Deformations of homogeneous associative submanifolds in nearly parallel $G_{2}$-manifolds

TL;DR

This paper investigates the deformations of homogeneous associative submanifolds within nearly parallel G2-manifolds, focusing on Cayley cone deformations in the cone over such manifolds, especially in the 7-sphere.

Contribution

It provides explicit analysis of Cayley cone deformations of homogeneous associative submanifolds in nearly parallel G2-manifolds, particularly in the 7-sphere.

Findings

Explicit deformation descriptions for homogeneous associative submanifolds.

Identification of conditions under which deformations are possible.

Insights into the geometric structure of associative submanifolds in G2-geometry.

Abstract

A nearly parallel $G_{2}$-manifold $Y$ is a Riemannian 7-manifold whose cone $C(Y) = \mathbb{R}_{>0} \times Y$ has the holonomy group contained in ${\rm Spin(7)}$. 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 $S^{7}$.