Geometry & Topology

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

Michael Jablonski

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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.

Article information

Geom. Topol., Volume 15, Number 2 (2011), 735-764.

Received: 26 September 2010
Revised: 13 January 2011
Accepted: 13 March 2011
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22E25: Nilpotent and solvable Lie groups 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C30: Homogeneous manifolds [See also 14M15, 14M17, 32M10, 57T15]

Einstein metric Ricci soliton solvsoliton nilsoliton solvable nilpotent Lie group left-invariant metric


Jablonski, Michael. Concerning the existence of Einstein and Ricci soliton metrics on solvable Lie groups. Geom. Topol. 15 (2011), no. 2, 735--764. doi:10.2140/gt.2011.15.735.

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