Apparently my last comment was redirected to the chat

I was wondering if I can choose a compact submanifold $N, \partial N \subset M$ where V does not vanish on the boundary $\partial N$. Then I would compute the Poincare-Hopf theorem for the submanifold $N, \partial N$. Since I know the behavior of the vector field at infinity, it would be sufficient for me to know the index of $V$ in a compact region $S^1\times[-H, H]$ where H is a larger number.

The manifold is the infinity cylinder $M=S^1\times \mathbb{R}$. I have a vector field $V$ defined all over $M$. I know, from the context of my problem, that this vector field does not have isolated zeroes for high values of height $H$. I want to compute some topological invariant related to the zeros of $V$.

I am trying to understand how one can adapt the Poincare-Hopf index theorem to a non-compact manifold. In particular, I am considering a vector field that does not vanish at the infinity of my non-compact manifold. The answer of Professor Bill Thurston is exactly what I am looking for. I would like to read a reference to improve my knowledge (I am self studying the Poincare-Hopf theorem and this topic is new to me).