An Isometric Embedding of the Impossible Triangle into the Euclidean Space of Lowest Dimension

2021 ◽  
pp. 438-457
Author(s):  
Zhenbing Zeng ◽  
Yaochen Xu ◽  
Zhengfeng Yang ◽  
Zhi-bin Li
Keyword(s):  
Euclidean Space ◽  
Isometric Embedding

On the global isometric embedding of pseudo-Riemannian manifolds

1970 ◽  
Vol 314 (1518) ◽  
pp. 417-428 ◽  
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Riemannian Manifolds ◽  
General Result ◽  
Isometric Embedding ◽  
Positive Definite ◽  
Space Time ◽  
Lower Dimension ◽  
Definite Case

It is shown that any pseudo-Riemannian manifold has (in Nash’s sense) a proper isometric embedding into a pseudo-Euclidean space, which can be made to be of arbitrarily high differentiability. The application of this to the positive definite case treated by Nash gives a new proof using a Euclidean space of substantially lower dimension. The general result is applied to the space-time of relativity, and the dimensions and signatures of the spaces needed to embed various cases are evaluated.

scholarly journals Isometric embeddings of polyhedra into Euclidean space

2015 ◽  
Vol 07 (04) ◽  
pp. 677-692 ◽  
Author(s):  
Barry Minemyer
Keyword(s):  
Euclidean Space ◽  
Metric Space ◽  
Isometric Embedding ◽  
Piecewise Linear ◽  
Embedding Theorem ◽  
Embedding Theorems ◽  
Dimensional Simplex ◽  
Isometric Embeddings ◽  
Lipschitz Map ◽  
Hyperbolic Polyhedra

In this paper we consider piecewise linear (pl) isometric embeddings of Euclidean polyhedra into Euclidean space. A Euclidean polyhedron is just a metric space [Formula: see text] which admits a triangulation [Formula: see text] such that each n-dimensional simplex of [Formula: see text] is affinely isometric to a simplex in 𝔼n. We prove that any 1-Lipschitz map from an n-dimensional Euclidean polyhedron [Formula: see text] into 𝔼3n is ϵ-close to a pl isometric embedding for any ϵ > 0. If we remove the condition that the map be pl, then any 1-Lipschitz map into 𝔼2n + 1 can be approximated by a (continuous) isometric embedding. These results are extended to isometric embedding theorems of spherical and hyperbolic polyhedra into Euclidean space by the use of the Nash–Kuiper C1 isometric embedding theorem ([9] and [13]).

scholarly journals Isometric Embeddings of Pro-Euclidean Spaces

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Barry Minemyer
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Metric Space ◽  
Inverse Limit ◽  
Isometric Embedding ◽  
Embedding Theorem ◽  
Euclidean Spaces ◽  
Compact Metric Space ◽  
Covering Dimension ◽  
Isometric Embeddings

Abstract In [12] Petrunin proves that a compact metric space X admits an intrinsic isometry into En if and only if X is a pro-Euclidean space of rank at most n, meaning that X can be written as a “nice” inverse limit of polyhedra. He also shows that either case implies that X has covering dimension at most n. The purpose of this paper is to extend these results to include both embeddings and spaces which are proper instead of compact. The main result of this paper is that any pro-Euclidean space of rank at most n is proper and admits an intrinsic isometric embedding into E2n+1. Since every n-dimensional Riemannian manifold is a pro-Euclidean space of rank at most n, this result is a partial generalization of (the C0 version of) the famous Nash isometric embedding theorem from [10].

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