Deformations of $G_2$ and $\Spin(7)$ Structures

http://dx.doi.org/10.4153/CJM-2005-039-x
Canad. J. Math. 57(2005), 1012-1055

  Published:2005-10-01
 Printed: Oct 2005

We consider some deformations of $G_2$-structures on $7$-manifolds. We discover a canonical way to deform a $G_2$-structure by a vector field in which the associated metric gets ``twisted'' in some way by the vector cross product. We present a system of partial differential equations for an unknown vector field $w$ whose solution would yield a manifold with holonomy $G_2$. Similarly we consider analogous constructions for $\Spin(7)$-structures on $8$-manifolds. Some of the results carry over directly, while others do not because of the increased complexity of the $\Spin(7)$ case.