On the rigidity of special and exceptional geometries with torsion a closed $3$-form

Abstract: We demonstrate that all Riemannian manifolds $(M, g, H)$ that admit a connection $\hat\nabla$ with torsion a 3-form $H$, which is both closed $d H=0$ and $\hat\nabla$-covariantly constant, are locally isometric to a product $N\times G$, where $G$ is a semisimple group and $N$ is a Riemannian manifold with $ι_V H=0$ for all tangent vectors $V \in T_pN\subset T_pM$, $p\in M$. If $M$ is simply connected and complete, then by the de Rham theorem $M=N\times G$ globally. We use this to simplify the proof of similar results for strong KT, CYT and HKT manifolds that obey the above hypotheses and extend them to strong $G_2$ and $\mathrm{Spin}(7)$ manifolds with torsion. As an application, we describe the geometry of all complete and simply connected $G_2$ and $\mathrm{Spin}(7)$ manifolds whose torsion satisfies the above conditions. We also demonstrate that all compact 8-dimensional manifolds with strong HKT structure are locally isometric to one of the following: 8-dimensional hyper-Kähler; $SU(3)$ equipped with the bi-invariant metric and 3-form; or the product $(U(1)\times SU(2))\times B<sup>4$, where $B<sup>4$ is either a hyper-Kähler manifold or $U(1)\times SU(2)$ equipped with an HKT structure.