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Geometry of surfaces in \(\mathbb R^5\) through projections and normal sections

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

We study the geometry of surfaces in \(\mathbb {R}^5\) by relating it to the geometry of regular and singular surfaces in \(\mathbb {R}^4\) obtained by orthogonal projections. In particular, we obtain relations between asymptotic directions, which are not second order geometry for surfaces in \(\mathbb {R}^5\) but are in \(\mathbb {R}^4\). We also relate the umbilic curvatures of each type of surface and their contact with spheres. We then consider the surfaces as normal sections of 3-manifolds in \(\mathbb {R}^6\) and again relate asymptotic directions and contact with spheres by defining an appropriate umbilic curvature for 3-manifolds.

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Acknowledgements

The authors thank their families for understanding, since this work was developed mostly during confinement. The authors also thank Farid Tari for useful conversations. The first author would like to express his gratitude to the Universitat de València, where this work was partially carried out, for its hospitality.

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Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal de Santa Catarina-UFSC, 89036-004, Blumenau, SC, Brazil

    J. L. Deolindo-Silva

  2. Departament de Matemàtiques, Universitat de València, Campus de Burjassot, 46100, Burjassot, Spain

    R. Oset Sinha

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  1. J. L. Deolindo-Silva
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  2. R. Oset Sinha
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Correspondence to R. Oset Sinha.

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J. L. D. Silva: Work of J. L. Deolindo-Silva partially supported by CAPES/JSPS Grant no. 88887.357189/2019–00. R. O. Sinha: Work of R. Oset Sinha partially supported by MICINN Grant PGC2018-094889-B-I00 and GVA Grant AICO 2019/024.

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Deolindo-Silva, J.L., Sinha, R.O. Geometry of surfaces in \(\mathbb R^5\) through projections and normal sections. RACSAM 115, 81 (2021). https://doi.org/10.1007/s13398-021-01019-1

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  • DOI: https://doi.org/10.1007/s13398-021-01019-1

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