The infinitesimal deformations of hypersurfaces that preserve the Gauss map

TL;DR

This paper classifies hypersurfaces in Euclidean space that admit infinitesimal deformations preserving the Gauss map, revealing that such hypersurfaces must be Kaehler, thus clarifying their geometric nature.

Contribution

The paper provides a parametric solution showing that hypersurfaces with infinitesimal Gauss map-preserving deformations are necessarily Kaehler, advancing the understanding of their geometric structure.

Findings

Hypersurfaces with infinitesimal Gauss map-preserving deformations are Kaehler.

The classical minimality condition is complemented by a Kaehler condition.

The parametric solution clarifies the geometric nature of such hypersurfaces.

Abstract

Classifying the nonflat hypersurfaces in Euclidean space $f\colon M^n\to\mathbb{R}^{n+1}$ that locally admit smooth infinitesimal deformations that preserve the Gauss map infinitesimally was a problem only considered by Schouten \cite{Sc} in 1928. He found two conditions that are necessary and sufficient, with the first one being the minimality of the submanifold. The second is a technical condition that does not clarify much about the geometric nature of the hypersurface. In that respect, the parametric solution of the problem given in this note yields that the submanifold has to be Kaehler.