Affine surfaces in \(\mathbb R^4\) with planar geodesics with respect to the affine metric (Q2567420)

The present paper characterizes affine surfaces in \(\mathbb R^4\) with planar geodesics with respect to the affine metric. The main theorem is: The only nondegenerate surfaces in \(\mathbb R^4\) whose geodesics with respect to the Levi-Civita connection of the affine metric are planar are the surfaces \(x(u,\nu)=(u,\nu,u\nu ,u^2-\nu^2)\) and \(x(u,\nu)=(u,\nu,u^2,\nu^2)\).