Cross fields are auxiliary in the generation of quadrangular meshes. A method to generate cross fields on surface manifolds is presented in this paper. Algebraic topology constraints on quadrangular meshes are first discussed. The duality between quadrangular meshes and cross fields is then outlined, and a generalization to cross fields of the Poincaré-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. It is first presented in the planar case, and then extended to a general surface manifold. Finally, application examples are solved and discussed.