Existence of left invariant Ricci flat metrics on nilpotent Lie groups

Ricci-flat and Einstein pseudoriemannian nilmanifolds

Complex Manifolds ◽

10.1515/coma-2019-0010 ◽

2019 ◽

Vol 6 (1) ◽

pp. 170-193 ◽

Cited By ~ 2

Author(s):

Diego Conti ◽

Federico A. Rossi

Keyword(s):

Lie Groups ◽

Lie Group ◽

Einstein Metrics ◽

Einstein Metric ◽

Nilpotent Lie Groups ◽

Nilpotent Lie Group ◽

Ricci Flat ◽

Open Questions ◽

Expository Paper ◽

New Criterion

AbstractThis is partly an expository paper, where the authors’ work on pseudoriemannian Einstein metrics on nilpotent Lie groups is reviewed. A new criterion is given for the existence of a diagonal Einstein metric on a nice nilpotent Lie group. Classifications of special classes of Ricci-˛at metrics on nilpotent Lie groups of dimension [eight.tf] are obtained. Some related open questions are presented.

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Indefinite Einstein metrics on nice Lie groups

Forum Mathematicum ◽

10.1515/forum-2020-0049 ◽

2020 ◽

Vol 32 (6) ◽

pp. 1599-1619

Author(s):

Diego Conti ◽

Federico A. Rossi

Keyword(s):

Lie Groups ◽

Lie Group ◽

Einstein Metrics ◽

Algebraic Characterization ◽

Nilpotent Lie Groups ◽

Nilpotent Lie Group ◽

Systematic Method ◽

Ricci Flat ◽

Nonzero Scalar

AbstractWe introduce a systematic method to produce left-invariant, non-Ricci-flat Einstein metrics of indefinite signature on nice nilpotent Lie groups. On a nice nilpotent Lie group, we give a simple algebraic characterization of non-Ricci-flat left-invariant Einstein metrics in both the class of metrics for which the nice basis is orthogonal and a more general class associated to order two permutations of the nice basis. We obtain classifications in dimension 8 and, under the assumption that the root matrix is surjective, dimension 9; moreover, we prove that Einstein nilpotent Lie groups of nonzero scalar curvature exist in every dimension \geq 8.

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