Binormal motion of curves with constant torsion in 3-spaces (Q1798472)

The authors use a geometrical approach to investigate an extension of the localized induction equation \(x_{t}=x_{s}\times x_{ss}\) which is a soliton equation. Under some suitable conditions a curve motion describes a curve \(\gamma\) evolving under the binormal flow, determines an immersed surface \(S_{\gamma}\) in a Riemannian or Lorentzian 3-space with constant curvature. Applying fundamental results of theory of submaifolds, they obtain the curvature and torsion of a geodesic foliation of \(S_{\gamma}\). Also, they study the binormal evolution surfaces, whose filaments have the same constant torsion and use the Gauss-Codazzi equations to construct solutions.