The Ricci curvature of finite dimensional approximations to loop and path groups

Abstract

Let W ( G ) and L ( G ) denote the path and loop groups respectively of a connected real unimodular Lie group G endowed with a left-invariant Riemannian metric. We study the Ricci curvature of certain finite dimensional approximations to these groups based on partitions of the interval [ 0 , 1 ] . We find that the Ricci curvatures of the finite dimensional approximations are bounded below independent of partition iff G is of compact type with an Ad-invariant metric.