Abstract
We present a criterion of local normal embedding of a semialgebraic (or definable in a polynomially bounded o-minimal structure) germ contained in \(\mathbb R^n\) in terms of orders of contact of arcs. Namely, we prove that a semialgebraic germ is normally embedded if and only if for any pair of arcs, coming to this point the inner order of contact is equal to the outer order of contact.
L. Birbrair—partially supported by CAPES-COFECUB and by CNPq-Brazil, grants no. 302655/2014-0.
R. Mendes—partially supported by Capes.
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Birbrair, L., Brasselet, J.-P.: Metric homology. Commun. Pure Appl. Math. LIII, 1434–1447 (2000)
Birbrair, L., Fernandes, A., Le, D.T., Sampaio, J.E.: Lipschitz regular complex algebraic sets are smooth. Proc. Amer. Math. Soc. 144(3), 983–987 (2016)
Birbrair, L., Fernandes, A., Neumann, W.D.: On normal embedding of complex algebraic surfaces. Real and complex singularities. Lond. Math. Soc. Lecture Note Ser. (380), 17–22. Cambridge University Press, Cambridge (2010)
Birbrair, L., Mostowski, T.: Normal embeddings of semialgebraic sets. Michigan Math. J. 47, 125–132 (2000)
Bröcker, L.: Families of semialgebraic sets and limits. In: Real Algebraic Geometry, Lect. Notes. Math. (1524) pp. 145–162. Springer, Berlin (1992)
Birbrair, L., Fernandes, A.: Metric theory of semialgebraic curves. Rev. Mat. Complut. 13(2), 369–382 (2000)
Katz, K., Katz, M., Kerner, D., Liokumovich, Y., Solomon, J.: Determinantal variety and bilipschitz equivalence. J. Topol. Anal. 10, 27 (2018)
Kurdyka, K., Orro, P.: Distance géodésique sur un sous-analytique. Revista Mat. Univ. Complutense de Madrid 10, 173–182 (1997)
Neumann, W.D., Pedersen, H.M., Pichon, A.: Minimal surface singularities are Lipschitz normally embedded (2015). arXiv:1503.03301
Pedersen, H.M.: Maria Aparecida Soares Ruas. Lipschitz Normal Embeddings and Determinantal Singularities (2016). arXiv:1607.07746
Valette, G.: Vanishing homology. Selecta Math. (N.S.) 16(2), 267–296 (2010)
Acknowledgements
We would like to thank Alexandre Fernandes, Edson Sampaio, Anne Pichon and Walter Neumann for useful discussions. We would like also to thank the anonymous referee for his patience and extremely useful comments and corrections.
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Birbrair, L., Mendes, R. (2018). Arc Criterion of Normal Embedding. In: Araújo dos Santos, R., Menegon Neto, A., Mond, D., Saia, M., Snoussi, J. (eds) Singularities and Foliations. Geometry, Topology and Applications. NBMS BMMS 2015 2015. Springer Proceedings in Mathematics & Statistics, vol 222. Springer, Cham. https://doi.org/10.1007/978-3-319-73639-6_19
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DOI: https://doi.org/10.1007/978-3-319-73639-6_19
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