On a construction of Burago and Zalgaller

TL;DR

This paper critically examines Burago and Zalgaller's proof on $PL$ isometric embeddings, showing limitations in higher dimensions and extending some results to noncompact surfaces, while exploring curvature relations.

Contribution

It identifies the limitations of existing $PL$ embedding proofs in higher dimensions and extends results to noncompact $PL$ 2-manifolds, proposing a $PL$ Gauss equation.

Findings

Burago and Zalgaller's proof does not extend directly to higher dimensions.

$PL$ manifolds of dimension ≥ 3 generally do not admit near-conformal embeddings in $ eal^{n+1}$.

Extension of Burago and Zalgaller's results to noncompact $PL$ 2-manifolds.

Abstract

The purpose of this note is to scrutinize the proof of Burago and Zalgaller regarding the existence of $PL$ isometric embeddings of $PL$ compact surfaces into $\mathbb{R}^3$. We conclude that their proof does not admit a direct extension to higher dimensions. Moreover, we show that, in general, $PL$ manifolds of dimension $n \geq 3$ admit no nontrivial $PL$ embeddings in $\mathbb{R}^{n+1}$ that are close to conformality. We also extend the result of Burago and Zalgaller to a large class of noncompact $PL$ 2-manifolds. The relation between intrinsic and extrinsic curvatures is also examined, and we propose a $PL$ version of the Gauss compatibility equation for smooth surfaces.