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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Infinitesimal Einstein deformations of nearly Kähler metrics


Authors: Andrei Moroianu and Uwe Semmelmann
Journal: Trans. Amer. Math. Soc. 363 (2011), 3057-3069
MSC (2010): Primary 58H15, 58E30, 53C10, 53C15
Published electronically: January 26, 2011
MathSciNet review: 2775798
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Abstract: It is well known that every 6-dimensional strictly nearly Kähler manifold $ (M,g,J)$ is Einstein with positive scalar curvature $ \operatorname{scal}>0$. Moreover, one can show that the space $ E$ of co-closed primitive $ (1,1)$-forms on $ M$ is stable under the Laplace operator $ \Delta$. Let $ E(\lambda)$ denote the $ \lambda$-eigenspace of the restriction of $ \Delta$ to $ E$. If $ M$ is compact, and has normalized scalar curvature $ \operatorname{scal}=30$, we prove that the moduli space of infinitesimal Einstein deformations of the nearly Kähler metric $ g$ is naturally isomorphic to the direct sum $ E(2)\oplus E(6)\oplus E(12)$. From Moroianu, Nagy, and Semmelmann (2008), the last summand is itself isomorphic with the moduli space of infinitesimal nearly Kähler deformations.


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Additional Information

Andrei Moroianu
Affiliation: CMLS, École Polytechnique, UMR 7640 du CNRS, 91128 Palaiseau, France
Email: am@math.polytechnique.fr

Uwe Semmelmann
Affiliation: Mathematisches Institut, Universität zu Köln, Weyertal 86-90, D-50931 Köln, Germany
Address at time of publication: Department of Mathematics, University of Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany
Email: uwe.semmelmann@math.uni-koeln.de, uwe.semmelmann@mathematik.uni-stuttgart.de

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05064-6
PII: S 0002-9947(2011)05064-6
Keywords: Einstein deformations, nearly Kähler manifolds, Gray structures
Received by editor(s): June 10, 2008
Received by editor(s) in revised form: March 30, 2009
Published electronically: January 26, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.