Anti-self-dual metrics on Lie groups

Smedt, Vivian De; Salamon, Simon

doi:10.1090/conm/308/05312

Cited by 17 publications

(28 citation statements)

References 9 publications

(19 reference statements)

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“…Our next goal is to show that (iii) is a non-trivial Bach-flat metric. Because of [6], we know that (iii) is not half conformally flat and this is verified by direct compution of W ± . From [10, Proposition 1] (see also [9]), a necessary condition for a 4-dimensional manifold to be locally conformally Einstein is the existence of a non-zero vector field T = …”

Section: Curvature Calculationsmentioning

confidence: 83%

“…Solutions (i) and (ii) correspond to those of [6]. The first is hyperhermitian (so W + = 0) and the second is the Einstein metric on CH 2 , so the vanishing of their respective Bach tensor is already known.…”

Section: Curvature Calculationsmentioning

confidence: 98%

“…Our first example generalizes one in [6]. Consider the Lie algebra g α,β defined by a basis (e i ) of where α and β are non-zero real numbers.…”

Section: A Family Of Metric Lie Algebrasmentioning

confidence: 99%

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Bach-flat Lie groups in dimension 4

Abbena

Garbiero

Salamon³

2013

Comptes Rendus. Mathématique

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Section: Curvature Calculationsmentioning

confidence: 83%

Section: Curvature Calculationsmentioning

confidence: 98%

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Bach-flat Lie groups in dimension 4

Abbena

Garbiero

Salamon³

2013

Comptes Rendus. Mathématique

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“…The scale-invariant Ricci-flat metric in dimension 4 coincides also with the natural metric on the cotangent bundle T * H of the upper half plane H induced from a special Kaehler metric on H [37].…”

Section: Introductionmentioning

confidence: 88%

Conformally parallel Spin(7) structures on solvmanifolds

Uğuz¹

2014

Turk J Math

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In this paper we review the Spin(7) geometry in relation to solvmanifolds. Starting from a 7 -dimensional nilpotent Lie group N endowed with an invariant G2 structure, we present an example of a homogeneous conformally parallel Spin(7) metric on an associated solvmanifold. It is thought that this paper could lead to very interesting and exciting areas of research and new results in the direction of (locally conformally) parallel Spin(7) structures.

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“…It is easy to check that this algebra is solvable. It is shown in [29] that g λ and g λ ′ are not isomorphic if λ ′ = ±λ. The algebras g λ and g −λ are isomorphic by the map which just changes the signs of E 3 and E 4 .…”

Section: Integrability Of Natural Almost Complex Structures On a Prodmentioning

confidence: 99%

Product twistor spaces and Weyl geometry

Davidov

2020

Proc. Amer. Math. Soc.

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Motivated by generalized geometry (à la Hitchin), we discuss the integrability conditions for four natural almost complex structures on the product bundle Z × Z → M , where Z is the twistor space of a Riemannian 4manifold M endowed with a metric connection D with skew-symmetric torsion. These structures are defined by means of the connection D and four (Kähler) complex structures on the fibres of this bundle. Their integrability conditions are interpreted in terms of Weyl geometry and this is used to supply examples satisfying the conditions. 2010 Mathematics Subject Classification 53C28; 53C15, 53D18.Remark. In order to define a complex structure on the horizontal spaces H J , J = (J 1 , J 2 ), we can use the structure J 2 instead of J 1 . Then we get four more almost complex structures on Z × Z. But, by symmetry, they do not differ essentially from the structures J m , m = 1, ..., 4.In this section, we shall find the integrability conditions for the restrictions of J m to the four connected components of Z × Z, the subbundles Z ± × Z ± , under the assumption that the torsion of D is skew-symmetric.Denote by N m the Nijenhuis tensor of the almost complex structure J m . Then we have the following analog of Lemma 1 (with a similar proof).Corollary 2. The restrictions of the almost complex structures J m , m = 3, 4, to the connected components of Z × Z are not integrable.

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Anti-self-dual metrics on Lie groups

Abstract: Abstract. The aim of the paper is to determine left-invariant, anti-self-dual, non conformally flat, Riemannian metrics on four-dimensional Lie groups.

Cited by 17 publications

References 9 publications

Bach-flat Lie groups in dimension 4

Bach-flat Lie groups in dimension 4

Conformally parallel Spin(7) structures on solvmanifolds

Product twistor spaces and Weyl geometry

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