On the Local Isometric Embedding in $R^3$ of Surfaces with Zero Sets of Gaussian Curvature Forming Cusp Domains

TL;DR

This paper investigates conditions under which a 2D Riemannian surface with zero Gaussian curvature along two tangent curves can be locally embedded into 3D Euclidean space, establishing existence results for such embeddings.

Contribution

It proves the existence of local smooth isometric embeddings for surfaces with zero Gaussian curvature along tangent curves, extending previous embedding results to cusp domain zero sets.

Findings

Existence of local isometric embeddings when Gaussian curvature vanishes to finite order

Embedding is possible if zero set consists of two smooth tangent curves

Provides conditions for embeddings in cusp domain scenarios

Abstract

We study the problem of isometrically embedding a two-dimensional Riemannian manifold into Euclidean three-space. It is shown that if Gaussian curvature vanishes to finite order and its zero set consists of two smooth curves tangent at a point, then local sufficiently smooth isometric embedding exists.