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Hopf cylinders, \(B\)-scrolls and solitons of the Betchov-Da Rios equation in the three-dimensional anti De Sitter space. (English. Abridged French version) Zbl 0836.53027
Summary: We use the natural Hopf fibration from \({\mathbf H}^3_1 (-1)\) over \({\mathbf H}^2_s (-1/4)\) \((s = 0,1)\) to give a geometric interpretation of the \(B\)-scrolls in terms of the Hopf cylinders shaped on non-null curves in \({\mathbf H}^2_s (-1/4)\). We also find those parametrizations of the Hopf cylinders which are solutions of the Betchov-Da Rios soliton equation in \({\mathbf H}^3_1 (-1)\). In particular, the soliton solutions are the null geodesics of the Lorentzian Hopf cylinders.

MSC:
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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