Deformations of nearly parallel $G_2$-structures

We study the infinitesimal deformations of a proper nearly parallel G_2-structure and prove that they are characterized by a certain first order differential equation. In particular we show that the space of infinitesimal deformations modulo the group of diffeomorphisms is isomorphic to a subspace o...

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Bibliographic Details
Published in:The Asian journal of mathematics 2012, Vol.16 (4), p.713-744
Main Authors: Alexandrov, Bogdan, Semmelmann, Uwe
Format: Article
Language:English
Subjects:
53C10
53C25
58H15
deformations
Nearly parallel G_2-structures
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Summary:We study the infinitesimal deformations of a proper nearly parallel G_2-structure and prove that they are characterized by a certain first order differential equation. In particular we show that the space of infinitesimal deformations modulo the group of diffeomorphisms is isomorphic to a subspace of co-closed \Lambda^3_{27}-eigenforms of the Laplace operator for the eigenvalue 8\mathrm{scal} /21. We give a similar description for the space of infinitesimal Einstein deformations of a fixed nearly parallel G_2-structure. Moreover we show that there are no deformations on the squashed S^7 and on \mathrm{SO}(5)/\mathrm{SO}(3), but that there are infinitesimal deformations on the Aloff-Wallach manifold N(1, 1) = \mathrm{SU}(3)/U(1).
ISSN:1093-6106
1945-0036
DOI:10.4310/AJM.2012.v16.n4.a6