Asian Journal of Mathematics

Volume 20 (2016)

Number 1

Deformations of $G_2$-structures with torsion

Pages: 123 – 156



Sergey Grigorian (School of Mathematical and Statistical Sciences, The University of Texas Rio Grande Valley, Edinburg, Texas, U.S.A.)


We consider non-infinitesimal deformations of $G_2$-structures on 7-dimensional manifolds and derive an exact expression for the torsion of the deformed $G_2$-structure. We then specialize to a case when the deformation is defined by a vector $v$ and we explicitly derive the expressions for the different torsion components of the new $G_2$-structure in terms of the old torsion components and derivatives of $v$. In particular this gives a set of differential equations for the vector $v$ which have to be satisfied for a transition between $G_2$-structures with particular torsions. For some specific torsion classes we find that these equations have no solutions.


$G_2$-structures, deformations, holonomy

2010 Mathematics Subject Classification

53C10, 53C25, 53C29

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