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Hypersurfaces with a canonical principal direction. (English) Zbl 1251.53012
Summary: Given a vector field \(X\) in a Riemannian manifold, a hypersurface is said to have a canonical principal direction relative to \(X\) if the projection of \(X\) onto the tangent space of the hypersurface gives a principal direction. We give different ways for building these hypersurfaces, as well as a number of useful characterizations. In particular, we relate them with transnormal functions and eikonal equations. With the additional condition of having constant mean curvature (CMC) we obtain a characterization of the canonical principal direction surfaces in Euclidean space as Delaunay surfaces. We also prove that CMC constant angle hypersurfaces in a product \(\mathbb R\times N\) are either totally geodesic or cylinders.

MSC:
53B25 Local submanifolds
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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[1] Barros, Manuel, General helices and a theorem of lancret, Proc. amer. math. soc., 125, 5, 1503-1509, (1997) · Zbl 0876.53035
[2] Berndt, Jürgen; Console, Sergio; Olmos, Carlos, Submanifolds and holonomy, Chapman & Hall/CRC res. notes math., vol. 434, (2003), Chapman & Hall/CRC Boca Raton, FL · Zbl 1043.53044
[3] Cho, Jong Taek; Inoguchi, Jun-ichi; Lee, Ji-Eun, On slant curves in Sasakian 3-manifolds, Bull. aust. math. soc., 74, 3, 359-367, (2006) · Zbl 1106.53013
[4] Di Scala, Antonio; Cermelli, P., Constant-angle surfaces in liquid crystals, Philos. mag., 87, (2007)
[5] Di Scala, Antonio J., Weak helix submanifolds of Euclidean spaces, Abh. math. semin. univ. hambg., 79, 1, 37-46, (2009) · Zbl 1190.53006
[6] Di Scala, Antonio J.; Ruiz-Hernández, Gabriel, Higher codimensional Euclidean helix submanifolds, Kodai math. J., 33, 2, 192-210, (2010) · Zbl 1211.53008
[7] Dillen, Franki; Fastenakels, Johan; Van der Veken, Joeri, Surfaces in \(\mathbb{S}^2 \times \mathbb{R}\) with a canonical principal direction, Ann. global anal. geom., 35, 4, 381-396, (2009) · Zbl 1176.53031
[8] Dillen, Franki; Munteanu, Marian Ioan; Nistor, Ana-Irina, Canonical coordinates and principal directions for surfaces in \(\mathbb{H}^2 \times \mathbb{R}\), Taiwanese J. math., 15, 5, 2265-2289, (2011) · Zbl 1241.53010
[9] Dillen, Franki; Munteanu, Marian Ioan; Van der Veken, Joeri; Vrancken, Luc, Classification of constant angle surfaces in a warped product, Balkan J. geom. appl., 16, 2, 35-47, (2011) · Zbl 1228.53021
[10] Gallot, Sylvestre; Hulin, Dominique; Lafontaine, Jacques, Riemannian geometry, Universitext, (2004), Springer-Verlag Berlin · Zbl 1068.53001
[11] Eugenio Garnica, Oscar Palmas, Gabriel Ruiz-Hernández, Classification of constant angle hypersurfaces in warped products via eikonal functions, Bol. Soc. Mat. Mex., in press. · Zbl 1280.53015
[12] Inoguchi, Jun-ichi; Lee, Sungwook, Null curves in Minkowski 3-space, Int. electron. J. geom., 1, 2, 40-83, (2008) · Zbl 1166.53011
[13] Montiel, Sebastián, Unicity of constant mean curvature hypersurfaces in some Riemannian manifolds, Indiana univ. math. J., 48, 2, 711-748, (1999) · Zbl 0973.53048
[14] Munteanu, Marian Ioan, From Golden spirals to constant slope surfaces, J. math. phys., 51, 7, 073507, (2010), 9 pp · Zbl 1311.14037
[15] Munteanu, Marian Ioan; Nistor, Ana-Irina, A new approach on constant angle surfaces in \(\mathbb{E}^3\), Turkish J. math., 33, 2, 169-178, (2009) · Zbl 1175.53006
[16] Munteanu, Marian Ioan; Nistor, Ana Irina, Complete classification of surfaces with a canonical principal direction in the Euclidean space \(\mathbb{E}^3\), Cent. eur. J. math., 9, 2, 378-389, (2011) · Zbl 1222.53009
[17] Sakai, Takashi, On Riemannian manifolds admitting a function whose gradient is of constant norm, Kodai math. J., 19, 1, 39-51, (1996) · Zbl 0881.53035
[18] Tojeiro, Ruy, On a class of hypersurfaces in \(\mathbb{S}^n \times \mathbb{R}\) and \(\mathbb{H}^n \times \mathbb{R}\), Bull. braz. math. soc. (N.S.), 41, 2, 199-209, (2010) · Zbl 1218.53061
[19] Ming Wang, Qi, Isoparametric functions on Riemannian manifolds. I, Math. ann., 277, 4, 639-646, (1987) · Zbl 0638.53053
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