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Constant angle surfaces in Minkowski space. (English) Zbl 1220.53024

P. Cermelli and A. J. di Scala introduced in [“Constant angle surfaces in liquid crystals”, Philosophical Magazine 87, 1871–1888 (2007), doi:10.1080/14786430601110364] the notion of surfaces with constant angle as surfaces in the usual three-dimensional Euclidean space whose tangent planes form a constant angle with a fixed constant vector field. This kind of surfaces is used to describe interesting phenomena in physics. In the present paper, the ambient space is considered to be the three-dimensional Lorentzian space. More precisely, these surfaces are studied and classified and it is proved that they are all flat. Also, it is proved that a tangent developable surface is a constant angle surface in this framework if and only if the generating curve is a helix. There are a lot of examples with figures.

MSC:

53B25 Local submanifolds
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
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Full Text: arXiv Euclid