We show that if a complete Riemannian manifold supports a vector field such that the Ricci tensor plus the Lie derivative of the metric with respect to the vector field has a positive lower bound, then the fundamental group is finite. In particular, it follows that complete shrinking Ricci solitons and complete smooth metric measure spaces with a positive… (More)
We define a gradient Ricci soliton to be rigid if it is a flat bundle N × Γ R k where N is Einstein. It is known that not all gradient solitons are rigid. Here we offer several natural conditions on the curvature that characterize rigid gradient solitons. Other related results on rigidity of Ricci solitons are also explained in the last section.
We show that the only shrinking gradient solitons with vanishing Weyl tensor are quotients of the standard ones S n , S n−1 × R, and R n. This gives a new proof of the Hamilton-Ivey-Perel'man classification of 3-dimensional shrinking gradient solitons. We also show that gradient solitons with constant scalar curvature and suitably decaying Weyl tensor when… (More)
We study gradient Ricci solitons with maximal symmetry. First we show that there are no non-trivial homogeneous gradient Ricci solitons. Thus the most symmetry one can expect is an isometric cohomogeneity one group action. Many examples of cohomogeneity one gradient solitons have been constructed. However, we apply the main result in  to show that there… (More)
Let (M, d) be a metric space. For 0 < r < R, let G(p, r, R) be the group obtained by considering all loops based at a point p ∈ M whose image is contained in the closed ball of radius r and identifying two loops if there is a homotopy betweeen them that is contained in the open ball of radius R. In this paper we study the asymptotic behavior of the G(p, r,… (More)