# Nearly parallel $G_2$-structures with large symmetry group

## TL;DR

This paper constructs a one-parameter family of nearly parallel G2-structures on S^3×R^4 with large symmetry, connecting two homogeneous structures via a cohomogeneity one action of SU(2)^3.

## Contribution

It demonstrates the existence of a continuous family of nearly parallel G2-structures with large symmetry group on a specific manifold, expanding understanding of G2-geometry.

## Key findings

- Existence of a one-parameter family of G2-structures on S^3×R^4.
- Structures are mutually non-isomorphic and invariant under SU(2)^3.
- Connects two homogeneous G2-structures on S^7.

## Abstract

We prove the existence of a one-parameter family of nearly parallel $G_2$-structures on the manifold $S^3\times \mathbb R^4$, which are mutually non isomorphic and invariant under the cohomogeneity one action of the group $SU(2)^3$. This family connects the two locally homogeneous nearly parallel $G_2$-structures which are induced by the homogeneous ones on the sphere $S^7$.