Cones of G manifolds and Killing spinors with skew torsion

TL;DR

This paper explores the cone construction for G manifolds with skew torsion, showing how Killing spinors induce parallel spinors on the cone and establishing a correspondence between geometric structures.

Contribution

It systematically develops the cone construction for G manifolds with skew torsion and relates geometric structures on the manifold to those on its cone.

Findings

Killing spinors with torsion induce parallel spinors on the cone.

Explicit correspondence between classes of structures on M and on the cone.

Examples demonstrating the cone correspondence with torsion.

Abstract

This paper is devoted to the systematic investigation of the cone construction for Riemannian $G$ manifolds M, endowed with an invariant metric connection with skew torsion $\nabla^c$, a `characteristic connection'. We show how to define a $\bar G$ structure on the cone $\bar M=M\x \R^+$ with a cone metric, and we prove that a Killing spinor with torsion on $M$ induces a spinor on $\bar M$ that is parallel w.\,r.\,t. the characteristic connection of the $\bar G$ structure. We establish the explicit correspondence between classes of metric almost contact structures on $M$ and almost hermitian classes on $\bar M$, resp. between classes of $G_2$ structures on $M$ and $\Spin(7)$ structures on $\bar M$. Examples illustrate how this `cone correspondence with torsion' works in practice.