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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.)