BI-LIPSCHITZ CONTACT INVARIANCE OF RANK

Authors

  • Nguyen Xuan Viet Nhan Basque Center for Applied Mathematics, Bilbao, Spain

DOI:

https://doi.org/10.37569/DalatUniversity.12.2.886(2022)

Keywords:

Bi-Lipschitz contact invariant, Rank, Thom-Boardman symbol.

Abstract

We address the question of whether the Thom-Boardman symbol of a map germ is an invariant with respect to bi-Lipschitz right equivalence. We give a counterexample showing that in general the answer is negative. We prove that the rank of a map germ is a bi-Lipschitz contact invariant. Consequently, the first Thom-Boardman symbol and its length are bi-Lipschitz contact invariants.

Downloads

Download data is not yet available.

References

Birbrair, L., Costa, J. C. F., Fernandes, A., & Ruas, M. A. S. (2007). K-bi-Lipschitz equivalence of real function-germs. Proceedings of the American Mathematical Society, 135(4), 1089-1095. https://doi.org/10.1090/S0002-9939-06-08566-2

Birbrair, L., & Mendes, R. (2018). Lipschitz contact equivalence and real analytic functions. arXiv:1801.05842v1. https://doi.org/10.48550/arXiv.1801.05842

Bivià-Ausina, C., & Fukui, T. (2017). Invariants for bi-Lipschitz equivalence of ideals. The Quarterly Journal of Mathematics, 68(3), 791-815. https://doi.org/10.1093/qmath/hax002

Boardman, J. M. (1967). Singularities of differentiable maps. Publications Mathématiques de l'I.H.É.S., 33, 21-57. https://doi.org/10.1007/BF02684585

Gibson, C. G. (1979). Singular points of smooth mappings (Research notes in mathematics, Vol. 25). Pitman.

Nguyen, N., Ruas, M. A. S., & Trivedi, S. (2020). Classification of Lipschitz simple function germs. Proceedings of the London Mathematical Society, 121(1), 51-82. https://doi.org/10.1112/plms.12310

Pham, T.-S. & Bui, N. T. N., (2019). Invariants of the bi-Lipschitz contact equivalence of continuous definable function germs. arXiv:1901.04479v1. https://doi.org/10.48550/arXiv.1901.04479

Ruas, M. A. S., & Valette, G. (2011). Co and bi-Lipschitz K-equivalence of mappings. Mathematische Zeitschrift, 269(12), 293-308. https://doi.org/10.1007/s00209-010-0728-z

Sampaio, J. E. (2016). Bi-Lipschitz homeomorphic subanalytic sets have bi-Lipschitz homeomorphic tangent cones. Selecta Mathematica: New Series, 22(2), 553-559. https://doi.org/10.1007/s00029-015-0195-9

Thom, R. (1962). La stabilité topologique des applications polynomiales. L'Enseignement Mathématique, 8, 24-33.

Downloads

Published

28-01-2022

Volume and Issues

Section

Natural Sciences and Technology

How to Cite

Nguyen, X. V. N. (2022). BI-LIPSCHITZ CONTACT INVARIANCE OF RANK. Dalat University Journal of Science, 12(2), 40-47. https://doi.org/10.37569/DalatUniversity.12.2.886(2022)