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The infinitesimal deformations of hypersurfaces that preserve the Gauss map

Marcos Dajczer, Miguel Ibieta Jimenez

TL;DR

Addresses the problem of classifying hypersurfaces in $\\mathbb{R}^{n+1}$ that locally admit infinitesimal deformations preserving the Gauss map. The authors develop a framework for infinitesimal isometric variations, derive the fundamental relations between the variation tensor and the second fundamental form, and prove that any nontrivial Gauss-map-preserving infinitesimal bending forces the hypersurface to be minimal and Kaehler (up to Euclidean cylinders); if the manifold is simply connected the bending aligns with the conjugate in the associated family. They provide a parametric description of the resulting Kaehler hypersurfaces, including a Weierstrass-type representation, and clarify the relationship to the Gauss parametrization and to the associated minimal immersion family. The results yield a complete local classification and link Schouten's 1928 condition to classical complex-analytic constructions.

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.

The infinitesimal deformations of hypersurfaces that preserve the Gauss map

TL;DR

Addresses the problem of classifying hypersurfaces in that locally admit infinitesimal deformations preserving the Gauss map. The authors develop a framework for infinitesimal isometric variations, derive the fundamental relations between the variation tensor and the second fundamental form, and prove that any nontrivial Gauss-map-preserving infinitesimal bending forces the hypersurface to be minimal and Kaehler (up to Euclidean cylinders); if the manifold is simply connected the bending aligns with the conjugate in the associated family. They provide a parametric description of the resulting Kaehler hypersurfaces, including a Weierstrass-type representation, and clarify the relationship to the Gauss parametrization and to the associated minimal immersion family. The results yield a complete local classification and link Schouten's 1928 condition to classical complex-analytic constructions.

Abstract

Classifying the nonflat hypersurfaces in Euclidean space 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.
Paper Structure (3 sections, 3 theorems, 23 equations)

This paper contains 3 sections, 3 theorems, 23 equations.

Table of Contents

  1. Infinitesimal isometric variations
  2. Euclidean Kaehler hypersurfaces
  3. The proof of Theorem \ref{['main']}

Key Result

Theorem 1

Let $f\colon M^n\to\mathbb{R}^{n+1}$, $n\geq 2$, be an isometric immersion that admits a nowhere trivial infinitesimal variation that infinitesimally preserves the Gauss map. Assume that $M^n$ is free of flat points and that $f$ is not part of a Euclidean cylinder when restricted to any open subset

Theorems & Definitions (4)

  • Theorem 1
  • Proposition 2
  • Proposition 3
  • Remark 4