Vector fields on non-compact manifolds

TL;DR

This paper proves a Poincaré-Hopf theorem for bounded vector fields on non-compact manifolds with group actions, showing such fields have infinitely many zeros under certain conditions.

Contribution

It establishes a new Poincaré-Hopf theorem for bounded vector fields on non-compact manifolds with group actions, extending classical results.

Findings

Bounded vector fields on certain non-compact manifolds must have infinitely many zeros.

The theorem applies when the acting group is amenable and the Euler characteristic of the quotient is non-zero.

Provides conditions under which vector fields on non-compact manifolds have zeros.

Abstract

Let $M$ be a non-compact connected manifold with a cocompact and properly discontinuous action of a discrete group $G$. We establish a Poincar\'{e}-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 non-zero.