On the Riemannian metrics in \(\mathbb{R}^ n\) which admit all hyperplanes as minimal hypersurfaces
type: Item
instance of: scholarly article
title: On the Riemannian metrics in \(\mathbb{R}^ n\) which admit all hyperplanes as minimal hypersurfaces
zbMATH Open document ID: 0888.53007
DOI: 10.1016/0393-0440(95)00007-0
author: Yanyan Li
published in: Journal of Geometry and Physics
publication date: 1996-08-21T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53B20
zbMATH DE Number: 880620
zbMATH Keywords: minimal hypersurface
zbMATH Keywords: Killing tensor
zbMATH Keywords: Heisenberg group
reviewed by: F. Tomi
MaRDI profile type: Publication
full work available at URL: 0393-0440(95)00007-0
OpenAlex ID: W2080984558
Wikidata QID: Q115363075
cites work: Q4274259
cites work: Q4303000
cites work: Q3322013
cites work: Q3772868
cites work: Q5730628
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description: scientific article; zbMATH DE number 880620