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${\mathrm G}_2$-structures with parallel skew-symmetric torsion

Andrei Moroianu, Uwe Semmelmann

TL;DR

The paper delivers a complete local classification of 7-dimensional Riemannian manifolds carrying a metric connection with skew-symmetric, parallel torsion and holonomy contained in $\mathrm{G}_2$, up to naturally reductive homogeneous spaces and nearly parallel $\mathrm{G}_2$-structures. The authors organize the analysis by the dimension of the space of $\nabla^\tau$-parallel vector fields and derive explicit geometric models: naturally reductive spaces, torsion-free $G_2$-manifolds, and a variety of fibrations and Sasaki-type constructions whose bases are Calabi–Yau, strictly nearly Kähler, or Kähler–Einstein, including circle and twistor space fibrations over ASD Einstein bases. They also obtain a parallel SU(3) classification as a corollary by examining the corresponding projections, thus extending Friedrich’s cocalibrated $G_2$ results and completing the broader program for parallel skew-symmetric torsion geometries. The work leverages the standard decomposition of the tangent bundle, reductions to base manifolds via submersions, and the interplay between $G_2$ and $SU(3)$ structures to produce a comprehensive local picture with explicit torsion and curvature data. Overall, the results provide a detailed map of how parallel torsion and $G_2$-holonomy constrain seven-dimensional geometries, with clear links to Sasaki, nearly Kähler, Calabi–Yau, and hyperkähler geometries.

Abstract

We classify $7$-dimensional Riemannian manifolds carrying a metric connection with parallel skew-symmetric torsion whose holonomy is contained in $\mathrm{G}_2$, up to naturally reductive homogeneous spaces and nearly parallel $\mathrm{G}_2$-structures. This extends and completes the classification initiated by Th. Friedrich in the cocalibrated case. Incidentally, we also obtain the list of $\mathrm{SU}(3)$ geometries with parallel skew-symmetric torsion, up to naturally reductive homogeneous spaces and nearly Kähler manifolds.

${\mathrm G}_2$-structures with parallel skew-symmetric torsion

TL;DR

The paper delivers a complete local classification of 7-dimensional Riemannian manifolds carrying a metric connection with skew-symmetric, parallel torsion and holonomy contained in , up to naturally reductive homogeneous spaces and nearly parallel -structures. The authors organize the analysis by the dimension of the space of -parallel vector fields and derive explicit geometric models: naturally reductive spaces, torsion-free -manifolds, and a variety of fibrations and Sasaki-type constructions whose bases are Calabi–Yau, strictly nearly Kähler, or Kähler–Einstein, including circle and twistor space fibrations over ASD Einstein bases. They also obtain a parallel SU(3) classification as a corollary by examining the corresponding projections, thus extending Friedrich’s cocalibrated results and completing the broader program for parallel skew-symmetric torsion geometries. The work leverages the standard decomposition of the tangent bundle, reductions to base manifolds via submersions, and the interplay between and structures to produce a comprehensive local picture with explicit torsion and curvature data. Overall, the results provide a detailed map of how parallel torsion and -holonomy constrain seven-dimensional geometries, with clear links to Sasaki, nearly Kähler, Calabi–Yau, and hyperkähler geometries.

Abstract

We classify -dimensional Riemannian manifolds carrying a metric connection with parallel skew-symmetric torsion whose holonomy is contained in , up to naturally reductive homogeneous spaces and nearly parallel -structures. This extends and completes the classification initiated by Th. Friedrich in the cocalibrated case. Incidentally, we also obtain the list of geometries with parallel skew-symmetric torsion, up to naturally reductive homogeneous spaces and nearly Kähler manifolds.

Paper Structure

This paper contains 21 sections, 21 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalities on multilinear algebra
  4. $\mathrm{SU}(3)$-structures
  5. $\mathrm{G}_2$-structures
  6. Sasaki-type structures
  7. Nearly Kähler and Calabi-Yau manifolds
  8. Metric connections with parallel skew-symmetric torsion
  9. $\nabla^\tau$-parallel vector fields
  10. $\mathrm{G}_2$-structures with torsion
  11. The case $\dim(\mathcal{H})=6$
  12. The case $\alpha=0$
  13. The case $\alpha\ne 0$ and $\sigma= 0$
  14. The case $\dim(\mathcal{H})=4$
  15. The case $a+2b=0$
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $(M^7, g, \tau, \varphi)$ be a Riemannian manifold with a $\mathrm{G}_2$-structure defined by the $3$-form $\varphi$ and a metric connection $\nabla^\tau$ with skew-symmetric and parallel torsion $\tau$, preserving the $\mathrm{G}_2$-structure, i.e. $\nabla^\tau \tau = 0$ and $\nabla^\tau \varph

Theorems & Definitions (40)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 30 more