M ar 2 00 3 3 - dimensional affine hypersurfaces admitting a pointwise SO ( 2 ) - or Z 3 - symmetry

3-dimensional affine hypersurfaces admitting a pointwise SO(2)-or Z 3-symmetry Abstract In (equi-)affine differential geometry, the most important algebraic invariants are the affine (Blaschke) metric h, the affine shape operator S and the difference tensor K. A hypersurface is said to admit a point-wise symmetry if at every point there exists a linear transformation preserving the affine metric, the affine shape operator and the difference tensor K. In this paper, we consider the 3-dimensional positive definite hypersurfaces for which at each point the group of symmetries is isomorphic to either Z 3 or SO(2). We classify such hypersurfaces and show how they can be constructed starting from 2-dimensional positive definite affine spheres.