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

TL;DR

This paper investigates the local and global rigidity of special geometries with torsion 3-forms that are closed and covariantly constant, extending results to G2 and Spin(7) manifolds and classifying certain compact HKT manifolds.

Contribution

It simplifies proofs of rigidity results for special geometries with torsion and extends these to G2 and Spin(7) manifolds, providing a classification of certain HKT manifolds.

Findings

Manifolds with torsion 3-forms are locally isometric to a product of a Riemannian manifold and a semisimple group.

Complete simply connected manifolds with these properties are globally isometric to such products.

Certain compact 8-dimensional HKT manifolds are either group manifolds or admit specific Lie group actions.

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 $K_3$.