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arXiv · 1411.5840

Some associative submanifolds of the squashed 7-sphere

Kotaro Kawai
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Abstract

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.

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Kotaro Kawai. 2014-11-21. Some associative submanifolds of the squashed 7-sphere. https://doi.org/10.1093/qmath/hav021

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