Cones of G manifolds and Killing spinors with skew torsion

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 ${\mathrm{\nabla }}^c$, 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.