Journal of Lie Theory 23 (2013), No. 1, 177--202
Copyright Heldermann Verlag 2013
Ricci Yang-Mills Solitons on Nilpotent Lie Groups
Dept. of Mathematics, University of Oklahoma, Norman, OK 73019-3103, U.S.A.
Dept. of Mathematics and Computer Science, Ripon College, 300 Seward Street, PO Box 248, Ripon, WI 54971, U.S.A.
The purpose of this paper is to introduce the Ricci Yang-Mills 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 Yang-Mills solitons on 2-step 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 Yang-Mills 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 Yang-Mills soliton metrics.
Keywords: Ricci Yang-Mills, soliton, nilpotent, Lie group, principal bundle.
MSC: 53C44, 22E25
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