Proceedings of the Japan Academy, Series A, Mathematical Sciences

Complete flat fronts as hypersurfaces in Euclidean space

Atsufumi Honda

Full-text: Open access


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

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

First available in Project Euclid: 28 February 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

Export citation


  • R. L. Bishop, There is more than one way to frame a curve, Amer. Math. Monthly 82 (1975), 246–251.
  • P. Hartman and L. Nirenberg, On spherical image maps whose Jacobians do not change sign, Amer. J. Math. 81 (1959), 901–920.
  • M. Hasegawa, A. Honda, K. Naokawa, K. Saji, M. Umehara and K. Yamada, Intrinsic properties of surfaces with singularities. Internat. J. Math. 26 (2015), no. 4, 1540008, 34 pp.
  • A. Honda, Isometric immersions of the hyperbolic plane into the hyperbolic space, Tohoku Math. J. (2) 64 (2012), no. 2, 171–193.
  • A. Honda, Weakly complete wave fronts with one principal curvature constant, Kyushu J. Math. 70 (2016), no. 2, 217–226.
  • A. Honda, Isometric immersions with singularities between space forms of the same positive curvature, J. Geom. Anal. 27 (2017), no. 3, 2400–2417.
  • G. Ishikawa, Developable of a curve and determinacy relative to osculation-type, Quart. J. Math. Oxford Ser. (2) 46 (1995), no. 184, 437–451.
  • W. S. Massey, Surfaces of Gaussian curvature zero in Euclidean 3-space, Tôhoku Math. J. (2) 14 (1962), 73–79.
  • S. Murata and M. Umehara, Flat surfaces with singularities in Euclidean 3-space, J. Differential Geom. 82 (2009), no. 2, 279–316.
  • K. Naokawa, Singularities of the asymptotic completion of developable Möbius strips, Osaka J. Math. 50 (2013), no. 2, 425–437.
  • B. O'Neill and E. Stiel, Isometric immersions of constant curvature manifolds, Michigan Math. J. 10 (1963), 335–339.
  • K. Saji, M. Umehara and K. Yamada, The geometry of fronts, Ann. of Math. (2) 169 (2009), no. 2, 491–529.
  • K. Saji, M. Umehara and K. Yamada, $A_{2}$-singularities of hypersurfaces with non-negative sectional curvature in Euclidean space, Kodai Math. J. 34 (2011), no. 3, 390–409.
  • K. Saji, M. Umehara and K. Yamada, Coherent tangent bundles and Gauss-Bonnet formulas for wave fronts, J. Geom. Anal. 22 (2012), no. 2, 383–409.
  • K. Saji, M. Umehara and K. Yamada, An index formula for a bundle homomorphism of the tangent bundle into a vector bundle of the same rank, and its applications, J. Math. Soc. Japan 69 (2017), no. 1, 417–457.