Thomsen surfaces in affine 4-space (Q1364384)

The author investigates the Thomsen problem for nondegenerate smooth surfaces \(x:M^2 \to \mathbb{R}^4\), namely the determination of all surfaces which are both Euclidean minimal and affine extremal. The latter refers to an equiaffine volume form \(\omega_g\) on \(M^2\) induced by the nondegenerate affine metric \(g(\cdot, \cdot) ={G_w (\cdot, \cdot) \over (\text{det}_w G_w)^{1/3}}\) where \(G_w(Y,Z) ={1\over 2} (\text{det} (X_1,X_2, D_YX_1, D_Z X_2) +\text{det} (X_1,X_2, D_ZX_1, D_YX_2))\) \((w= \{X_1,X_2\}\) is a local, differentiable frame on \(M^2)\). Introducing suitable transversal planes for \(x\), defined in [\textit{K. Nomizu} and \textit{L. Vrancken}, Int. J. Math. 4, 127-165 (1993; Zbl 0810.53006)], the main theorem says that every nondegenerate Thomsen surface \(x\) in \(\mathbb{R}^4\) (Euclidean minimal and affine extremal) is affinely congruent to the complex surface \(x(u,v)=(u,v, {1\over 2} (u^2-v^2), uv)\).