# Complete classification of surfaces with a canonical principal direction in the Euclidean space $\mathbb{E}$ 3

Open Mathematics (2011)

- Volume: 9, Issue: 2, page 378-389
- ISSN: 2391-5455

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topMarian Munteanu, and Ana Nistor. "Complete classification of surfaces with a canonical principal direction in the Euclidean space \[ \mathbb {E} \] 3." Open Mathematics 9.2 (2011): 378-389. <http://eudml.org/doc/269808>.

@article{MarianMunteanu2011,

abstract = {In the present paper we classify all surfaces in \[ \mathbb \{E\} \]
3 with a canonical principal direction. Examples of this type of surfaces are constructed. We prove that the only minimal surface with a canonical principal direction in the Euclidean space \[ \mathbb \{E\} \]
3 is the catenoid.},

author = {Marian Munteanu, Ana Nistor},

journal = {Open Mathematics},

keywords = {Canonical coordinates; Minimal surface; Euclidean 3-space; canonical coordinates; minimal surface},

language = {eng},

number = {2},

pages = {378-389},

title = {Complete classification of surfaces with a canonical principal direction in the Euclidean space \[ \mathbb \{E\} \] 3},

url = {http://eudml.org/doc/269808},

volume = {9},

year = {2011},

}

TY - JOUR

AU - Marian Munteanu

AU - Ana Nistor

TI - Complete classification of surfaces with a canonical principal direction in the Euclidean space \[ \mathbb {E} \] 3

JO - Open Mathematics

PY - 2011

VL - 9

IS - 2

SP - 378

EP - 389

AB - In the present paper we classify all surfaces in \[ \mathbb {E} \]
3 with a canonical principal direction. Examples of this type of surfaces are constructed. We prove that the only minimal surface with a canonical principal direction in the Euclidean space \[ \mathbb {E} \]
3 is the catenoid.

LA - eng

KW - Canonical coordinates; Minimal surface; Euclidean 3-space; canonical coordinates; minimal surface

UR - http://eudml.org/doc/269808

ER -

## References

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- [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 Zbl1176.53031
- [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 Zbl1241.53010
- [6] Morita S., Geometry of Differential Forms, Transl. Math. Monogr., 201, American Mathematical Society, Providence, 2001 Zbl0987.58002
- [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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