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Generalized constant ratio surfaces in \(\mathbb E^3\). (English) Zbl 1310.53006

The constant angle surfaces in the Euclidean 3-space, i.e., oriented surfaces whose normals make a constant angle with some fixed direction, have been intensively studied in the last years. One possible generalization for this topic is to consider surfaces with the property that the angle between the normal and the position vector is constant, these surface are called constant slope surfaces and they were classified by the second author in [“From golden spirals to constant slope surfaces”, J. Math. Phys. 51, No. 7, Article ID 073507 (2010; doi:10.1063/1.3459064)]. A remarkable property of these special surfaces is that the tangential component of the position vector in the tangent plane of the surface is a principal direction.
In this work the authors extend the concept of constant slope surfaces by using this property and call them generalized constant ratio surfaces (GCR surfaces) in order to point out the connection with the constant ratio hypersurfaces in the Euclidean \(n\)-space defined by B.-Y. Chen in [Soochow J. Math. 27, No. 4, 353–362 (2001; Zbl 1007.53006)]. They obtain interesting results about flat and constant mean curvature GCR surfaces and give some pictures using MATLAB.

MSC:

53A05 Surfaces in Euclidean and related spaces
53B25 Local submanifolds
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)

Citations:

Zbl 1007.53006

Software:

Matlab
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] K.N. Boyadzhiev. Equiangular Surfaces, Self-Similar Surfaces, and the Geometry of Seashells. Coll. Math. J., 38(4) (2007), 265-271. · Zbl 1245.53009
[2] P. Cermelli and A.J. Di Scala. Constant-angle surfaces in liquid crystals. Philos. Magazine, 87 (2007), 1871-1888.
[3] B.-Y. Chen. Geometry of Submanifolds.Marcel Dekker, New York (1973). · Zbl 0262.53036
[4] B.-Y. Chen. Constant-ratio hypersurfaces. Soochow J. Math., 27(4) (2001), 353-362. · Zbl 1007.53006
[5] B.-Y. Chen. Geometry of position functions of Riemannian submanifolds in pseudo-Euclidean space. J. Geom., 74 (2002), 61-77. · Zbl 1031.53043
[6] B.-Y. Chen. Convolution of Riemannian manifolds and its applications. Bull. Austral. Math. Soc., 66 (2002), 177-191. · Zbl 1041.53012
[7] B.-Y. Chen. More on Convolution of Riemannian Manifolds.Beiträge zur Algebra und Geometrie, 44(1) (2003), 9-24. · Zbl 1031.53099
[8] B.-Y. Chen. When Does the Position Vector of a Space Curve Always Lie in Its Rectifying Plane? Amer. Math. Monthly, 110(2) (2003), 147-152. · Zbl 1035.53003
[9] B.-Y. Chen. Constant-ratio space-like submanifolds in pseudo-Euclidean space. Houston J. Math., 29(2) (2003), 281-294. · Zbl 1044.53040
[10] F. Dillen, J. Fastenakels, J. Van der Veken and L. Vrancken. Constant angle surfaces in \(\mathbb{S}^2 \times \mathbb{R} \) × ℝ. Monatsh. Math., 152(2) (2007), 89-96. · Zbl 1140.53006
[11] F. Dillen, J. Fastenakels and J. Van der Veken. Surfaces in \(\mathbb{S}^2 \times \mathbb{R} \) × ℝ with a canonical principal direction. Ann. Global Anal. Geom., 35(4) (2009), 381-396. · Zbl 1176.53031
[12] F. Dillen and M.I. Munteanu. Constant angle surfaces in ℍ2 × ℝ. Bull. Braz. Math. Soc., 40(1) (2009), 85-97. · Zbl 1173.53012
[13] F. Dillen, M.I. Munteanu and A.I. Nistor. Canonical coordinates and principal directions for surfaces in ℍ2 × ℝ. Taiwanese J. Math., 15(5) (2011), 2265-2289. · Zbl 1241.53010
[14] F. Dillen, M.I. Munteanu, J. Van der Veken and L. Vrancken. Constant angle surfaces in a warped product. Balkan J. Geom. Appl., 16(2) (2011), 35-47. · Zbl 1228.53021
[15] J. Eells. The surfaces of Delaunay. Math. Intelligencer, 9 (1997), 53-57. · Zbl 0605.53002
[16] J. Fastenakels, M.I. Munteanu and J. Van der Veken. Constant angle surfaces in the Heisenberg group. Acta Math. Sin. (Engl. Ser.), 27(4) (2011), 747-756. · Zbl 1218.53019
[17] Y. Fu and A.I. Nistor. Constant angle property and canonical principal directions for surfaces in \(\mathbb{M}^2 \)(c) × ℝ1. Mediter. J. Math., 10(2) (2013), 1035-1049. · Zbl 1277.53018
[18] Y. Fu and D. Yang. On constant slope spacelike surfaces in 3-dimensional Minkowski space. J. Math.Anal. Appl., 385(1) (2012), 208-220. · Zbl 1226.53012
[19] Y. Fu and X. Wang. Classification of timelike constant slope surfaces in 3-dimensional Minkowski space. Result. Math., 63(3-4) (2013), 1095-1108. · Zbl 1384.53017
[20] E. Garnica, O. Palmas and G. Ruiz-Hernández. Hypersurfaces with a canonical principal direction. Differ. Geom. Appl., 30(5) (2012), 382-391. · Zbl 1251.53012
[21] S. Haesen, A.I. Nistor and L. Verstraelen. On Growth and Form and Geometry. I. Kragujevac J. Math., 36(1) (2012), 5-25. · Zbl 1349.53032
[22] K. Kenmotsu. Surfaces of revolution with prescribed mean curvature. Tôhoku Math. J., 32 (1980), 147-153. · Zbl 0431.53005
[23] R. López and M.I. Munteanu. On the geometry of constant angle surfaces in Sol3. Kyushu J. Math., 65(2) (2011), 237-249. · Zbl 1236.53011
[24] M.I. Munteanu and A.I. Nistor. A new approach on constant angle surfaces in \(\mathbb{E}^3 \). Turkish J. Math., 33(1) (2009), 169-178.
[25] M.I. Munteanu. From golden spirals to constant slope surfaces. J. Math. Phys., 51(7) (2010), 073507. · Zbl 1311.14037
[26] M.I. Munteanu and A.I. Nistor.Complete classification of surfaces with a canonical principal direction in the Euclidean space \(\mathbb{E}^3 \). Cent. Eur. J. Math., 9(2) (2011), 378-389. · Zbl 1222.53009
[27] A.I. Nistor. A note on spacelike surfaces in Minkowski 3-space, Filomat., 27(5) (2013), 843-849. · Zbl 1451.53095
[28] D’Arcy Thompson. On growth and form. Cambridge Univ. Press (1948).
[29] R. Tojeiro. On a class of hypersurfaces in \(\mathbb{S}^n \) × ℝ and ℍn × ℝ. Bull. Braz. Math. Soc., 41(2) (2010), 199-209. · Zbl 1218.53061
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