## Proceedings of the Japan Academy, Series A, Mathematical Sciences

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

### Complete flat fronts as hypersurfaces in Euclidean space

#### 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