Volume 15, issue 2 (2011)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Concerning the existence of Einstein and Ricci soliton metrics on solvable Lie groups

Michael Jablonski

Geometry & Topology 15 (2011) 735–764
Abstract

In this work we investigate solvable and nilpotent Lie groups with special metrics. The metrics of interest are left-invariant Einstein and algebraic Ricci soliton metrics. Our main result shows that one may determine the existence of a such a metric by analyzing algebraic properties of the Lie algebra and infinitesimal deformations of any initial metric.

Our second main result concerns the isometry groups of such distinguished metrics. Among the completely solvable unimodular Lie groups (this includes nilpotent groups), if the Lie group admits such a metric, we show that the isometry group of this special metric is maximal among all isometry groups of left-invariant metrics.

Keywords
Einstein metric, Ricci soliton, solvsoliton, nilsoliton, solvable, nilpotent, Lie group, left-invariant metric
Mathematical Subject Classification 2010
Primary: 22E25, 53C25, 53C30
References
Publication
Received: 26 September 2010
Revised: 13 January 2011
Accepted: 13 March 2011
Published: 14 May 2011
Proposed: Dmitri Burago
Seconded: Frances Kirwan, Simon Donaldson
Authors
Michael Jablonski
Department of Mathematics
University of Oklahoma
Norman OK 73019-0315
USA
http://www.math.ou.edu/~mjablonski/