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Contact Cylindrical Surfaces and a Projection of a Surface Around a Parabolic Point

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Abstract

We investigate differential geometric properties of a parabolic point of a surface in the Euclidean three space. We introduce the contact cylindrical surface which is a cylindrical surface having a degenerate contact type with the original surface at a parabolic point. Furthermore, we show that such a contact property gives a characterization to the \(\mathcal {A}\)-singularity of the orthogonal projection of a surface from the asymptotic direction.

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References

  1. Arnold, V.I.: Critical points of functions on a manifold with boundary, the simple Lie groups \(B_{k}\), \(C_{k}\), \(F_{4}\) and singularities of evolutes. Russ. Math. Surv. 33, 99 (1978)

    Article  Google Scholar 

  2. Banchoff, T., Gaffney, T., McCrory, C.: Cusps of Gauss mappings, Research Notes in Math., vol. 55. Pitman, Boston–London (1982)

    Google Scholar 

  3. Bruce, J.W.: Projections and reflections of generic surfaces in \({\mathbb{R} }^{3}\). Math. Scand. 54(2), 262–278 (1984)

    Article  MathSciNet  Google Scholar 

  4. Bruce, J., Tari, F.: Frame and direction mappings for surfaces in \({\mathbb{R} }^{3}\). Proc. Roy. Soc. Edinb. Sect. A Math. 149, 795–830 (2019)

    Article  Google Scholar 

  5. Gaffney, T.: The structure of \(T{\cal{A}}(f)\), classification and an application to differential geometry. In: Singularities, Part I, Proc. Sympos. in Pure Math., vol. 40, pp. 409-427. Amer. Math. Soc. (1983)

  6. Fukui, T., Hasegawa, M.: Singularities of parallel surfaces. Tohoku Math. J. 64, 387–408 (2012)

    Article  MathSciNet  Google Scholar 

  7. Fukui, T., Hasegawa, M., Nakagawa, K.: Contact of a regular surface in Euclidean \(3\)-space with cylinders and cubic binary differential equations. J. Math. Soc. Jpn. 69(2), 819–847 (2017)

    Article  MathSciNet  Google Scholar 

  8. Honda, S.: Versality of central projections of regular surfaces. Topol. Appl. 313, 107982 (2021). https://doi.org/10.1016/j.topol.2021.107982

    Article  MathSciNet  Google Scholar 

  9. Izumiya, S., Romero-Fuster, M.C., Ruas, M.A.S., Tari, F.: Differential Geometry from a Singularity Theory Viewpoint. World Scientific Pub. Co Inc., Singapore (2015)

    Book  Google Scholar 

  10. Kabata, Y.: Recognition of plane-to-plane map-germs. Topol. Appl. 202, 216–238 (2016)

    Article  MathSciNet  Google Scholar 

  11. Martins, L.F., Nuño-Ballesteros, J.J.: Contact properties of surfaces in \({\mathbb{R} }^{3}\) with corank 1 singularities. Tohoku Math. J. 67, 105–124 (2015)

    Article  MathSciNet  Google Scholar 

  12. Montaldi, J.: On contact between submanifolds. Mich. Math. J. 33, 195–199 (1986)

    Article  MathSciNet  Google Scholar 

  13. Platonova, O.A.: Projections of smooth surfaces. J. Sov. Math. 35(6), 2796–2808 (1986). [Tr. Sem. I. G. Petvoskii 10 (1984), 135-149 in Russian]

  14. Rieger, J.H.: Families of maps from the plane to the plane. J. Lond. Math. Soc. (2) 36(2), 351–369 (1987)

    Article  MathSciNet  Google Scholar 

  15. Whitney, H.: On singularities of mappings of Euclidean spaces. I. Mappings of the plane into the plane. Ann. Math. (2) 62, 374–410 (1955)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to thank the referee for helpful comments to improve the original manuscript. The authors also thank Toshizumi Fukui and Farid Tari for helpful discussions. This work is partially supported by JSPS KAKENHI Grant numbers 21K03230, 20K14312, 18K03301, Japan-Brazil bilateral project JPJSBP1 20190103 and Japan-Russia Research Cooperative Program 120194801.

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Correspondence to Yutaro Kabata.

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Hasegawa, M., Kabata, Y. & Saji, K. Contact Cylindrical Surfaces and a Projection of a Surface Around a Parabolic Point. Arnold Math J. (2024). https://doi.org/10.1007/s40598-024-00251-y

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  • DOI: https://doi.org/10.1007/s40598-024-00251-y

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