Ricci curvature of left invariant metrics on solvable unimodular Lie groups

Miatello, Isabel Dotti

doi:10.1007/bf01318909

Cited by 41 publications

(22 citation statements)

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“…Let S be an Einstein solvmanifold with Ric D cI . We can assume that S is not unimodular by using the following fact proved by I. Dotti [DM82]: a unimodular Einstein solvmanifold must be flat and consequently standard (see [Heb98,Prop. 4.9]).…”

Section: ])mentioning

confidence: 99%

Einstein solvmanifolds are standard

Lauret

2010

Ann. Math.

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We study Einstein manifolds admitting a transitive solvable Lie group of isometries (solvmanifolds). It is conjectured that these exhaust the class of noncompact homogeneous Einstein manifolds. J. Heber has shown that under a simple algebraic condition (he calls such a solvmanifold standard), Einstein solvmanifolds have many remarkable structural and uniqueness properties. In this paper, we prove that any Einstein solvmanifold is standard, by applying a stratification procedure adapted from one in geometric invariant theory due to F. Kirwan.

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Section: ])mentioning

confidence: 99%

Einstein solvmanifolds are standard

Lauret

2010

Ann. Math.

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“…However by [Do1], any left-invariant Einstein metric on a unimodular solvable Lie group is flat and then, so is M .…”

Section: Structure Of the Proofmentioning

confidence: 99%

On harmonic and asymptotically harmonic homogeneous spaces

Heber

2006

GAFA, Geom. funct. anal.

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We classify noncompact homogeneous spaces which are Einstein and asymptotically harmonic. This completes the classification of Riemannian harmonic spaces in the homogeneous case: Any simply connected homogeneous harmonic space is flat, or rank-one symmetric, or a nonsymmetric Damek-Ricci space. Independently, Y. Nikolayevsky has obtained the latter classification under the additional assumption of nonpositive sectional curvatures [N2].

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“…On the other hand, it is proved by Miatello [8] that no unimodular solvable Lie algebra admits inner product of strictly negative Ricci curvature. Hence, since Q 0 is Einstein, we have Ric…”

Section: Proofmentioning

confidence: 99%

Homogeneous Kähler Einstein manifolds of nonpositive curvature operator

Obata¹

2007

Tohoku Math. Publ.

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Ricci curvature of left invariant metrics on solvable unimodular Lie groups

Cited by 41 publications

References 4 publications

Einstein solvmanifolds are standard

Einstein solvmanifolds are standard

On harmonic and asymptotically harmonic homogeneous spaces

Homogeneous Kähler Einstein manifolds of nonpositive curvature operator

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