Bi-Lipschitz embedding of the generalized Grushin plane into Euclidean spaces. (English) Zbl 1385.53019
Summary: We show that, for all \(\alpha \geq 0\), the generalized Grushin plane \(\mathbb{G}_{\alpha}\) is bi-Lipschitz homeomorphic to a 2-dimensional quasiplane in the Euclidean space \(\mathbb{R}^{\lfloor \alpha \rfloor +2}\), where \(\lfloor \alpha \rfloor\) is the integer part of \(\alpha\). The target dimension is sharp. This generalizes a recent result of J.-M. Wu [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 14, No. 2, 633–644 (2015; Zbl 1329.53048)].
MSC:
53C17 | Sub-Riemannian geometry |
30L05 | Geometric embeddings of metric spaces |
30L10 | Quasiconformal mappings in metric spaces |