# zbMATH — the first resource for mathematics

Constant angle property and canonical principal directions for surfaces in $$\mathbb M^2(c)\times\mathbb R_1$$. (English) Zbl 1277.53018
In recent years a lot of progress has been made in the study of surfaces in a product space of a constant curvature surface with a real line. Of course, by changing the sign of the metric on the line component, one obtains an example of a Lorentzian space. In the paper under review, the authors study similar questions for space-like and time-like surfaces in the Lorentzian space obtained in this way. In these spaces, the authors obtain a classification of constant angle surfaces and surfaces with a principal canonical direction. Most of the statements of the theorems and its proofs are quite similar to the Riemannian case.

##### MSC:
 53B25 Local submanifolds 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
Full Text:
##### References:
 [1] Albujer A.L., Alís L.J.: Calabi–Bernstein results for maximal surfaces in Lorentzian product spaces. J. Geom. Phys. 59, 620–631 (2009) · Zbl 1173.53025 · doi:10.1016/j.geomphys.2009.01.008 [2] A. L. Albujer and L. J. Alís, Parabolicity of maximal surfaces in Lorentzian product spaces, Math. Z. 267 (2011), 453–464. · Zbl 1218.53062 [3] Cermelli P., Di Scala A.J.: Constant–angle surfaces in liquid crystals. Philosophical Magazine 87, 1871–1888 (2007) · doi:10.1080/14786430601110364 [4] B. Y. Chen, Geometry of Submanifolds, Dekker, New York, 1973. · Zbl 0262.53036 [5] F. Dillen, J. Fastenakels, J. Van der Veken and L. Vrancken, Constant angle surfaces in $${{$$\backslash$$mathbb{S}\^{2} $$\backslash$$times $$\backslash$$mathbb{R}}}$$ , Monatsh. Math. 152 (2007), no. 2, 89–96. [6] F. Dillen, J. Fastenakels and J. Van der Veken, Surfaces in $${{$$\backslash$$mathbb{S}\^{2} $$\backslash$$times $$\backslash$$mathbb{R}}}$$ with a canonical principal direction, Ann. Global Anal. Geom. 35 (2009), no. 4, 381–396. · Zbl 1176.53031 [7] F. Dillen and M. I. Munteanu, Constant angle surfaces in $${{$$\backslash$$mathbb{H}\^{2} $$\backslash$$times $$\backslash$$mathbb{R}}}$$ , Bull. Braz. Math. Soc. (N.S.) 40 (2009), no. 1, 85–97. · Zbl 1173.53012 [8] F. Dillen, M. I. Munteanu and A. I. Nistor, Canonical coordinates and principal directions for surfaces in $${{$$\backslash$$mathbb{H}\^{2} $$\backslash$$times $$\backslash$$mathbb{R}}}$$ , Taiwanese J. Math. 15 (2011), no. 5, 2265–2289. · Zbl 1241.53010 [9] F. Dillen, M. I. Munteanu, J. van der Veken and L. Vrancken, Constant angle surfaces in a warped product, Balkan J. Geom. Appl. 16 (2011), no. 2, 35–47. · Zbl 1228.53021 [10] J. Fastenakels, M. I. Munteanu and J. Van der Veken, Constant angle surfaces in the Heisenberg group, Acta Math. Sin. (Engl. Ser.) 27 (2011), no. 4, 747–756. · Zbl 1218.53019 [11] F. Güler, G. Şaffak and E. Kasap, Timelike constant angle surfaces in Minkowski space $${{$$\backslash$$mathbb{R}_{1}\^{3}}}$$ , Int. J. Contemp. Math. Sci. 6 (2011), no. 44, 2189–2200. · Zbl 1243.53024 [12] López R., Munteanu M.I.: Constant angle surfaces in Minkowski space. Bull. Belg. Math. Soc. Simon Stevin 18(2), 271–286 (2011) · Zbl 1220.53024 [13] R. López and M. I. Munteanu, On the geometry of constant angle surfaces in Sol 3, Kyushu J. Math. 65 (2011), no. 2, 237–249. · Zbl 1236.53011 [14] M. I. Munteanu and A. I. Nistor, A new approach on constant angle surfaces in $${{$$\backslash$$mathbb{E}\^{3}}}$$ , Turkish J. Math. 33 (2009), no. 1, 169–178. · Zbl 1175.53006 [15] M. I. Munteanu and A. I. Nistor, Complete classification of surfaces with a canonical principal direction in the Euclidean space $${{$$\backslash$$mathbb{E}\^{3}}}$$ , Cent. Eur. J. Math. 9 (2011), no. 2, 378–389. · Zbl 1222.53009 [16] A. I. Nistor, A note on spacelike surfaces in Minkowski 3–space, preprint 2012. · Zbl 1451.53095 [17] O’Neill B.: Semi–Riemannian geometry with applications to relativity. Academic Press, New York (1982) [18] R. Tojeiro, On a class of hypersurfaces in $${{$$\backslash$$mathbb{S}\^{n} $$\backslash$$times $$\backslash$$mathbb{R}}}$$ and $${{$$\backslash$$mathbb{H}\^{n} $$\backslash$$times $$\backslash$$mathbb{R}}}$$ , Bull. Braz. Math. Soc. (N.S.) 41 (2010), no. 2, 199–209. [19] I. van de Woestijne , Minimal surfaces in the 3–dimensional Minkowski space, In: M. Boyom et al. (eds), Geometry and Topology of Submanifolds II, World Scientific, 1990, 344–369.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.