Ricci structure and volume growth for left invariant Riemannian metrics on nilpotent and some solvable Lie groups

Abstract

We discuss left invariant Riemannian metrics on Lie groups, and the Ricci structures they induce. A computational approach is used to manipulate the curvature tensors, and construct perturbations which preserve the Ricci structure but not the (algebraic) Lie structure. We show that this method can lead to significant changes in the growth of volume. We also show that this approach may be used to reduce the complexity of some curvature computations in Lie groups.