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

Georgios Papadopoulos

TL;DR

The paper proves a rigidity result for Riemannian manifolds $(M,g,H)$ equipped with a connection having torsion the closed 3-form $H$ that is $\widehat{\nabla}$-covariantly constant: such manifolds are locally isometric to a product $N\times G$ with $G$ a semisimple group and $N$ carrying a metric annihilated by $H$ in the $N$-directions. This local splitting extends to KT, CYT, HKT, G$_2$, and Spin$(7)$ geometries, providing a unified framework and global statements when $M$ is simply connected and complete. The paper then applies these results to classify complete, simply connected strong $G_2$ and Spin$(7)$ manifolds with the stated torsion parallelism, and shows that compact 8D strong HKT manifolds must be locally isometric to either hyper-Kähler, a bi-invariant $SU(3)$-model, or a product involving $U(1) imes SU(2)$ over a 4-manifold, with topological constraints narrowing the possibilities. It further discusses weakening the parallel-torsion condition (e.g., constant $|H|$) and connects to generalized Ricci solitons, principal-bundle constructions, and explicit examples such as the HKT structure on $SU(3)$, highlighting both the geometric rigidity and the remaining open space for nontrivial compact examples.

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 $(U(1)\times SU(2))\times B^4$, where $B^4$ is either a hyper-Kähler manifold or $U(1)\times SU(2)$ equipped with an HKT structure.

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

TL;DR

The paper proves a rigidity result for Riemannian manifolds equipped with a connection having torsion the closed 3-form that is -covariantly constant: such manifolds are locally isometric to a product with a semisimple group and carrying a metric annihilated by in the -directions. This local splitting extends to KT, CYT, HKT, G, and Spin geometries, providing a unified framework and global statements when is simply connected and complete. The paper then applies these results to classify complete, simply connected strong and Spin manifolds with the stated torsion parallelism, and shows that compact 8D strong HKT manifolds must be locally isometric to either hyper-Kähler, a bi-invariant -model, or a product involving over a 4-manifold, with topological constraints narrowing the possibilities. It further discusses weakening the parallel-torsion condition (e.g., constant ) and connects to generalized Ricci solitons, principal-bundle constructions, and explicit examples such as the HKT structure on , highlighting both the geometric rigidity and the remaining open space for nontrivial compact examples.

Abstract

We demonstrate that all Riemannian manifolds that admit a connection with torsion a 3-form , which is both closed and -covariantly constant, are locally isometric to a product , where is a semisimple group and is a Riemannian manifold with for all tangent vectors , . If is simply connected and complete, then by the de Rham theorem 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 and manifolds with torsion. As an application, we describe the geometry of all complete and simply connected and 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; equipped with the bi-invariant metric and 3-form; or the product , where is either a hyper-Kähler manifold or equipped with an HKT structure.

Paper Structure

This paper contains 15 sections, 16 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Manifolds with parallel 3-form torsion
  3. Riemannian manifolds with parallel 3-form torsion
  4. KT and CYT manifolds with parallel 3-form torsion
  5. HKT manifolds with parallel 3-form torsion
  6. $G_2$ and $\mathrm{Spin}(7)$ manifolds with torsion
  7. $G_2$ manifolds
  8. $\mathrm{Spin}(7)$
  9. Geometric structures and restrictions on torsion
  10. Generalised Ricci solitons and compact strong HKT manifolds
  11. Generalised Ricci solitons and strong CYT structures
  12. Principal bundles and strong HKT structures
  13. An example
  14. Compact strong 8-dimensional HKT manifolds
  15. A refinement of theorems \ref{['th:pfibre']} and \ref{['th:pfibre2']}

Key Result

Theorem 1.1

Let $(M^n, g, H)$ be a connected Riemannian manifold, $n\geq 3$, with metric $g$ and 3-form $H$. If $H$ is closed and $\widehat{\nabla}$-covariantly constant with respect to the metric connection with torsion $H$, $\widehat{\nabla} H=0$, then $M$ is locally isometric to $N\times G$, where $G$ is a s

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Remark
  • Lemma 2.2
  • proof
  • proof
  • Corollary 2.1
  • proof
  • ...and 43 more