Illinois Journal of Mathematics

Helix, shadow boundary and minimal submanifolds

Gabriel Ruiz-Hernández

Full-text: Open access


We give conditions for the shadow boundary of a Riemannian submanifold $M$ to be regular. We prove that a helix hypersurface is ruled. By studying some relations between these natural submanifolds, we show that a minimal helix shadow boundary hypersurface of $M$ is totally geodesic in $M$.

Article information

Illinois J. Math., Volume 52, Number 4 (2008), 1385-1397.

First available in Project Euclid: 18 November 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C40: Global submanifolds [See also 53B25] 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]


Ruiz-Hernández, Gabriel. Helix, shadow boundary and minimal submanifolds. Illinois J. Math. 52 (2008), no. 4, 1385--1397. doi:10.1215/ijm/1258554368.

Export citation


  • M. Barros, General helices and a theorem of Lancret, Proc. Amer. Math. Soc. 125 (1997), 1503–1509.
  • B. Y. Chen, Geometry of submanifolds, Pure and Applied Mathematics, Vol. 22, 1973.
  • A. Besse, Einstein manifolds, Springer, Berlin, 1987.
  • J. Choe, Index, vision number and stability of complete minimal surfaces, Arch. Rational Mech. Anal. 109 (1990), 195–212.
  • A. Di Scala and G. Ruiz-Hernández, Helix submanifolds of Euclidean space, Monatsh. Math., available at
  • F. Dillen and M. I. Munteanu, Constant angle surfaces in $\mathbb{H}^2 \times \mathbb{R}$, available at arXiv:0705.3744, 2007.
  • F. Dillen, J. Fastenakels, J. Van der Veken and L. Vrancken, Constant angle surfaces in $\mathbb{S}^2\times \mathbb{R}$, Monatsh. Math. 152 (2007), 89–96.
  • N. Ekmekci and K. Ilarslan, Null general helices and submanifolds, Bol. Soc. Mat. Mexicana (3) 9 (2003), 279–286.
  • M. Ghomi, Shadows and convexity of surfaces, Ann. of Math. 155 (2002), 281–293.
  • V. Guillemin and A. Pollack, Diferential Topology, Prentice-Hall, Englewood Cliffs, NJ, 1974.
  • T. Hasanis, A. Savas-Halilaj and T. Vlachos, Minimal hypersurfaces with zero Gauss–Kronecker curvature, Illinois J. Math. 49 (2005), 523–529.
  • K. Nomizu and T. Sasaki, Affine differential geometry, Cambridge Univ. Press, 1994.
  • G. Ruiz-Hernández, Totally geodesic shadow boundary submanifolds and a characterization for $\mathbb{S}^n$, Arch. Math. 90 (2008), 374–384.
  • A. Schwenk, Affinsphären mit ebenen schattengrenzen, Global differential geometry and global analysis 1984 (Berlin, 1984), 296–315, Lecture Notes in Math., Vol. 1156, Springer, Berlin, 1985.
  • D. J. Welsh, On the existence of complete parallel vector fields, Proc. Amer. Math. Soc. 97 (1986), 311–314.
  • D. J. Welsh, Manifolds that admit parallel vector fields, Illinois J. Math. 30 (1986), 9–18.