Self-dual centroaffine surfaces of codimension two with constant affine mean curvature (Q1420754)

Let \(f\) be a centroaffine immersion of an \(n\)-dimensional manifold \(M\) into the space \(\mathbb{R}^{n+2}\setminus\{0\}\) where \(f(x)\) is transversal to \(f_*T_xM\) for every \(x\in M\). To such a nondegenerate \(f\) a uniquely determined generalized Blaschke normal vector field \(\xi\) belongs which induces a nondegenerate symmetric \((0,2)\)-tensor field \(h\) on \(M\). Then \(f\) is called self-dual if there exists a volume preserving linear map \(L\) from \(\mathbb{R}^{n+2}\) into the dual space \(\mathbb{R}_{n+2}\) of \(\mathbb{R}^{n+2}\) such that \(f^*:= Lf\) has the properties \(f^*(x)(f_* X)= 0\), \(f^*(x)(\xi(x))= 1\) and every \(x\in M\) and \(X\in T_x M\). The main results of the paper are: a) A self-dual minimal centroaffine surface in \(\mathbb{R}^4\) with indefinite tensor field \(h\) is locally the tensor product of two centroaffine plane curves, and b) Also each such surface with nonvanishing constant affine mean curvature may be explicitly represented in a simple manner. Moreover, the authors indicate several complex supplements to this topic.