Abstract
A Riemannian homogeneous manifold admitting a strict nearly-Kähler structure is 3-symmetric. We actually classify them in dimension 6 and use previous results of Swann, Cleyton and Nagy to prove the conjecture in higher dimensions. The six-dimensional homogeneous spaces, S3 × S3, S6, CP(3) and the flag manifold F(1, 2) have a unique (after a change of scale) nearly-Kähler, invariant structure. For the first one we solve a differential equation on the SU(3)-structure given by Reyes Carrión. For the last two it is obtained by canonical variation of the Kähler structure of the twistor space over a four-dimensional manifold. Finally, from Bär, a nearly-Kähler structure on the sphere S6 corresponds to a constant 3-form on the Riemannian cone R7.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bär, C.: Real Killing spinors and holonomy, Comm. Math. Phys. 154 (1993), 509–521.
Baum, H., Friedrich, Th., Grunewald, R. and Kath, I.: Twistors and Killing Spinors on Riemannian Manifolds, Teubner-Verlag, Stuttgart/Leipzig, 1991.
Bryant, R. L.: Metrics with special holonomy, Ann. Math. 126 (1987), 525–576.
Cleyton, R.: G-structures and Einstein metrics, PhD Thesis, Odense, 2001.
Cleyton, R. and Swann, A.: Einstein metrics via intrinsic or parallel torsion, arXiv:math.DG/0211446 (2002).
Friedrich, Th. and Kurke, H.: Compact four-dimensional self-dual Einstein manifolds with positive scalar curvature, Math. Nachr. 106 (1982), 271–299.
Friedrich, Th. and Grunewald, R.: On Einstein metrics on the twistor space of a four-dimensional Riemannian manifold, Math. Nachr. 123 (1985), 55–60.
Gray, A.: Riemannian manifolds with geodesic symmetries of order 3, J. Differential Geom. 7 (1972), 343–369.
Gray, A.: The structure of nearly Kähler manifolds, Math. Ann. 223 (1976), 233–248.
Gray, A. and Hervella, L. M.: The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl. 123 (1980), 35–58.
Grunewald, R.: Six-dimensional Riemannian manifold with real Killing spinors, Ann. Global Anal. Geom. 8 (1990), 43–59.
Hitchin, N.: Kählerian twistor spaces, Proc. London Math. Soc. 43(3) (1981), 133–150.
Kobayashi, S. and Nomizu, K.: Foundations of Differential Geometry, Vol. 2, Wiley-Interscience, New York, 1963.
Ledger, A. J. and Obata, M.: Affine and Riemannian s-manifolds, J. Differential Geom. 2 (1968), 451–459.
Nagy, P. A.: On nearly Kähler geometry, Ann. Global Anal. Geom. 22 (2002), 167–178.
Nagy, P. A.: Nearly Kähler geometry and Riemannian foliations, Asian J. Math. 6 (2002), 481–504.
Reyes Carrión, R.: Some special geometries defined by Lie groups, PhD Thesis, Oxford, 1993.
Wolf, J. A. and Gray, A.: Homogeneous spaces defined by Lie group automorphisms I, II, J. Differential Geom. 2 (1968), 77–114, 115–159.
Author information
Authors and Affiliations
Corresponding author
Additional information
Mathematics Subject Classifications (2000): 53C15, 53C25, 53C30, 53C56.
Rights and permissions
About this article
Cite this article
Butruille, JB. Classification des variété approximativement kähleriennes homogénes. Ann Glob Anal Geom 27, 201–225 (2005). https://doi.org/10.1007/s10455-005-1581-x
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10455-005-1581-x