Vector fields on non-compact manifolds

Let $M$ be a non-compact connected manifold with a cocompact and properly discontinuous action of a discrete group $G$. We establish a Poincaré-Hopf theorem for a bounded vector field on $M$ satisfying a mild condition on zeros. As an application, we show that such a vector field must have infinitely many zeros whenever $G$ is amenable and the Euler characteristic of $M\mathrm{/}G$ is non-zero.