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Singularities of affine equidistants: extrinsic geometry of surfaces in 4-space

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Abstract

For a generic embedding of a smooth closed surface M into R4, the subset of R4 which is the affine λ−equidistant of M appears as the discriminant set of a stable mapping M × M → R4, hence their stable singularities are A k , k = 2, 3, 4, and C 2,2 ±. In this paper, we characterize these stable singularities of λ−equidistants in terms of the bi-local extrinsic geometry of the surface, leading to a geometrical study of the set of weakly parallel points on M.

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Authors and Affiliations

  1. Faculty of Mathematics and Information Science, Warsaw University of Technology, ul. Koszykowa 75, 00-662, Warszawa, Poland

    W. Domitrz & S. Janeczko

  2. Institute of Mathematics, Polish Academy of Sciences, ul. Sniadeckich 8, 00-956, Warsaw, Poland

    S. Janeczko

  3. Departamento de Matemática, ICMC, Universidade de São Paulo, 13560-970, São Carlos, SP, Brazil

    P. de M. Rios & M. A. S. Ruas

Authors
  1. W. Domitrz
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  2. S. Janeczko
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  3. P. de M. Rios
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  4. M. A. S. Ruas
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Corresponding author

Correspondence to M. A. S. Ruas.

Additional information

W. Domitrz and S. Janeczko were partially supported by NCN grant no. DEC-2013/11/B/ST1/03080.

P. de M. Rios was partially supported by FAPESP grants no. 2013/04630-9 and 2015/02029-1.

M.A.S. Ruas was partially supported by FAPESP grant no. 2014/00304-2 and CNPq grant no. 305651/2011-0.

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Domitrz, W., Janeczko, S., de M. Rios, P. et al. Singularities of affine equidistants: extrinsic geometry of surfaces in 4-space. Bull Braz Math Soc, New Series 47, 1155–1179 (2016). https://doi.org/10.1007/s00574-016-0208-0

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  • DOI: https://doi.org/10.1007/s00574-016-0208-0

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