Before revision:  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 Poincar\'e-Hopf theorem is proposed, which highlights some fundamental and important topological constraints on cross fields.
After revision:  A method to generate cross fields on surface manifolds is presented in this paper is to exploit this theoretical background for crossfield computation on arbitrary surfaces. Such crossfields are instrumental in the generation of meshes with quadrangular elements. The relation between the topological properties of 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.