On the Riemannian metrics in \(\mathbb{R}^ n\) which admit all hyperplanes as minimal hypersurfaces
It is shown that all those Riemannian metrics \((g_{ij})\) on some open subset \(D\) of \(\mathbb{R}^n\) which have all hyperplanes as minimal surfaces are precisely of the form \(g_{ij} = (\text{det} G)^{2/(1-n)} G_{ij}\) where \(G=(G_{ij})\) is a Euclidean Killing tensor field which in this case means that \(G\) is a linear combination of symmetric tensor products of the Euclidean Killing 1-forms \(dx^i\) and \(x^i dx^i- x^jdx^i\), \(i,j=1, \dots, n\). Thus, the \(G_{ij}\) are quadratic polynomials. For the special case that \(n=3\) and \(\text{det} (g_{ij})\) has the constant value 1 or, equivalently, that \(\text{det} (G_{ij})\) is 1, the author shows that, up to an Euclidean isometry, \(g=dx^2+ dy^2+ (dz-{1\over 2} k(xdy- ydx))^2\) with \(k\in\mathbb{R} \setminus \{0\}\), i.e., \(g\) is a left-invariant metric on the Heisenberg group \(H_1\) (the model used here is \(\mathbb{R}^3\) with the multiplication \(\mu_k: \mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R}^3\), \(\mu_k ((x,y,z)\), \((x',y',z')) =(x+x', y+y', z+z'+ {1\over 2} kxy'- {1\over 2} kx'y))\). The proofs of these results are by direct calculation.
scientific article; zbMATH DE number 880620
scientific article