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On the geometry of folded cuspidal edges

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Abstract

We study the geometry of cuspidal \(S_k\) singularities in \({\mathbb {R}}^3\) obtained by folding generically a cuspidal edge. In particular we study the geometry of the cuspidal cross-cap M, i.e. the cuspidal \(S_0\) singularity. We study geometrical invariants associated to M and show that they determine it up to order 5. We then study the flat geometry (contact with planes) of a generic cuspidal cross-cap by classifying submersions which preserve it and relate the singularities of the resulting height functions with the geometric invariants.

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Acknowledgements

The authors would like to thank Farid Tari for helpful discussions and the referees for valuable suggestions which improved the scope and presentation of the results.

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Authors and Affiliations

  1. Departament de Matemàtiques, Universitat de València, c/ Dr Moliner no 50, 46100, Burjassot, Valencia, Spain

    Raúl Oset Sinha

  2. Department of Mathematics, Kobe University, Rokko 1-1, Nada, Kobe, 657-8501, Japan

    Kentaro Saji

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  1. Raúl Oset Sinha
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  2. Kentaro Saji
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Correspondence to Raúl Oset Sinha.

Additional information

Raúl Oset Sinha: Partially supported by DGICYT Grant MTM2015–64013–P.

Kentaro Saji: Supported by JSPS KAKENHI Grant Number JP26400087.

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Oset Sinha, R., Saji, K. On the geometry of folded cuspidal edges. Rev Mat Complut 31, 627–650 (2018). https://doi.org/10.1007/s13163-018-0257-6

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  • DOI: https://doi.org/10.1007/s13163-018-0257-6

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