A generalization of Lelieuvre's formula
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.
scientific article; zbMATH DE number 57249