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A new approach on constant angle surfaces in E3

In this paper we study constant angle surfaces in Euclidean 3–space. Even that the result is a consequence of some classical results involving the Gauss map (of the surface), we give another approach to classify all surfaces for which the unit normal makes a constant angle with a fixed direction. | Turk J Math 33 (2009) , 169 – 178. ¨ ITAK ˙ c TUB doi:10.3906/mat-0802-32 A New Approach on Constant Angle Surfaces in E3 Marian Ioan Munteanu, Ana–Irina Nistor Abstract In this paper we study constant angle surfaces in Euclidean 3–space. Even that the result is a consequence of some classical results involving the Gauss map (of the surface), we give another approach to classify all surfaces for which the unit normal makes a constant angle with a fixed direction. Key Words: Constant angle surfaces, Euclidean space. 1. Introduction Recently, constant angle surfaces were studied in product spaces S2 ×R in [2] or H2 ×R in [3, 4], where S2 and H2 represent the unit 2-sphere and the hyperbolic plane, respectively. The angle was considered between the unit normal of the surface M and the tangent direction to R. The idea of studying surfaces with different geometric properties in product spaces was initiated by H. Rosenberg and W. Meeks in [6] and [10], where they have considered the general case of a surface M2 and they have looked for minimal surfaces properties in the product space M2 × R. In this article we study the problem of constant angle surfaces in Euclidean 3-space. So, we want to find a classification of all surfaces in Euclidean 3-space for which the unit normal makes a constant angle with a fixed vector direction being the tangent direction to R. The applications of constant angle surfaces in the theory of liquid crystals and of layered fluids were considered by P. Cermelli and A. J. Di Scala in [1], but they used for their study of surfaces another method different from ours, the Hamilton-Jacobi equation, correlating the surface and the direction field. In [5], R. Howard explains how shadow boundaries are formed when the light source is situated at an infinite distance from the surface M using the geometric model of constant angle surfaces. 2000 AMS Mathematics Subject Classification: 53B25. 169 MUNTEANU, NISTOR 2. Preliminaries be its Levi Civita .