Projectively flat affine surfaces

Abstract.

We study non-degenerate affine surfaces in A ³ with a projectively flat induced connection. The curvature of the affine metric $\hat{K}$, the affine mean curvature H, and the Pick invariant J are related by $\hat{K} = H + J$. Depending on the rank of the span of the gradients of these functions, a local classification of three groups is given. The main result is the characterization of the projectively flat but not locally symmetric surfaces as a solution of a system of ODEs. In the final part, we classify projectively flat and locally symmetric affine translation surfaces.