
Journal of Lie Theory 23 (2013), No. 1, 177202 Copyright Heldermann Verlag 2013 Ricci YangMills Solitons on Nilpotent Lie Groups Michael Jablonski Dept. of Mathematics, University of Oklahoma, Norman, OK 730193103, U.S.A. mjablonski@math.ou.edu Andrea Young Dept. of Mathematics and Computer Science, Ripon College, 300 Seward Street, PO Box 248, Ripon, WI 54971, U.S.A. younga@ripon.edu The purpose of this paper is to introduce the Ricci YangMills soliton equations on nilpotent Lie groups N. As in the case of Ricci solitons, we demonstrate that such metrics arise from automorphisms of N/Z, where Z is the center of N. Additionally, using techniques from Geometric Invariant Theory, we produce a characterization of Ricci YangMills solitons on 2step nilpotent Lie groups as critical points of a natural functional. Applying our work on nilpotent Lie groups, we study compact torus bundles over tori with locally (nilpotent) homogeneous metrics. On such spaces, we prove that Ricci YangMills solitons are precisely the metrics whose Ricci tensor is invariant under the geodesic flow. We finish this note by producing examples of Lie groups that do not admit Ricci soliton metrics but that do admit Ricci YangMills soliton metrics. Keywords: Ricci YangMills, soliton, nilpotent, Lie group, principal bundle. MSC: 53C44, 22E25 [ Fulltextpdf (360 KB)] for subscribers only. 