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Deformations of G2 and Spin(7) Structures
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- Journal:
- Canadian Journal of Mathematics / Volume 57 / Issue 5 / 01 October 2005
- Published online by Cambridge University Press:
- 20 November 2018, pp. 1012-1055
- Print publication:
- 01 October 2005
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- Article
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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.
