An affine characterization of the Veronese surface (Q1900082)

On every nondegenerate surface \(M^2\) in a 4-dimensional Riemannian manifold \(\widetilde{M}\) there is a natural affine metric \(g\). On the other hand, there is the induced Riemannian metric, say \(g_1\) on \(M^2\). It is shown that \(g\), \(g_1\) are conformal iff \(M^2\) is a minimal Riemannian submanifold of \(\widetilde {M}\). Especially the cases are treated when \(\widetilde {M}\) is the Euclidean space \(\mathbb{R}^4\), the hyperbolic space \(\mathbb{H}^4\) or the standard 4-sphere \(S^4\). In the Euclidean and the hyperbolic case the conformal factor cannot be constant, i.e. \(g\) and \(g_1\) are never homothetic. In the case of \(S^4\) the metrics \(g\) and \(g_1\) are homothetic iff \(M^2\) is a Veronese surface.