Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Locally conformal Kähler structures
in quaternionic geometry

Authors: Liviu Ornea and Paolo Piccinni
Journal: Trans. Amer. Math. Soc. 349 (1997), 641-655
MSC (1991): Primary 53C15, 53C25, 53C55
MathSciNet review: 1348155
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider compact locally conformal quaternion Kähler manifolds $M$. This structure defines on $M$ a canonical foliation, which we assume to have compact leaves. We prove that the local quaternion Kähler metrics are Ricci-flat and allow us to project $M$ over a quaternion Kähler orbifold $N$ with fibers conformally flat 4-dimensional real Hopf manifolds. This fibration was known for the subclass of locally conformal hyperkähler manifolds; in this case we make some observations on the fibers' structure and obtain restrictions on the Betti numbers. In the homogeneous case $N$ is shown to be a manifold and this allows a classification. Examples of locally conformal quaternion Kähler manifolds (some with a global complex structure, some locally conformal hyperkähler) are the Hopf manifolds quotients of $\mathbb H^n-\{0\}$ by the diagonal action of appropriately chosen discrete subgroups of $CO^+(4)$.

References [Enhancements On Off] (What's this?)

  • [Be] A. Besse, Einstein manifolds, Springer-Verlag, Berlin and New York, 1987. MR 88f:53087
  • [Bo] Ch. P. Boyer, A note on hyperhermitian four-manifolds, Proc. Amer. Math. Soc. 102 (1988), 157-164. MR 89c:53049
  • [Bo-Ga-Ma1] Ch. P. Boyer, K. Galicki, and B. Mann, The geometry and topology of 3-Sasakian manifolds, J. Reine Angew. Math. 455 (1994), 183-220. MR 96e:53057
  • [Bo-Ga-Ma2] -, Hypercomplex structures on Stiefel manifolds, Ann. of Global Anal. Geom. 14 (1996), 81-105. CMP 96:08
  • [Ch-Pi] B. Y. Chen and P. Piccinni, The canonical foliations of a locally conformal Kähler manifold, Ann. Mat. Pura Appl. 141 (1985), 289-305. MR 87e:53054
  • [Gau1] P. Gauduchon, La 1-forme de torsion d'une variété hermitienne compacte, Math. Ann. 267 (1984), 495-518. MR 87a:53101
  • [Gau2] -, Structures de Weyl-Einstein, espaces de twisteurs et variétés de type $S^1\times S^3$, J. Reine Angew. Math. 469 (1995), 1-50. CMP 96:05
  • [Gr] A. Gray, A note on manifolds whose holonomy group is $Sp(n)Sp(1)$, Michigan Math. J. 16 (1969), 125-128. MR 39:6226
  • [Ka1] Ma. Kato, Topology of Hopf surfaces, J. Math. Soc. Japan 27 (1975), 222-238. MR 53:5949
  • [Ka2] -, Compact differentiable 4-folds with quaternionic structures, Math. Ann. 248 (1980), 79-96. MR 81h:53037
  • [Ku] Y. Y. Kuo, On almost contact 3-structures, Tôhoku Math. J. 22 (1970), 325-332. MR 43:3956
  • [La] J. Lafontaine, Remarques sur les variétés conformément plates, Math. Ann. 259 (1982), 313-319. MR 84a:53053
  • [Ma] S. Marchiafava, Sulle varietà a struttura quaternionale generalizzata, Rend. Mat. 3 (1970), 529-545. MR 43:2632
  • [Ma-Ro] S. Marchiafava and G. Romani, Sui fibrati con struttura quaternionale generalizzata, Ann. Mat. Pura Appl. 107 (1975), 131-157. MR 53:6558
  • [Mo] P. Molino, Riemannian foliations, Birkhäuser, Boston, 1988. MR 89b:53054
  • [Or] L. Ornea, Locally conformal Kähler manifolds. A survey, Quaderno n.12, Dip. Mat., Univ. di Roma ``La Sapienza'', 1994.
  • [Pa] R. S. Palais, A global formulation of the Lie theory of transformation groups Mem. Amer. Math. Soc. 22 (1957). MR 22:12162
  • [Pe-Po-Sw] H. Pedersen, Y. S. Poon, and A. Swann, The Einstein-Weyl equations in complex and quaternionic geometry, Diff. Geom. Appl. 3 (1993), 309-321. MR 94j:53058
  • [Pi] P. Piccinni, On the infinitesimal automorphisms of quaternionic structures, J. Math. Pures Appl. 72 (1993), 593-605. MR 94k:53048
  • [Sa] S. Salamon, Quaternionic Kähler manifolds, Invent. Math. 67 (1982), 143-171. MR 83k:53054
  • [Sat] I. Satake, The Gauss-Bonnet theorem for $V$-manifolds, J. Math. Soc. Japan 9 (1957), 464-476. MR 20:2022
  • [Ta] S. Tanno, Killing vectors on contact Riemannian manifolds and fiberings related to the Hopf fibration, Tôhoku Math. J. 23 (1971), 313-333. MR 44:4681
  • [Ud] C. Udri\c{s}te, Structures presque coquaternioniennes, Bull. Math. Soc. Sci. Math. R.S. Roumanie 13 (1969), 487-507. MR 45:5908
  • [Va1] I. Vaisman, On locally conformal almost Kähler manifolds, Israel J. Math. 24 (1976), 338-351. MR 54:6048
  • [Va2] -, A geometric condition for an l.c.K. manifold to be Kähler, Geom. Dedicata 10 (1981), 129-134. MR 83b:53059
  • [Va3] -, Generalized Hopf manifolds, Geom. Dedicata 13 (1982), 231-255. MR 84g:53096
  • [Va4] -, A survey of generalized Hopf manifolds, in Differential Geometry on Homogeneous Spaces (Proc. Conf. Torino, Italy, 1983), Rend. Sem. Mat. Torini, Fasc. Spec., 205-221. MR 87g:53100
  • [Va-Re] I. Vaisman and C. Reischer, Local similarity manifolds, Ann. Mat. Pura Appl. 135 (1983), 279-291. MR 86b:53050

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 53C15, 53C25, 53C55

Retrieve articles in all journals with MSC (1991): 53C15, 53C25, 53C55

Additional Information

Liviu Ornea
Affiliation: Faculty of Mathematics, University of Bucharest, 14, Academiei str., 70109 Bucha- rest, Romania

Paolo Piccinni
Affiliation: Dipartimento di Matematica “G. Castelnuovo”, Università di Roma “La Sapienza”, P.le A. Moro, 2, I-00185 Roma, Italy

Keywords: Locally conformal hyperk\"ahler manifold, locally conformal quaternion K\"ahler manifold, Einstein-Weyl structure
Received by editor(s): September 1, 1994
Additional Notes: The first author was supported by C.N.R. of Italy, the second author by M.U.R.S.T. of Italy and by the E. Schrödinger Institute in Vienna
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society