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

Published:2005-10-01
Printed: Oct 2005
• Spiro Karigiannis
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## Abstract

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.
 Keywords: $G_2 \Spin(7)$, holonomy, metrics, cross product
 MSC Classifications: 53C26 - Hyper-Kahler and quaternionic Kahler geometry, special'' geometry 53C29 - Issues of holonomy

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