Concerning the existence of Einstein and Ricci soliton metrics on solvable Lie groups

Jablonski, Michael¹

¹ Department of Mathematics, University of Oklahoma, Norman OK 73019-0315, USA

Geometry & topology, Tome 15 (2011) no. 2, pp. 735-764.

Résumé

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.

Détail

DOI : 10.2140/gt.2011.15.735

Classification : 22E25, 53C25, 53C30
Keywords: Einstein metric, Ricci soliton, solvsoliton, nilsoliton, solvable, nilpotent, Lie group, left-invariant metric

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