PreprintCC BY 4.0arXiv
On Strong Bi-Lipschitz Triviality of Deformations
Abstract
The Thom-Levine theorem is a classical method for proving the triviality of deformations by integrating suitable vector fields. In this note, we show that the converse of this criterion does not hold in the bi-Lipschitz category. More precisely, we give an example of a smooth one-parameter deformation of a real function germ in two variables which is bi-Lipschitz trivial but not strongly bi-Lipschitz trivial. The central germ has an isolated critical point, though it is not finitely determined.
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BibTeX
@misc{chakraborty2026strong,
title = {{On Strong Bi-Lipschitz Triviality of Deformations}},
author = {Debomita Chakraborty and Saurabh Trivedi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17034},
eprint = {2609.17034},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17034},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On Strong Bi-Lipschitz Triviality of Deformations
AU - Debomita Chakraborty
AU - Saurabh Trivedi
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17034
UR - https://arxiv.org/abs/2609.17034
AB - The Thom-Levine theorem is a classical method for proving the triviality of deformations by integrating suitable vector fields. In this note, we show that the converse of this criterion does not hold in the bi-Lipschitz category. More precisely, we give an example of a smooth one-parameter deformation of a real function germ in two variables which is bi-Lipschitz trivial but not strongly bi-Lipschitz trivial. The central germ has an isolated critical point, though it is not finitely determined.
ER -