Proceedings of the Japan Academy, Series A, Mathematical Sciences

Complete flat fronts as hypersurfaces in Euclidean space

Atsufumi Honda

Full-text: Open access

Abstract

By Hartman–Nirenberg’s theorem, any complete flat hypersurface in Euclidean space muast be a cylinder over a plane curve. However, if we admit some singularities, there are many non-trivial examples. {\it Flat fronts} are flat hypersurfaces with admissible singularities. Murata–Umehara gave a representation formula for complete flat fronts with non-empty singular set in Euclidean 3-space, and proved the four vertex type theorem. In this paper, we prove that, unlike the case of $n=2$, there do not exist any complete flat fronts with non-empty singular set in Euclidean $(n+1)$-space $(n\geq 3)$.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 94, Number 3 (2018), 25-30.

Dates
First available in Project Euclid: 28 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.pja/1519808414

Digital Object Identifier
doi:10.3792/pjaa.94.25

Mathematical Reviews number (MathSciNet)
MR3769187

Zentralblatt MATH identifier
06916912

Subjects
Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]
Secondary: 57R45: Singularities of differentiable mappings

Keywords
Flat hypersurface flat front Hartman–Nirenberg’s theorem singular point wave front coherent tangent bundle

Citation

Honda, Atsufumi. Complete flat fronts as hypersurfaces in Euclidean space. Proc. Japan Acad. Ser. A Math. Sci. 94 (2018), no. 3, 25--30. doi:10.3792/pjaa.94.25. https://projecteuclid.org/euclid.pja/1519808414


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