Seven-Dimensional Geometries With Special Torsion

Abstract

I use the methods of exterior differential systems and the moving frame to study two geometric structures in seven dimensions related to $G_2$-geometry, and linked by the idea of special torsion. The torsion tensor of a geometric structure is a basic first-order invariant of the structure, and both of the geometries I study have special torsion, meaning that the image of their torsion tensor is constrained to lie in a smaller than usual subset.In part 1, I study quadratic closed $G_2$-structures. A closed $G_2$-structure on a 7-manifold $M$ is given by a closed nondegenerate 3-form $\varphi$, and the quadratic condition, introduced by Bryant, says that $\varphi$ satisfies one of a particular natural one-parameter family of second order equations. The torsion tensor associated to a closed $G_2$-structure $\varphi$ takes values in $\mathfrak{g}_2$, and I study the cases where the image of this map lies in an exceptional $G_2$-orbit. A closed $G_2$-structure $\varphi$ induces a metric, and I give a classification of closed $G_2$-structures with conformally flat induced metric.In part 2, I study $G_2$-structures endowed with a distribution of calibrated planes. In this situation there is an induced $SO(4)$-structure, and I invesitigate the cases where the $G_2$-structure is torsion-free and the induced $SO(4)$-structure has torsion tensor taking values in an irreducible $SO(4)$-module. Additionally, I give a classification of $SO(4)$-structures with invariant torsion, meaning that their torsion tensor takes values in a direct sum of trivial $SO(4)$-modules.