On the finiteness of the fundamental group of a compact shrinking Ricci soliton

Abstract

Myers's classical theorem says that a compact Riemannian manifold with positive Ricci curvature has finite fundamental group. Using Ambrose's compactness crite- rion or J. Lott's results, M. Fernandez-Lopez and E. Garc´oa-R´oo showed that the finiteness of the fundamental group remains valid for a compact shrinking Ricci soliton. We give a self-contained proof of this fact by estimating the lengths of shortest geodesic loops in each homotopy class. Myers's classical theorem says that a compact Riemannian manifold with positive Ricci curvature has finite fundamental group. Fernopez and Garc´oa-R´oo (3) provided two methods to generalize this result to the Ricci soliton case: one used Ambrose's criterion for the compactness of a mani- fold under some Ricci curvature conditions (1), and the other used Lott's results (4). On the other hand, Derdzinski (2) proved the finiteness of the first homology group of a compact shrinking Ricci soliton by estimating the lengths of closed geodesics. Here we optimize Derdzinski's method to give another proof of the finiteness of the fundamental group of a compact shrinking Ricci soliton. Recall that a Riemannian manifold (M,g) is a shrinking Ricci soliton if there exist c > 0 and a C ∞ vector field X such that