Deformations of homogeneous associative submanifolds in nearly parallel $G_2$-manifolds

A nearly parallel $G_2$-manifold $Y$ is a Riemannian $7$-manifold whose cone $C(Y) = \mathbb{R}_{\gt 0} \times Y$ has the holonomy group contained in $\mathrm{Spin}(7)$. In other words, it is a spin $7$-manifold with a real Killing spinor.

We have a special class of calibrated submanifolds called Cayley submanifolds in $C(Y)$. An associative submanifold in $Y$ is a minimal $3$-submanifold whose cone is Cayley. We study its deformations, namely, Cayley cone deformations, explicitly when it is homogeneous in the $7$-sphere $S^7$.

Keywords

associative submanifolds, nearly parallel $G_2$-manifolds, Cayley cones

2010 Mathematics Subject Classification

53C30, 53C38

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The author is supported by Grant-in-Aid for JSPS fellows (26-7067).

Received 30 July 2014

Published 5 July 2017