Counterexamples for Local Isometric Embedding

2002/08/16 by Nikolai Nadirashvili, Nikolaï Nadirashvili, Yu Yuan +2
Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #math.AP #math.DG

paper · pdf · doi:10.48550/arxiv.math/0208127

submitted

arxiv created 2002/08/16 · openalex publication_date 2002/08/16 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We construct smooth metrics on 2-manifold with nonpositive Gauss curvature which cannot be (C³) locally isometrically embedded in R³. Moreover, the Gauss curvature of the metric can be made negative except for one point.

Related

• On the Local Isometric Embedding in R³ of Surfaces with Zero Sets of Gaussian Curvature Forming Cusp Domains
• The Local Isometric Embedding of Two-Metrics of Low Differentiability in Euclidean Three Space
• A Proof of the Isometric Embedding Theorem in Three Dimensional Euclidean Space
• Isometric Euclidean submanifolds with isometric Gauss maps
• Isometric embedding with nonnegative Gauss curvature under the graph setting