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Singularities of discrete improper indefinite affine spheres

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Abstract

In this paper we consider discrete improper affine spheres based on asymptotic nets. In this context, we distinguish the discrete edges and vertices that must be considered singular. The singular edges can be considered as discrete cuspidal edges, while some of the singular vertices can be considered as discrete swallowtails. The classification of singularities of discrete nets is quite a difficult task, and our results can be seen as a first step in this direction. We also prove some characterizations of ruled discrete improper affine spheres which are analogous to the smooth case.

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Acknowledgements

The authors are thankful to CAPES and CNPq for financial support during the preparation of this paper. Both authors thank Pontifical Catholic University of Rio de Janeiro and the first author also thanks Colégio Pedro II.

Funding

Both authors had the support of CNPq (Conselho Nacional de Desenvolvimento Científico e Teconológico - Brazil) and CAPES (Coordenação de Aperefeiçoamento de Pessoal de Nível Superior - Brazil).

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Authors and Affiliations

  1. Colégio Pedro II, Rio de Janeiro, Brazil

    Anderson Reis de Vargas

  2. Pontifical Catholic University of Rio de Janeiro, Rio de Janeiro, Brazil

    Marcos Craizer

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  1. Anderson Reis de Vargas
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  2. Marcos Craizer
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Both authors wrote the article, prepared the figures and reviewed the article.

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Correspondence to Marcos Craizer.

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de Vargas, A.R., Craizer, M. Singularities of discrete improper indefinite affine spheres. J. Geom. 114, 34 (2023). https://doi.org/10.1007/s00022-023-00698-y

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  • DOI: https://doi.org/10.1007/s00022-023-00698-y

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