Comptes Rendus
no. 7-8
Differential Geometry/Lie Algebras
Bach-flat Lie groups in dimension 4

Présenté par : the Editorial Board

Elsa Abbena1 ; Sergio Garbiero1 ; Simon Salamon2
1 Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy
2 Department of Mathematics, Kingʼs College London, Strand, London WC2L 2RS, UK
Comptes Rendus. Mathématique, Volume 351 (2013) no. 7-8, pp. 303-306.

Nous montrons lʼexistence de groupes de Lie résolubles de dimension 4 et de métriques riemanniennes invariantes à gauche, dont le tenseur de Bach est nul et qui ne sont ni conformément Einstein, ni semi-conformément plates.

We establish the existence of solvable Lie groups of dimension 4 and left-invariant Riemannian metrics with zero Bach tensor which are neither conformally Einstein nor half conformally flat.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2013.04.011
Elsa Abbena 1 ; Sergio Garbiero 1 ; Simon Salamon 2

1 Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy
2 Department of Mathematics, Kingʼs College London, Strand, London WC2L 2RS, UK 
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Elsa Abbena; Sergio Garbiero; Simon Salamon. Bach-flat Lie groups in dimension 4. Comptes Rendus. Mathématique, Volume 351 (2013) no. 7-8, pp. 303-306. doi : 10.1016/j.crma.2013.04.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.04.011/

[1] V. Apostolov; D.M.J. Calderbank; P. Gauduchon Ambitoric geometry I: Einstein metrics and extremal ambikähler structures | arXiv

[2] R. Bach Zur Weylschen Relativitätstheorie und der Weylschen Erweiterung des Krümmungstensorbegriffs, Math. Z., Volume 9 (1921), pp. 110-135

[3] M.L. Barberis Hypercomplex structures on four-dimensional Lie groups, Proc. Am. Math. Soc., Volume 125 (1997), pp. 1043-1054

[4] L. Bérard Bergery Les espaces homogènes riemanniens de dimension 4, Paris, 1978/1979 (Textes Math.), Volume vol. 3, CEDIC, Paris (1981), pp. 40-60

[5] A.L. Besse Einstein Manifolds, Springer-Verlag, Berlin, 1986

[6] H.-D. Cao; G. Catino; Q. Chen; C. Mantegazza; L. Mazzieri Bach-flat gradient steady Ricci solitons | arXiv

[7] A. Derziński Self-dual Kähler manifolds and Einstein manifolds of dimension four, Compos. Math., Volume 49 (1983), pp. 405-433

[8] V. De Smedt; S. Salamon Anti-self-dual metrics on Lie groups, Contemp. Math., Volume 308 (2002), pp. 63-75

[9] G.R. Jensen Homogeneous Einstein spaces of dimension four, J. Differ. Geom., Volume 3 (1969), pp. 309-349

[10] C.N. Kozameh; E.T. Newman; K.P. Tod Conformal Einstein spaces, Gen. Relativ. Gravit., Volume 17 (1985), pp. 343-352

[11] M. Listing Conformal Einstein spaces in N-dimensions, Ann. Glob. Anal. Geom., Volume 20 (2001), pp. 183-197

[12] T.B. Madsen; A. Swann Invariant strong KT geometry on four-dimensional solvable Lie groups, J. Lie Theory, Volume 21 (2011), pp. 55-70

[13] J. Milnor Curvature of left invariant metrics on Lie groups, Adv. Math., Volume 21 (1976), pp. 293-329

[14] P. Nurowski; J.F. Plebański Non-vacuum twisting type-N metrics, Class. Quantum Gravity, Volume 18 (2001), pp. 341-351

[15] H.-J. Schmidt Non-trivial solutions of the Bach equation exist, Ann. Phys., Volume 41 (1984), pp. 435-436

[16] G. Tian; J. Viaclovsky Bach-flat asymptotically locally Euclidean metrics, Invent. Math., Volume 160 (2005), pp. 357-415

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