A characterisation of Manhart's relative normal vector fields (Q2891066)

Consider an oriented surface \(M\) in three dimensional affine space \({\mathbb A}^3\) with nowhere degenerate Blaschke metric. The set of all tangent and \({\mathbb A}^3\)-valued vector fields along \(M\) are denoted by \(X(M)\) and \({\bar X}(M)\) respectively. A vector field \(Y\in{\bar X}(M)\) is called a \textit{relative normalization} if for all \(p\in M\) holds \(Y(p)\not\in T_p M\) and \(D_V Y\in X(M)\) for all \(V\in X(M)\). Here, \(D\) denotes the standard connection in \({\mathbb A}^3\).NEWLINENEWLINELet \(f:D\rightarrow {\mathbb R}:(u,v)\mapsto f(u,v)\) be an arbitrary smooth, nowhere vanishing function of two variables, where \(D\) contains the set\(\{(u,v)\in {\mathbb R}^2\mid u^2\geq v \) and \(v\neq 0\}\). Then for any non degenerate surface in Euclidean space the vector field given by NEWLINE\[NEWLINEN^M_f=f(H,K)N-grad_{II}(f(H,K))NEWLINE\]NEWLINE is a relative normal vector field. Here \(N, H\) and \(K\) denote the Euclidean normal vector field, mean curvature and Gaussian curvature respectively and \(grad_{II}\) stands for the gradient with respect to the Euclidean second fundamental form.NEWLINENEWLINEThe critical points of the relative area element are the relative minimal surfaces characterized by vanishing of the relative mean curvature. On the other hand, one can consider the critical points of the curvature functionals \(\int f(H,K)d\Omega\) with \(\Omega\) denoting the classical area element. It is proved that the two classes of surfaces coincide exactly in case of \(f(u,v)=|v|^{\alpha}\) \((\alpha\neq 0)\), characterizing a one parameter family of relative normal vector fields defined by the reviewer. In addition two characterizations of the Euclidean sphere by means of (Euclidean) curvature invariants are given.