A note on the nullity of homotopy group for complete three dimensional manifolds with Ricci \(\geq 0\) (Q1387500)

Let \(M\) be a Riemannian manifold of dimension 3 and \(V(r)\) denote the geodesic ball. In this paper, it is shown that if the Ricci curvature is nonnegative and \(\lim \sup V(r)/r^2= \infty\) as \(r\to \infty\), then \(M\) is contractible. The author also points out that the volume growth condition is optimal, which is shown by the examples \(S^1 \times \mathbb{R}^2\) and \(S^2 \times \mathbb{R}\).