Vector fields on noncompact manifolds

Let M be a noncompact 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=G is nonzero.