine hypersurfaces.