Projectively flat affine surfaces that are not locally symmetric

Abstract:

By studying affine rotation surfaces (ARS), we prove that any surface affine congruent to ${x^2} + \epsilon {y^2} = {z^r}$ or ${y^2} = z(x + \epsilon z\log z)$ is projectively flat but is neither locally symmetric nor an affine sphere, where $\epsilon$ is 1 or $- 1, r \in {\mathbf {R}} - \{ - 1,0,1,2\}$, and $z > 0$. The significance of these surfaces is due to the fact that until now ${x^2} + \epsilon {y^2} = {z^{ - 1}}$ are the only known surfaces which are projectively flat but not locally symmetric. Although Podestà recently proved the existence of an affine surface satisfying the above italicized conditions, he did not construct any concrete example.