Abstract
The moduli space \({\mathcal {NK}}\) of infinitesimal deformations of a nearly Kähler structure on a compact 6-dimensional manifold is described by a certain eigenspace of the Laplace operator acting on co-closed primitive (1, 1) forms (cf. Moroianu et al. in Pacific J Math 235:57–72, 2008). Using the Hermitian Laplace operator and some representation theory, we compute the space \({\mathcal {NK}}\) on all 6-dimensional homogeneous nearly Kähler manifolds. It turns out that the nearly Kähler structure is rigid except for the flag manifold F(1, 2) = SU3/T 2, which carries an 8-dimensional moduli space of infinitesimal nearly Kähler deformations, modeled on the Lie algebra \({\mathfrak{su}_3}\) of the isometry group.
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References
Baum H., Friedrich Th., Grunewald R., Kath I.: Twistor and Killing Spinors on Riemannian Manifolds. Teubner–Verlag, Stuttgart–Leipzig (1991)
Belgun F., Moroianu A.: Nearly Kähler 6-manifolds with reduced holonomy. Ann. Global Anal. Geom. 19, 307–319 (2001)
Besse A.: Einstein manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 10. Springer-Verlag, Berlin (1987)
Borel A., Hirzebruch F.: Characteristic classes and homogeneous spaces I. 80, 458–538 (1958)
Butruille J.-B.: Classification des variétés approximativement kähleriennes homogènes. Ann. Global Anal. Geom. 27, 201–225 (2005)
Cleyton R., Swann A.: Einstein metrics via intrinsic or parallel torsion. Math. Z. 247, 513–528 (2004)
Friedrich Th.: Nearly Kähler and nearly parallel G 2-structures on spheres. Arch. Math. (Brno) 42, 241–243 (2006)
Gray A.: The structure of nearly Kähler manifolds. Math. Ann. 223, 233–248 (1976)
Moroianu A., Nagy P.-A., Semmelmann U.: Unit Killing Vector Fields on Nearly Kähler Manifolds. Internat. J. Math. 16, 281–301 (2005)
Moroianu A., Nagy P.-A., Semmelmann U.: Deformations of Nearly Kähler Structures. Pacific J. Math. 235, 57–72 (2008)
Moroianu, A., Semmelmann, U.: Infinitesimal Einstein Deformations of Nearly Kähler Metrics. to appear in Trans. Amer. Math. Soc., 2009
Nagy P.-A.: Nearly Kähler geometry and Riemannian foliations. Asian J. Math. 3, 481–504 (2002)
Wolf J., Gray A.: Homogeneous spaces defined by Lie group automorphisms I, II. J. Differ. Geom. 2, 77–114 (1968) 115–159
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Communicated by A. Connes
This work was supported by the French-German cooperation project Procope no. 17825PG.
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Moroianu, A., Semmelmann, U. The Hermitian Laplace Operator on Nearly Kähler Manifolds. Commun. Math. Phys. 294, 251–272 (2010). https://doi.org/10.1007/s00220-009-0903-4
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DOI: https://doi.org/10.1007/s00220-009-0903-4