Abstract
Any generalized distance-squared mapping of equidimensional case has singularities, and their singularity types are wrapped into mystery in higher dimensional cases. Any generalized distance-squared mapping of equidimensional case is not injective. Nevertheless, in this paper, it is shown that the immersed property or the injective property of a mapping is preserved by composing a generic generalized distance-squared mapping of equidimensional case.
The first author is Research Fellow DC1 of Japan Society for the Promotion of Science.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Abraham, R.: Transversality in manifolds of mappings. Bull. Am. Math. Soc. 69, 470–474 (1963)
Bruce, J.W., Giblin, P.J.: Curves and Singularities, 2nd edn. Cambridge University Press, Cambridge (1992)
Golubitsky, M., Guillemin, V.: Stable Mappings and Their Singularities. Graduate texts in mathematics, vol. 14. Springer, New York (1973)
Ichiki, S., Nishimura, T.: Distance-squared mappings. Topol. Appl. 160, 1005–1016 (2013)
Ichiki, S., Nishimura, T.: Recognizable classification of Lorentzian distance-squared mappings. J. Geom. Phys. 81, 62–71 (2014)
Ichiki, S., Nishimura, T.: Generalized distance-squared mappings of \({\mathbb{R}}{^{n+1}}\) into \({\mathbb{R}}^{2n+1}\) Contemporary Mathematics. Am. Math. Soc. Providence RI, 675, 121–132 (2016)
Ichiki, S., Nishimura, T., Sinha, R.O., Ruas, M.A.S.: Generalized distance-squared mappings of the plane into the plane. Adv. Geom. 16, 189–198 (2016)
Mather, J.N.: Generic projections. Ann. Math. 98(2), 226–245 (1973)
Spanier, E.H.: Algebraic Topology. McGraw-Hill Book Company, New York (1966)
Acknowledgements
The authors are most grateful to the anonymous reviewer for his/her careful reading of the first manuscript of this paper and invaluable suggestions. The first author is supported by JSPS KAKENHI Grant Number 16J06911.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Ichiki, S., Nishimura, T. (2018). Preservation of Immersed or Injective Properties by Composing Generic Generalized Distance-Squared Mappings. 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_18
Download citation
DOI: https://doi.org/10.1007/978-3-319-73639-6_18
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-73638-9
Online ISBN: 978-3-319-73639-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)