I'm studying the Gauss-Bonnet theorem flor a compact surfaces using the ideas in the paper of Chern (1944). My principal source of this approach is the master thesis of Otto Romero "El Teorema de Gauss-Bonnet en Variedades Suaves". There he choose a point $p$ in a compact, orientable surface $M$, take a neighborhood of $p$ and then choose a vector field $X$ whose unique singular point is $p$.
I do not understand why we can choose such vector field. I know that in a compact manifold we can choose a vector field with a finite numbers of singular points but it doesn't need to be necessarily one with only a singular point.
Such construction reduce significantly calculations but I don'k know if he do that only for that or if there exist an argument who allow me to do that.
I will be grateful if you help to solve this doubt.