On affine surfaces whose cubic forms are parallel relative to the affine metric (Q917937)

The authors consider the case of a nondegenerate affine surface \(M^ 2\) in affine space \(R^ 3\). Denoting by \(\nabla\) of the induced connection, h the affine metric, \({\hat \nabla}\) the Levi-Civita connection for h, \(C=\nabla h\) the cubic form, J the Pick invariant, they prove the following classification result: Theorem. If a nondegenerate affine surface in \(R^ 3\) satisfies \({\hat \nabla}C=0\), \(C\neq 0\), then it is equiaffinely congruent to a piece of one of the following surfaces: 1) the graph of \(z=1/xy\) (h: elliptic); 2) the graph of \(z=1/(x^ 2+y^ 2)\) (h: hyperbolic and \(J\neq 0)\); 3) Cayley surface, namely \(z=xy-y^ 3/3\), (h: hyperbolic and \(J=0)\).