Revisiting 3-Sasakian and $G_2$-structures

TL;DR

This paper explores the algebraic structure of differential forms on 3-Sasakian 7-manifolds, focusing on nearly-parallel G_2 forms and their invariance under specific symmetry group actions, advancing understanding of special geometric structures.

Contribution

It provides a detailed analysis of the exterior algebra on 3-Sasakian manifolds and examines invariant 3-forms under cohomogeneity-one actions, offering new insights into G_2-structures.

Findings

Characterization of the algebra of forms on 3-Sasakian manifolds

Identification of invariant 3-forms under SO(4) actions

Applications to nearly-parallel G_2 structures

Abstract

The algebra of exterior differential forms on a regular 3-Sasakian 7-manifold is investigated, with special reference to nearly-parallel $G_2$ 3-forms. This is applied to the study of 3-forms invariant under cohomogeneity-one actions by $SO(4)$ on the 7-sphere and on Berger's space $SO(5)/SO(3)$.