On the Local Isometric Embedding in R^3 of Surfaces with Gaussian Curvature of Mixed Sign

TL;DR

This paper investigates conditions under which surfaces with mixed Gaussian curvature can be locally embedded into three-dimensional Euclidean space, showing that certain zero-curvature configurations allow for smooth isometric embeddings.

Contribution

It establishes the existence of local smooth isometric embeddings for surfaces with Gaussian curvature vanishing to finite order along intersecting Lipschitz curves.

Findings

Existence of embeddings when curvature vanishes to finite order

Embeddings possible with zero set of two intersecting Lipschitz curves

Results extend understanding of isometric embedding conditions

Abstract

We study the old problem of isometrically embedding a 2-dimensional Riemannian manifold into Euclidean 3-space. It is shown that if the Gaussian curvature vanishes to finite order and its zero set consists of two Lipschitz curves intersecting transversely at a point, then local sufficiently smooth isometric embeddings exist.