The squashed 7-sphere $S^{ 7}$ is a 7-sphere with an Einstein metric given by the canonical variation and its cone $\mathbb{ R}^{ 8} - \{ 0 \}$ has full holonomy ${ \rm Spin}(7)$. There is a canonical calibrating 4-form $\Phi$ on $\mathbb{ R}^{ 8} - \{ 0 \}$. A minimal 3-submanifold in $S^{ 7}$ is called associative if its cone is calibrated by $\Phi$. In this paper, we classify two types of fundamental associative submanifolds in the squashed $S^{ 7}$. One is obtained by the intersection with a 4-plane and the other is homogeneous. Then we study their infinitesimal associative deformations and explicitly show that all of them are integrable.
Categories: math.DG