The Poincare-Hopf theorem for a manifold with boundary states that the sum of the indices of zeros of a complex vector field $v$ equals the Euler characteristic of the manifold M if we impose 'outward' orientation along the boundary: $$ \Sigma \text{ind}(v) = \chi(M). $$ On the other hand, the Gauss-Bonnet theorem says that the gaussian curvature area integral + the geodesic curvature boundary line integral $= 2\pi\chi(M)$.
Now, consider a 2D manifold X with parameters $(x,y)$, with known Euler character $\chi(X)$. Over X, I can define various complex vector fields $v1, v2, v3, ...$ (all dependent on $(x,y)$). Does $\Sigma_j \text{ind}(v_j)$ always equal $\chi(X)$? Or are the Euler characteristics here $=\chi(M_{v1}),\chi(M_{v1}),\chi(M_{v1})...$ where $M_{vi}$ are some other manifold of each $v_i$.
My confusion is coming from a physics problem, where I have to choose a closed loop in $(x,y)$-space, and the definition of $v_i$ depends on the choice of loop. I am trying to understand whether the index of the zeros of these $v_i$ relate at all to $= \chi(X)$.