Exploring surfaces through methods from the theory of integrable systems: The Bonnet problem

1.1. Differential equations of surfaces Let F be a smooth orientable surface in 3-dimensional Euclidean space. The Euclidean metric induces a metric Ω on this surface, which in turn generates the complex structure of a Riemann surface R. Under such a parametrization, which is called conformal, the surface F is given by an immersion F = (F1, F2, F3) : R → R, and the metric is conformal: Ω = e dzdz̄, where z is a local coordinate on R. One should keep in mind that a complex coordinate is defined up to holomorphic z → w(z) transformation. This freedom will be used to simplify the corresponding equations. The conformal parametrization gives the following normalization of F (z, z̄):