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Complete classification of surfaces with a canonical principal direction in the Euclidean space \(\mathbb{E}^{3}\). (English) Zbl 1222.53009
A constant angle surface is defined as a surface whose unit normal forms a constant angle with a fixed direction \(\vec{k}\). The projection of its fixed direction \(\vec{k}\) on the tangent plane (denoted by \(U\)) is a principal direction with vanishing corresponding principal curvature.
The study of constant angle surfaces in \(\mathbb{R}^3\) can be generalized for surfaces whose angle function is no longer constant, provided that certain properties are preserved. For example, when \(U\) is a principal direction with corresponding principal curvature different from \(0\), these surfaces are called surfaces with a canonical principal direction. In the paper, the authors classify them, construct some examples and prove that the only minimal surface with a canonical principal direction in the Euclidean space \(\mathbb{R}^3\) is the catenoid.

53A05 Surfaces in Euclidean and related spaces
Full Text: DOI arXiv
[1] Blair D.E., On a generalization of the catenoid, Canad. J. Math., 1975, 27, 231-236 http://dx.doi.org/10.4153/CJM-1975-028-8 · Zbl 0307.53003
[2] do Carmo M.P., Dajczer M., Rotation hypersurfaces in spaces of constant curvature, Trans. Amer. Math. Soc, 1983, 277(2), 685-709 http://dx.doi.org/10.2307/1999231 · Zbl 0518.53059
[3] Cermelli P., Di Scala A.J., Constant-angle surfaces in liquid crystals, Philosophical Magazine, 2007, 87(12), 1871-1888 http://dx.doi.org/10.1080/14786430601110364
[4] Dillen F., Fastenakels J., Van der Veken J., Surfaces in \( \mathbb{S} \) 2×ℝ with a canonical principal direction, Ann. Global Anal. Geom., 2009, 35(4), 381-396 http://dx.doi.org/10.1007/s10455-008-9140-x · Zbl 1176.53031
[5] Dillen F., Munteanu M.I., Nistor A.I., Canonical coordinates and principal directions for surfaces in \( \mathbb{H} \) 2×ℝ, Taiwanese J. Math., 2011 (in press), preprint available at http://arxiv.org/abs/0910.2135 · Zbl 1241.53010
[6] Morita S., Geometry of Differential Forms, Transl. Math. Monogr., 201, American Mathematical Society, Providence, 2001 · Zbl 0987.58002
[7] Munteanu M.I., Nistor A.-I., A new approach on constant angle surfaces in \( \mathbb{H} \) 3, Turkish J. Math., 2009, 33(2), 169-178
[8] O’Neill B., Elementary Differential Geometry, 2nd ed. revised, Academic Press, Amsterdam, 2006
[9] Tojeiro R., On a class of hypersurfaces in \( \mathbb{S} \) n×ℝ and ℍn×ℝ, Illinois J. Math., 2011 (in press), preprint available at http://arxiv.org.abs/0909.2265
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