Isometric Euclidean submanifolds with isometric Gauss maps

2024/06/15 by Marcos Dajczer, Dajczer, Marcos, Miguel I. Jimenez +3
Engineering · Mathematics · #3D Shape Modeling and Analysis #Advanced Numerical Analysis Techniques #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2406.10620

openalex publication_date 2024/06/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We investigate isometric immersions f\colon Mⁿ→\Rⁿ⁺², n≥ 3, of Riemannian manifolds into Euclidean space with codimension two that admit isometric deformations that preserve the metric of the Gauss map. In precise terms, the preservation of the third fundamental form of the submanifold must be ensured throughout the deformation. For minimal isometric deformations of minimal submanifolds this is always the case. Our main result is of a local nature and states that if f is neither minimal nor reducible, then it is a hypersurface of an isometrically deformable hypersurface F\colonMⁿ⁺¹→\Rⁿ⁺² such that the deformations of F induce those of f. Moreover, for a particular class of such submanifolds, a complete local parametric description is provided.

Related

• Euclidean hypersurfaces with genuine deformations in codimension two
• Isometric immersions of Riemannian manifolds in k-codimensional Euclidean space
• The Gauss map of minimal surfaces in \mathbbS²×ℝ
• The infinitesimal deformations of hypersurfaces that preserve the Gauss map
• Counterexamples for Local Isometric Embedding