A generalization of Lelieuvre's formula
type: Item
instance of: scholarly article
title: A generalization of Lelieuvre's formula
zbMATH Open document ID: 0758.53004
DOI: 10.1007/BF03323204
author: Katsumi Nomizu
author: An-Min Li
author: Changping Wang
published in: Results in Mathematics
publication date: 1992-09-27T00:00:00Z
review text: Let \(x: M\to\mathbb{R}^ 3\) be a hyperbolic surface with equiaffine normalization. Let \((\xi,\eta)\) be an asymptotic coordinate system with respect to the Blaschke metric. Denote by \(U(\xi,\eta)\) the affine conormal vector field of \(x(M)\). The classical Lelieuvre formula states that \(x=\int U\times U_ \xi d_ \xi-U\times U_ \eta d\eta\). This formula is very important in the study of affine minimal surfaces in \(\mathbb{R}^ 3\). In this paper we generalize the Lelieuvre formula to hypersurfaces in \(\mathbb{R}^ n\) (\(n\geq 2\)) with a more general (relative) normalization and study the integrability condition concerning this formula.
Mathematics Subject Classification ID: 53A15
zbMATH DE Number: 57249
zbMATH Keywords: hyperbolic surface
zbMATH Keywords: equiaffine normalization
zbMATH Keywords: Blaschke metric
zbMATH Keywords: Lelieuvre formula
zbMATH Keywords: affine minimal surfaces
reviewed by: Changping Wang
MaRDI profile type: Publication
cites work: On the geometry of affine immersions
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description: scientific article; zbMATH DE number 57249