Inserted: 28 dec 2022
Last Updated: 28 dec 2022
Nonparametric $g$-surfaces in Euclidean space have recently been characterized by Bildhauer-Fuchs in terms of closure of a 1-form associated to the so called asymptotic normal. This 1-form can be written by means of the pull-back of a canonical vector-valued 1-form through a suitable map depending on the asymptotic normal, that in the minimal surfaces case agrees with the Gauss graph map. We show that a similar characterization holds true for g-hypersurfaces of any high dimension $N$, but this time in terms of a canonical vector valued form of degree $N-1$. In the minimal hypersurfaces case, we finally discuss the lack of a relationship between the previous result and existence of good parameterizations, when $N$ is greater than two.
Keywords: Generalized surfaces, Canonical forms, Parameterizations