One of the main goals of this project is to study the index of vector fields in compact surfaces and present the Poincar´e-Hopf Theorem, remarking that such a result connects topological elements to geometric objects.

In this work, we study the concepts of degree and local degree for maps between manifolds. Furthermore, we define the Poincaré-Hopf index and study some relevant proprieties of this index such as the famous Poincaré-Hopf ...