Some associative submanifolds of the squashed 7-sphere

TL;DR

This paper classifies fundamental associative submanifolds in the squashed 7-sphere, analyzing their deformations and demonstrating their integrability, contributing to understanding special submanifolds in Einstein and Spin(7) geometries.

Contribution

It introduces a classification of associative submanifolds in the squashed 7-sphere, including explicit deformation analysis and integrability results.

Findings

Classified two types of associative submanifolds in the squashed 7-sphere.

Proved all infinitesimal associative deformations are integrable.

Identified explicit examples of associative submanifolds.

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.