Before revision:  The duality between quadrangular meshes and cross fields is then outlined, and a generalization to cross fields of the Poincar\'e-Hopf theorem is proposed, which highlights some fundamental and important topological constraints on cross fields.A finite element formulation for the computation of cross fields is then presented, which is based on Ginzburg-Landau equations and makes use of edge-based Crouzeix-Raviart interpolation functions.
After revision:  The duality between quadrangular meshes and crossfields are hence first recalled. It is then shown that a crossfield on a surface can be represented by a complex function of unit norm with a number of critical points, i.e., a nearly everywhere smooth function taking its values in the unit circle of the complex plane. As maximal smoothness of the crossfield is equivalent with minimal energy, the crossfield problem is equivalent to an optimization problem based on Ginzburg-Landau equations and makes use of edge-based Crouzeix-Raviart interpolation functions.