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2012 | 10 | 5 | 1710-1720
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Transitive conformal holonomy groups

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For (M, [g]) a conformal manifold of signature (p, q) and dimension at least three, the conformal holonomy group Hol(M, [g]) ⊂ O(p + 1, q + 1) is an invariant induced by the canonical Cartan geometry of (M, [g]). We give a description of all possible connected conformal holonomy groups which act transitively on the Möbius sphere S p,q, the homogeneous model space for conformal structures of signature (p, q). The main part of this description is a list of all such groups which also act irreducibly on ℝp+1,q+1. For the rest, we show that they must be compact and act decomposably on ℝp+1,q+1, in particular, by known facts about conformal holonomy the conformal class [g] must contain a metric which is either Einstein (if p = 0 or q = 0) or locally isometric to a so-called special Einstein product.
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  • University of the Witwatersrand
  • [1] Alt J., Fefferman Constructions in Conformal Holonomy, PhD thesis, Humboldt University, Berlin, 2008
  • [2] Alt J., On quaternionic contact Fefferman spaces, Differential Geom. Appl., 2010, 28(4), 376–394
  • [3] Alt J., Notes of “Transitive conformal holonomy groups”, available at
  • [4] Armstrong S., Definite signature conformal holonomy: a complete classification, J. Geom. Phys., 2007, 57(10), 2024–2048
  • [5] Armstrong S., Leitner F., Decomposable conformal holonomy in Riemannian signature, Math. Nachr. (in press), DOI: 10.1002/mana.201000055
  • [6] Bailey T.N., Eastwood M.G., Gover A.R., Thomas’s structure bundle for conformal, projective and related structures, Rocky Mountain J. Math., 1994, 24(4), 1191–1217
  • [7] Bryant R.L., Conformal geometry and 3-plane fields on 6-manifolds, Developments of Cartan Geometry and Related Mathematical Problems, Kyoto, October 24–27, 2005, RIMS Symposium Proceedings, 1502, Kyoto University, Kyoto, 2006, 1–15
  • [8] Čap A., Gover A.R., CR-tractors and the Fefferman space, Indiana Univ. Math. J., 2008, 57(5), 2519–2570
  • [9] Čap A., Gover A.R., A holonomy characterisation of Fefferman spaces, Ann. Glob. Anal. Geom., 2010, 38(4), 399–412
  • [10] Čap A., Gover A.R., Hammerl M., Holonomy reductions of Cartan geometries and curved orbit decompositions, preprint available at
  • [11] Čap A., Slovák J., Parabolic Geometries. I, Math. Surveys Monogr., 154, American Mathematical Society, Providence, 2009
  • [12] Di Scala A.J., Leistner T., Connected subgroups of SO(2; n) acting irreducibly on R2;n , Israel J. Math., 2011, 182, 103–121
  • [13] Di Scala A.J., Leistner T., Neukirchner T., Geometric applications of irreducible representations of Lie groups, In: Handbook of Pseudo-Riemannian Geometry and Supersymmetry, IRMA Lect. Math. Theor. Phys., 16, European Mathematical Society, Zürich, 2010, 629–651
  • [14] Di Scala A.J., Olmos C., The geometry of homogeneous submanifolds of hyperbolic space, Math. Z., 2001, 237(1), 199–209
  • [15] Fefferman C.L., Monge-Ampère equations, the Bergman kernel and geometry of pseudoconvex domains, Ann. of Math., 1976, 103(2), 395–416
  • [16] Graham C.R., Willse T., Parallel tractor extension and ambient metrics of holonomy split G 2, preprint available at
  • [17] Hammerl M., Sagerschnig K., Conformal structures associated to generic rank 2 distributions on 5-manifolds - characterization and Killing-field decomposition, SIGMA Symmetry Integrability Geom. Methods Appl., 2009, 5, #081
  • [18] Hammerl M., Sagerschnig K., The twistor spinors of generic 2- and 3-distributions, Ann. Global Anal. Geom., 2011, 39(4), 403–425
  • [19] Iwahori N., On real irreducible representations of Lie algebras, Nagoya Math. J., 1959, 14, 59–83
  • [20] Kamerich B.N.P., Transitive Transformation Groups on Products of Two Spheres, Krips Repro, Meppel, 1977
  • [21] Knapp A.W., Lie Groups Beyond an Introduction, Progr. Math., 140, Birkhäuser, Boston, 1996
  • [22] Kramer L., Homogeneous Spaces, Tits Buildings, and Isoparametric Hypersurfaces, Mem. Amer. Math. Soc., 158(752), American Mathematical Society, Providence, 2002
  • [23] Leistner T., Conformal holonomy of C-spaces, Ricci-flat and Lorentzian manifolds, Differential Geom. Appl., 2006, 24(5), 458–478
  • [24] Leitner F., A remark on unitary conformal holonomy, In: Symmetries and Overdetermined Systems of Partial Differential Equations, Minneapolis, July 17-August 4, 2006, IMA Vol. Math. Appl., 144, Springer, New York, 2008, 445–460
  • [25] Leitner F., Normal conformal Killing forms, preprint available at
  • [26] Montgomery D., Simply connected homogeneous spaces, Proc. Amer. Math Soc., 1950, 1(4), 467–469
  • [27] Nurowski P., Differential equations and conformal structures, J. Geom. Phys., 2005, 55(1), 19–49
  • [28] Onishchik A.L., Topology of Transitive Transformation Groups, Johann Ambrosius Barth, Leipzig, 1994
  • [29] Onishchik A.L., Vinberg E.B., Lie Groups and Algebraic Groups, Springer Ser. Soviet Math., Springer, Berlin-Heidelberg, 1990
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