Indefinite affine hyperspheres admitting a pointwise symmetry

An affine hypersurface M is said to admit a pointwise symmetry, if there exists a subgroup G of Aut(T p M) for all p ∈ M , which preserves (pointwise) the affine metric h, the difference tensor K and the affine shape operator S. Here, we consider 3-dimensional indefinite affine hyperspheres, i. e. S = HId (and thus S is trivially preserved). First we solve an algebraic problem. We determine the non-trivial stabilizers G of a traceless cubic form on a Lorentz-Minkowski space R 3 1 under the action of the isometry group SO(1, 2) and find a representative of each SO(1, 2)/G-orbit. Since the affine cubic form is defined by h and K, this gives us the possible symmetry groups G and for each G a canonical form of K. Next, we classify hyperspheres admitting a pointwise G-symmetry for all non-trivial stabilizers G (apart from Z 2). Besides well-known hyperspheres (for Z 2 × Z 2 resp. R the hyperspheres have constant sectional curvature and Pick invariant J < 0 resp. J = 0) we obtain rich classes of new examples e.g. warped product structures of two-dimensional affine spheres (resp. quadrics) and curves. Moreover, we find a way to construct indefinite affine hyperspheres out of 2-dimensional quadrics or positive definite affine spheres.