Affine hypersurfaces admitting a pointwise symmetry

Abstract

An affine hypersurface is said to admit a pointwise symmetry, if there exists a subgroup of the automorphism group of the tangent space, which preserves (pointwise) the affine metric h, the difference tensor K and the affine shape operator S. In this paper, we deal with positive definite affine hypersurfaces of dimension three. First we solve an algebraic problem. We determine the non-trivial stabilizers G of the pair (K,S) under the action of SO(3) on an Euclidean vectorspace (V,h) and find a representative (canonical form of K and S) of each (SO(3)/G)-orbit. Then, we classify hypersurfaces admitting a pointwise G-symmetry for all non-trivial stabilizers G (apart of Z_2). Besides well-known hypersurfaces we obtain e.g. warped product structures of two-dimensional affine spheres (resp. quadrics) and curves.