Bi-Lipschitz Embeddability of the Grushin Plane into Euclidean Space

TL;DR

This paper demonstrates that the Grushin plane can be bi-Lipschitz embedded into Euclidean space, contrasting with other sub-Riemannian manifolds like the Heisenberg group, by extending embeddings via Whitney decomposition.

Contribution

It provides a novel bi-Lipschitz embedding construction for the Grushin plane into Euclidean space, expanding understanding of embeddability of sub-Riemannian manifolds.

Findings

Grushin plane admits bi-Lipschitz embedding into Euclidean space

Extension of embedding uses Whitney decomposition of the singular line

Contrasts with non-embeddability of the Heisenberg group

Abstract

Many sub-Riemannian manifolds like the Heisenberg group do not admit bi- Lipschitz embedding into any Euclidean space. In contrast, the Grushin plane admits a bi-Lipschitz embedding into some Euclidean space. This is done by extending a bi-Lipschitz embedding of the singular line, using a Whitney decomposition of its complement.