The complex 2-sphere in ℂ3 and Schrödinge flows

By using holomorphic Riemannian geometry in ℂ3, the coupled Landau-Lifshitz equation (CLL) is proved to be exactly the equation of Schrodinger flows from ℝ1 to the complex 2-sphere ℂS2(1) ↪ ℂ3. Furthermore, regarded as a model of moving complex curves in ℂ3, CLL is shown to preserve the \(\mathcal{PT}\) symmetry if the initial data is of the \(\mathcal{P}\) symmetry. As a consequence, the nonlocal nonlinear Schrodinger equation (NNLS) proposed recently by Ablowitz and Musslimani is proved to be gauge equivalent to CLL with initial data being restricted by the \(\mathcal{P}\) symmetry. This gives an accurate characterization of the gauge-equivalent magnetic structure of NNLS described roughly by Gadzhimuradov and Agalarov (2016).