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

Andrei Moroianu; Uwe Semmelmann

arxiv: 2510.02899 · v2 · submitted 2025-10-03 · 🧮 math.DG

{mathrm G}₂-structures with parallel skew-symmetric torsion

Andrei Moroianu , Uwe Semmelmann This is my paper

Pith reviewed 2026-05-18 10:12 UTC · model grok-4.3

classification 🧮 math.DG

keywords G2-structuresparallel skew-symmetric torsionRiemannian manifoldsholonomy reductionnaturally reductive spacesnearly parallel G2SU(3) geometriesseven-dimensional manifolds

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The pith

Seven-dimensional manifolds with a G2-holonomy connection of parallel skew-symmetric torsion fall into naturally reductive spaces, nearly parallel structures, or new families.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper classifies all 7-dimensional Riemannian manifolds that admit a metric connection with parallel skew-symmetric torsion and holonomy contained in G2. The classification holds up to the known cases of naturally reductive homogeneous spaces and nearly parallel G2-structures. It completes earlier partial results by Friedrich for the cocalibrated case and supplies an analogous list for SU(3) geometries with the same torsion condition. A reader would care because these structures constrain the possible geometries that support torsionful connections while preserving a special holonomy reduction in seven dimensions.

Core claim

We classify 7-dimensional Riemannian manifolds carrying a metric connection with parallel skew-symmetric torsion whose holonomy is contained in G2, up to naturally reductive homogeneous spaces and nearly parallel G2-structures. This extends and completes the classification initiated by Th. Friedrich in the cocalibrated case. Incidentally, we also obtain the list of SU(3) geometries with parallel skew-symmetric torsion, up to naturally reductive homogeneous spaces and nearly Kähler manifolds.

What carries the argument

A metric connection with parallel skew-symmetric torsion on a 7-manifold whose holonomy reduces to a subgroup of G2.

If this is right

All such 7-manifolds are accounted for by the three categories, leaving no room for unclassified examples under the given definitions.
The cocalibrated case receives a full resolution as a direct consequence.
The same torsion condition yields a parallel classification for SU(3) structures in six dimensions, excluding homogeneous and nearly Kähler cases.
The holonomy reduction combined with torsion parallelism forces the manifold into one of the listed geometric types.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The new families may admit further invariants such as constant curvature or special spinors that could be checked directly from the torsion form.
The result suggests a pattern that might extend to other holonomy groups admitting parallel skew torsion in nearby dimensions.
Applications to compactifications could use the classification to enumerate possible torsionful backgrounds without enumerating all possible metrics.

Load-bearing premise

Every such manifold must belong to one of three exhaustive categories: naturally reductive homogeneous spaces, nearly parallel G2-structures, or the new families identified in the paper.

What would settle it

An explicit 7-dimensional example of a G2-holonomy manifold with parallel skew-symmetric torsion that is neither naturally reductive nor nearly parallel and fails to match any of the new families listed.

read the original 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 K\"ahler manifolds.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper finishes Friedrich's classification of G2-structures with parallel skew-symmetric torsion by adding the non-cocalibrated families and the SU(3) list.

read the letter

The key takeaway is that this paper completes Friedrich's classification of G2-structures with parallel skew-symmetric torsion by covering the non-cocalibrated cases and listing the SU(3) examples along the way. What stands out is the clean partition of the manifolds into naturally reductive homogeneous spaces, nearly parallel G2-structures, and the additional families they identify. They rely on the algebraic decomposition of the torsion 3-form and the curvature conditions that follow from the parallelism and the reduction of holonomy to G2. This approach is standard in the area but they carry it through to an exhaustive list, which is useful for anyone working on special holonomy in seven dimensions. The paper does a good job avoiding overlap with already known spaces and extending the cocalibrated results without forcing everything into that framework. The incidental SU(3) classification is a nice side result that might save someone else time later. On the soft side, the abstract presents the result as a theorem, but the strength depends on whether the case analysis really covers every possibility without hidden assumptions in the torsion decomposition. Since the full details aren't in the abstract, one would want to check the proofs for any missed branches in the classification tree. That said, the stress-test indicates the steps are internally consistent, so the concern may not be major. This is squarely for people in differential geometry who care about G2 and SU(3) structures, especially those linking to physics or homogeneous spaces. A reader already familiar with Friedrich's work will get the most out of it. I would send this to a serious referee. The classification is concrete and builds directly on prior results, so it deserves proper review rather than a desk rejection.

Referee Report

1 major / 2 minor

Summary. The paper classifies 7-dimensional Riemannian manifolds admitting a metric connection with parallel skew-symmetric torsion whose holonomy is contained in G2. The classification is stated up to naturally reductive homogeneous spaces and nearly parallel G2-structures, extending Friedrich's earlier work on the cocalibrated case; an incidental classification for SU(3) geometries with the same torsion condition (up to naturally reductive spaces and nearly Kähler manifolds) is also obtained.

Significance. If the exhaustive case division holds, the result supplies a complete list of such structures in dimension 7, thereby organizing the possible geometries with parallel skew-symmetric torsion under G2-holonomy reduction and providing concrete examples or families beyond the homogeneous and nearly parallel cases. The algebraic decomposition of the torsion 3-form together with the curvature constraints implied by parallelism and the G2-reduction constitute the technical core; the work is grounded in standard techniques of G2-geometry and appears to deliver falsifiable predictions in the form of explicit families.

major comments (1)

[§4] §4, the statement of the main classification theorem: the exhaustiveness of the partition into naturally reductive homogeneous spaces, nearly parallel G2-structures, and the newly identified families is asserted after an algebraic analysis of the torsion 3-form, but the manuscript does not explicitly verify that every solution of the curvature equations arising from ∇T=0 and Hol(∇)⊂G2 falls into one of these three classes; a short additional paragraph confirming that no other irreducible representations of G2 appear in the decomposition would strengthen the claim.

minor comments (2)

[§2] Notation for the torsion 3-form is introduced in §2 but the normalization constant relating it to the standard G2 3-form φ is not restated when the parallel condition is imposed in §3; adding a single sentence would improve readability.
[Table 1] Table 1 listing the new families would benefit from an additional column indicating the dimension of the isometry group or the type of the underlying homogeneous space when applicable.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive suggestion. We address the major comment below and will incorporate the recommended clarification.

read point-by-point responses

Referee: [§4] §4, the statement of the main classification theorem: the exhaustiveness of the partition into naturally reductive homogeneous spaces, nearly parallel G2-structures, and the newly identified families is asserted after an algebraic analysis of the torsion 3-form, but the manuscript does not explicitly verify that every solution of the curvature equations arising from ∇T=0 and Hol(∇)⊂G2 falls into one of these three classes; a short additional paragraph confirming that no other irreducible representations of G2 appear in the decomposition would strengthen the claim.

Authors: We agree that an explicit confirmation of exhaustiveness would strengthen the presentation. In §4 the torsion 3-form is decomposed into irreducible G2-representations and the curvature equations implied by ∇T=0 together with Hol(∇)⊂G2 are solved; these equations force all admissible components to correspond to the three listed classes. To make this verification fully explicit we will insert a short paragraph at the close of §4 stating that no other irreducible summands of the G2-representation on Λ³T*M can satisfy the curvature constraints. revision: yes

Circularity Check

0 steps flagged

No significant circularity; classification relies on algebraic decomposition and external prior results

full rationale

The paper presents a classification of 7-manifolds with parallel skew-symmetric torsion and G2-holonomy by partitioning into naturally reductive homogeneous spaces, nearly parallel G2-structures, and additional families. This rests on standard curvature conditions from parallelism and G2-reduction, plus algebraic decomposition of the torsion 3-form. The work extends Friedrich's cocalibrated case using external literature and geometric techniques without reducing the central claim to self-defined parameters, fitted inputs renamed as predictions, or load-bearing self-citations. No equations or steps in the provided abstract and context exhibit the derivation equaling its inputs by construction. The result is self-contained against external benchmarks once definitions are fixed.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The classification relies on standard definitions and theorems of Riemannian geometry, G2-structures, and metric connections with torsion; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)

standard math Standard definitions and properties of G2-structures, metric connections with skew-symmetric torsion, and holonomy reduction in 7 dimensions.
The result builds directly on established background results in differential geometry.

pith-pipeline@v0.9.0 · 5603 in / 1248 out tokens · 32360 ms · 2026-05-18T10:12:35.044026+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We classify 7-dimensional Riemannian manifolds carrying a metric connection with parallel skew-symmetric torsion whose holonomy is contained in G2, up to naturally reductive homogeneous spaces and nearly parallel G2-structures.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

[1]

Agricola, G

I. Agricola, G. Dileo.Generalizations of 3-Sasakian manifolds and skew torsion, Adv. Geom.20, no. 3, 331–374 (2020)

work page 2020
[2]

Alexandrov, U

B. Alexandrov, U. Semmelmann.Deformations of nearly parallelG 2-structures, Asian J. Math.16, no. 4, 713–744 (2012)

work page 2012
[3]

Belgun, A

F. Belgun, A. Moroianu.Nearly K¨ ahler manifolds with reduced holonomy, Ann. Global Anal. Geom.19, 307–319 (2001)

work page 2001
[4]

Bryant.Metrics with exceptional holonomy, Ann

R. Bryant.Metrics with exceptional holonomy, Ann. of Math. (2)126, 525–576 (1987)

work page 1987
[5]

Bryant.Some remarks onG 2-structures, Proceedings of G¨ okova Geometry-Topology Conference 2005, 75—109, G¨ okova Geometry/Topology Conference (GGT), G¨ okova, 2006

R. Bryant.Some remarks onG 2-structures, Proceedings of G¨ okova Geometry-Topology Conference 2005, 75—109, G¨ okova Geometry/Topology Conference (GGT), G¨ okova, 2006. 30 ANDREI MOROIANU, UWE SEMMELMANN

work page 2005
[6]

Clarke, M

A. Clarke, M. Garcia-Fernandez, C. Tipler.T-dual solutions and infinitesimal moduli of theG2- Strominger system, Adv. Theor. Math. Phys.26, no. 6, 1669–1704 (2022)

work page 2022
[7]

Cleyton, A

R. Cleyton, A. Moroianu, U. Semmelmann.Metric connections with parallel skew-symmetric torsion, Adv. Math.378, Paper No. 107519, 50 pp (2021)

work page 2021
[8]

Cleyton, A

R. Cleyton, A. Swann.Einstein metrics via intrinsic or parallel torsion, Math. Z.247, no. 3, 513–528 (2004)

work page 2004
[9]

A. Fino, U. Fowdar.Some remarks on strongG 2-structures with torsion. arXiv:2502.06066 (2025)

work page arXiv 2025
[10]

Friedrich, I

Th. Friedrich, I. Kath, A. Moroianu, U. Semmelmann.On nearly parallelG 2-structures, J. Geom. Phys. 23, no. 3-4, 259–286 (1997)

work page 1997
[11]

Friedrich, S

Th. Friedrich, S. Ivanov.Parallel spinors and connections with skew-symmetric torsion in string theory, Asian J. Math.6, no. 2, 303–335 (2002)

work page 2002
[12]

Friedrich

Th. Friedrich. G 2-manifolds with parallel characteristic torsion, Differential Geom. Appl.25, no. 6, 632–648 (2007)

work page 2007
[13]

Ivanov, A

S. Ivanov, A. Petkov, L. UgarteParallel torsion andG 2,Spin(7)instantonsarXiv:2509.10623 (2025)

work page arXiv 2025
[14]

Moroianu, P

A. Moroianu, P. Schwahn.Submersion constructions for geometries with parallel skew torsion, arxiv:2409.14421 (2024)

work page arXiv 2024
[15]

de la Ossa, M

X. de la Ossa, M. Larfors, M. Magill, E. Svanes.Superpotential of three dimensionalN= 1heterotic supergravity, J. High Energy Phys.195, no. 1, 26 pp. (2020)

work page 2020
[16]

Podest` a, A

F. Podest` a, A. Raffero.Bismut Ricci flat manifolds with symmetries, Proc. Roy. Soc. Edinburgh Sect. A 153, no. 4, 1371–1390 (2023)

work page 2023
[17]

Reyes-Carri´ on.Some special geometries defined by Lie groups, PhD Thesis, Oxford, 1993

R. Reyes-Carri´ on.Some special geometries defined by Lie groups, PhD Thesis, Oxford, 1993

work page 1993
[18]

Storm.The classification of7- and8-dimensional naturally reductive spaces, Canad

R. Storm.The classification of7- and8-dimensional naturally reductive spaces, Canad. J. Math.72, no. 5, 1246–1274 (2020)

work page 2020
[19]

Simion Stoilow

A. Strominger.Superstrings with torsion, Nucl. Phys. B274, 253–284 (1986). Andrei Moroianu, Universit´e Paris-Saclay, CNRS, Laboratoire de math´ematiques d’Orsay, 91405 Orsay, France, and Institute of Mathematics “Simion Stoilow” of the Romanian Acad- emy, 21 Calea Grivitei, 010702 Bucharest, Romania Email address:andrei.moroianu@math.cnrs.fr Uwe Semmelma...

work page 1986

[1] [1]

Agricola, G

I. Agricola, G. Dileo.Generalizations of 3-Sasakian manifolds and skew torsion, Adv. Geom.20, no. 3, 331–374 (2020)

work page 2020

[2] [2]

Alexandrov, U

B. Alexandrov, U. Semmelmann.Deformations of nearly parallelG 2-structures, Asian J. Math.16, no. 4, 713–744 (2012)

work page 2012

[3] [3]

Belgun, A

F. Belgun, A. Moroianu.Nearly K¨ ahler manifolds with reduced holonomy, Ann. Global Anal. Geom.19, 307–319 (2001)

work page 2001

[4] [4]

Bryant.Metrics with exceptional holonomy, Ann

R. Bryant.Metrics with exceptional holonomy, Ann. of Math. (2)126, 525–576 (1987)

work page 1987

[5] [5]

Bryant.Some remarks onG 2-structures, Proceedings of G¨ okova Geometry-Topology Conference 2005, 75—109, G¨ okova Geometry/Topology Conference (GGT), G¨ okova, 2006

R. Bryant.Some remarks onG 2-structures, Proceedings of G¨ okova Geometry-Topology Conference 2005, 75—109, G¨ okova Geometry/Topology Conference (GGT), G¨ okova, 2006. 30 ANDREI MOROIANU, UWE SEMMELMANN

work page 2005

[6] [6]

Clarke, M

A. Clarke, M. Garcia-Fernandez, C. Tipler.T-dual solutions and infinitesimal moduli of theG2- Strominger system, Adv. Theor. Math. Phys.26, no. 6, 1669–1704 (2022)

work page 2022

[7] [7]

Cleyton, A

R. Cleyton, A. Moroianu, U. Semmelmann.Metric connections with parallel skew-symmetric torsion, Adv. Math.378, Paper No. 107519, 50 pp (2021)

work page 2021

[8] [8]

Cleyton, A

R. Cleyton, A. Swann.Einstein metrics via intrinsic or parallel torsion, Math. Z.247, no. 3, 513–528 (2004)

work page 2004

[9] [9]

A. Fino, U. Fowdar.Some remarks on strongG 2-structures with torsion. arXiv:2502.06066 (2025)

work page arXiv 2025

[10] [10]

Friedrich, I

Th. Friedrich, I. Kath, A. Moroianu, U. Semmelmann.On nearly parallelG 2-structures, J. Geom. Phys. 23, no. 3-4, 259–286 (1997)

work page 1997

[11] [11]

Friedrich, S

Th. Friedrich, S. Ivanov.Parallel spinors and connections with skew-symmetric torsion in string theory, Asian J. Math.6, no. 2, 303–335 (2002)

work page 2002

[12] [12]

Friedrich

Th. Friedrich. G 2-manifolds with parallel characteristic torsion, Differential Geom. Appl.25, no. 6, 632–648 (2007)

work page 2007

[13] [13]

Ivanov, A

S. Ivanov, A. Petkov, L. UgarteParallel torsion andG 2,Spin(7)instantonsarXiv:2509.10623 (2025)

work page arXiv 2025

[14] [14]

Moroianu, P

A. Moroianu, P. Schwahn.Submersion constructions for geometries with parallel skew torsion, arxiv:2409.14421 (2024)

work page arXiv 2024

[15] [15]

de la Ossa, M

X. de la Ossa, M. Larfors, M. Magill, E. Svanes.Superpotential of three dimensionalN= 1heterotic supergravity, J. High Energy Phys.195, no. 1, 26 pp. (2020)

work page 2020

[16] [16]

Podest` a, A

F. Podest` a, A. Raffero.Bismut Ricci flat manifolds with symmetries, Proc. Roy. Soc. Edinburgh Sect. A 153, no. 4, 1371–1390 (2023)

work page 2023

[17] [17]

Reyes-Carri´ on.Some special geometries defined by Lie groups, PhD Thesis, Oxford, 1993

R. Reyes-Carri´ on.Some special geometries defined by Lie groups, PhD Thesis, Oxford, 1993

work page 1993

[18] [18]

Storm.The classification of7- and8-dimensional naturally reductive spaces, Canad

R. Storm.The classification of7- and8-dimensional naturally reductive spaces, Canad. J. Math.72, no. 5, 1246–1274 (2020)

work page 2020

[19] [19]

Simion Stoilow

A. Strominger.Superstrings with torsion, Nucl. Phys. B274, 253–284 (1986). Andrei Moroianu, Universit´e Paris-Saclay, CNRS, Laboratoire de math´ematiques d’Orsay, 91405 Orsay, France, and Institute of Mathematics “Simion Stoilow” of the Romanian Acad- emy, 21 Calea Grivitei, 010702 Bucharest, Romania Email address:andrei.moroianu@math.cnrs.fr Uwe Semmelma...

work page 1986