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

Abstract

Under some suitable assumptions 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 $H\vert_N=0$. 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 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 that satisfy the above conditions. Compact, strong, 8-dimensional HKT manifolds, which are not hyper-K\"ahler, admit an either $\oplus^4 \mathfrak{u}(1)$ or a $\mathfrak{u}(1)\oplus \mathfrak{su}(2)$ locally free action, otherwise, they are group manifolds. We find that if these Lie algebra actions can be integrated to an appropriate free action of $T^4$ or $S(U(1)\times U(2))$ Lie groups that preserves the span of three complex structures, then these HKT manifolds are either locally isometric and tri-holomorphic to $\mathbb{R}\times S^3\times B^4$ or diffeomorphic to $SU(3)$, where $B^4= \mathbb{R}\times S^3$, $\mathbb{R}^4$ or $K_3$.