Projectively flat affine surfaces (Q1879089)

While in dimension greater than two the affine hypersurfaces with projectively flat induced connection \(\Delta\) are exactly the affine hyperspheres, in the two-dimensional case there exist more examples of such projectively flat affine surfaces \(F\) (e.g. affine ruled surfaces). In all examples known to the author the affine mean curvature \(H\) or the (normed) scalar curvature \(\widehat K\) of the affine metric or the Pick invariant \(J\) of \(F\) are constant. Therefore he looks now for projectively flat surfaces where none of the functions \(H\), \(\widehat K\) or \(J\) is constant. His main result says that if the gradients of these functions (forming a parallelogramm as \(\widehat K=H+J)\) span a plane resp. a line then such an \(F\) must be (locally) symmetric resp. \(F\) is given by the solution of a system of five ODE's of first order. Finally, the symmetric as well as the projectively flat affine translation surfaces are classified.