Abstract
A submanifold is said to be tangentially biharmonic if the bitension field of the isometric immersion that defines the submanifold has vanishing tangential component. The purpose of this paper is to prove that a surface in Euclidean 3-space has tangentially biharmonic normal bundle if and only if it is either minimal, a part of a round sphere, or a part of a circular cylinder.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Balmuş A., Montaldo S., Oniciuc C.: Classification results for biharmonic submanifolds in spheres. Israel J. Math. 168, 201–220 (2008)
Chen B.Y.: Some open problems and conjectures on submanifolds of finite type. Soochow J. Math. 17, 169–188 (1991)
Eells J., Sampson J.H.: Harmonic mappings of Riemannian manifolds. Amer. J. Math. 86, 109–160 (1964)
Harvey R., Lawson H.B.: Calibrated geometries. Acta Math. 148, 47–157 (1982)
Jiang G.Y.: 2-harmonic maps and their first and second variational formulas (Chinese). Chinese Ann. Math. A 7, 389–402 (1986)
Sakaki M.: Hamiltonian stationary normal bundles of surfaces in R 3. Proc. Amer. Math. Soc. 127, 1509–1515 (1999)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sasahara, T. Surfaces in Euclidean 3-space whose normal bundles are tangentially biharmonic. Arch. Math. 99, 281–287 (2012). https://doi.org/10.1007/s00013-012-0410-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-012-0410-2