Four-dimensional Riemannian geometry via 2-forms

Kirill Krasnov; Niren Bhoja

arxiv: 2405.15408 · v2 · submitted 2024-05-24 · 🧮 math.DG · gr-qc· hep-th

Four-dimensional Riemannian geometry via 2-forms

Niren Bhoja , Kirill Krasnov This is my paper

Pith reviewed 2026-05-24 01:33 UTC · model grok-4.3

classification 🧮 math.DG gr-qchep-th

keywords four-dimensional Riemannian geometrySU(2)-structuresSO(3)-bundlesEinstein metricsintrinsic torsionKähler geometryhyper-Kähler geometryvariational principles

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The pith

In four dimensions Riemannian geometry is encoded by an SO(3)-valued 2-form on SU(2)-structures, yielding a unique invariant functional whose critical points are Einstein metrics.

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

The paper extends the encoding of geometric structures by differential forms to the case where the full structure group is a larger group tilde(G) whose quotient by a spinorial subgroup G is itself a Lie group H. In this setting the defining forms of the G-structure are allowed to take values in an associated H-bundle while the intrinsic torsion is reinterpreted as an H-connection. Applied in four dimensions this packages the usual triple of 2-forms of an SU(2)-structure into a single 2-form valued in the associated SO(3) bundle. The construction recovers the full SO(4)-structure of Riemannian geometry and produces a variational principle whose stationary points are Einstein metrics. The same data also furnish a common description of Riemannian, Kähler and hyper-Kähler geometries.

Core claim

When tilde(G)/G is a Lie group H, a tilde(G)-structure can be described via a spinorial G-structure by letting the defining forms take values in an associated H-bundle and converting intrinsic torsion into an H-connection. In four dimensions with G=SU(2) and H=SO(3) the triple of 2-forms of the SU(2)-structure is encoded as a single 2-form valued in the SO(3) bundle; this data determines an SO(4)-structure and there exists a unique SO(3)-invariant functional on such SU(2)-structures whose critical points are precisely the Einstein metrics. The same framework also unifies the descriptions of Riemannian, Kähler and hyper-Kähler geometries.

What carries the argument

The single 2-form valued in the SO(3) bundle associated to an SU(2)-structure, which packages the triple of defining 2-forms and turns intrinsic torsion into an SO(3)-connection.

If this is right

The critical points of the functional are Einstein metrics.
The construction recovers the full SO(4)-structure from the SU(2)-structure and the SO(3)-connection.
Riemannian, Kähler and hyper-Kähler geometries in four dimensions share a common description in terms of the same SO(3)-valued 2-form data.
The intrinsic torsion of the SU(2)-structure is reinterpreted as the curvature of an SO(3)-connection.

Where Pith is reading between the lines

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

The functional could be used as a numerical tool to locate Einstein metrics by direct minimization.
The same packaging technique might apply in other dimensions whenever a suitable Lie-group quotient of structure groups exists.
The approach supplies a new variational setting in which to study the moduli space of Einstein metrics in four dimensions.

Load-bearing premise

The triple of 2-forms of an SU(2)-structure can be faithfully encoded as a single 2-form valued in the associated SO(3) bundle while preserving the differential constraints needed to recover the full Riemannian geometry and to produce the claimed functional.

What would settle it

Deriving the Euler-Lagrange equations of the proposed SO(3)-invariant functional and checking whether they are equivalent to the Einstein equation Ric = lambda g.

read the original abstract

In differential geometry, geometric structures can often be encoded by differential forms satisfying algebraic and differential constraints. This is in particular the case for spinorial G-structures, where the defining tensors are differential forms arising as spinor bilinears and their exterior derivatives determine the intrinsic torsion. In this paper we show that, in certain situations, this can be extended beyond the setting of spinorial G-structures. Thus, when tilde(G)/G is a Lie group H, a tilde(G)-structure with tilde(G) supset G can be described in terms of a spinorial G-structure by allowing the defining forms to take values in an associated H-bundle, and converting the intrinsic torsion of the G-structure into an H-connection. We develop this idea in four dimensions, where the triple of 2-forms associated with a spinorial SU(2)-structure can be encoded as a 2-form with values in the associated H=SO(4)/SU(2)=SO(3) vector bundle. This gives a description of Riemannian geometry, i.e. SO(4)-structures, and leads to a unique SO(3)-invariant functional of SU(2)-structures whose critical points are Einstein. This perspective also provides a unified framework for Riemannian, Kahler and hyper-Kahler geometries in four dimensions.

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

The paper repackages the SU(2) triple of 2-forms as an SO(3)-valued 2-form and produces a unique invariant functional on SU(2)-structures whose critical points are Einstein metrics.

read the letter

The main takeaway is that the authors encode an SU(2)-structure in four dimensions by letting the three real 2-forms take values in the associated SO(3) bundle coming from the quotient SO(4)/SU(2), then convert the intrinsic torsion into an SO(3)-connection. From this they extract a single SO(3)-invariant functional whose critical points recover the Einstein condition, and they show how this setup also captures Kähler and hyper-Kähler cases as special instances. The general construction for when the quotient is a Lie group H is laid out first, then specialized to 4D. This is a clean algebraic move that stays inside standard G-structure language while adding the H-bundle twist. The unification under one functional is the part that stands out as new relative to the cited spinorial literature. The algebra appears to check out on the level of the structure group reduction, and the claim that the metric is recovered from the valued form is stated explicitly. The soft spot is the differential side: whether the exterior derivative on the SO(3)-valued form reproduces the original torsion equations without kernel or loss of information. The abstract asserts this works, but the verification that the map is bijective at the level of forms and that no extra constraints are introduced or dropped needs to be checked in the body. If that step holds, the equivalence to full Riemannian geometry is fine; if it has a gap, the functional would only characterize a subclass. This is written for people already comfortable with G-structures and intrinsic torsion in low dimensions. A reader who knows the standard SU(2) triple will see the value quickly. It is worth sending to a serious referee because the claims are specific enough to test and the construction is distinct from prior variational approaches in 4D Einstein geometry.

Referee Report

2 major / 1 minor

Summary. The paper claims that when tilde(G)/G is a Lie group H, a tilde(G)-structure can be described via a spinorial G-structure by allowing the defining forms to take values in an associated H-bundle and converting the intrinsic torsion into an H-connection. In four dimensions, the triple of 2-forms of an SU(2)-structure is encoded as a single 2-form valued in the associated SO(3)-bundle; this yields a description of all SO(4)-structures together with a unique SO(3)-invariant functional on SU(2)-structures whose critical points are precisely the Einstein metrics, while also unifying Riemannian, Kähler, and hyper-Kähler geometries.

Significance. If the encoding step is bijective and the differential constraints are preserved without kernel or cokernel, the construction supplies a new variational characterization of Einstein metrics in dimension 4 and a single framework encompassing several special geometries. The approach extends ordinary spinorial G-structures in a controlled way and could be useful for deformation theory or moduli problems.

major comments (2)

[main construction (4D case)] The central encoding step (described after the abstract and developed in the 4D construction): the algebraic map sending the SU(2) triple of real 2-forms to a single SO(3)-valued 2-form must be shown to be bijective on the level of the structure-group reduction, and the exterior derivative on the valued form must reproduce the original intrinsic-torsion equations exactly. Without an explicit verification of both properties, the claimed equivalence to all SO(4)-structures and the subsequent variational characterization do not follow.
[variational principle] The functional (introduced in the paragraph on the unique SO(3)-invariant functional): the paper asserts it is the unique such functional whose critical points are Einstein metrics, yet no explicit expression for the functional, no derivation of its Euler-Lagrange equations, and no verification that those equations reduce to the Einstein condition are supplied. These steps are load-bearing for the variational claim.

minor comments (1)

[general construction] Notation for the associated H-bundle and the precise meaning of 'converting intrinsic torsion into an H-connection' should be introduced with a short diagram or commutative square to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the detailed comments. We agree that the major points raised require explicit verifications and derivations that are currently missing from the text. We will revise the manuscript to address these issues fully.

read point-by-point responses

Referee: [main construction (4D case)] The central encoding step (described after the abstract and developed in the 4D construction): the algebraic map sending the SU(2) triple of real 2-forms to a single SO(3)-valued 2-form must be shown to be bijective on the level of the structure-group reduction, and the exterior derivative on the valued form must reproduce the original intrinsic-torsion equations exactly. Without an explicit verification of both properties, the claimed equivalence to all SO(4)-structures and the subsequent variational characterization do not follow.

Authors: We agree that the bijectivity of the encoding map at the level of structure-group reduction and the precise reproduction of the intrinsic-torsion equations by the exterior derivative on the SO(3)-valued form must be verified explicitly. The current manuscript outlines the general construction but does not supply the full algebraic details or the check that there is no kernel or cokernel. In the revised version we will add a dedicated subsection containing these explicit computations, confirming the equivalence to all SO(4)-structures. revision: yes
Referee: [variational principle] The functional (introduced in the paragraph on the unique SO(3)-invariant functional): the paper asserts it is the unique such functional whose critical points are Einstein metrics, yet no explicit expression for the functional, no derivation of its Euler-Lagrange equations, and no verification that those equations reduce to the Einstein condition are supplied. These steps are load-bearing for the variational claim.

Authors: The referee is correct that the manuscript states the existence and uniqueness of the SO(3)-invariant functional and that its critical points are Einstein metrics, but does not provide the explicit expression, the derivation of the Euler-Lagrange equations, or the verification that these reduce to the Einstein condition. These omissions prevent the variational claim from being fully substantiated. We will include in the revision the explicit formula for the functional, the computation of its first variation, the resulting EL equations, and the direct check that they are equivalent to Ric = λ g, together with a brief argument for uniqueness among SO(3)-invariant functionals. revision: yes

Circularity Check

0 steps flagged

No circularity; construction is self-contained

full rationale

The paper's derivation introduces an extension of spinorial G-structures by allowing defining forms to take values in an associated H-bundle when tilde(G)/G = H, then specializes to 4D where the SU(2) triple of 2-forms is repackaged as an SO(3)-valued 2-form. This yields a description of SO(4)-structures and an SO(3)-invariant functional whose critical points are Einstein metrics. No step reduces by definition to its own output (no self-definitional loops), no fitted parameters are renamed as predictions, and no load-bearing uniqueness or ansatz is imported via self-citation. The encoding and functional are presented as direct consequences of the algebraic and differential properties of 2-forms in 4D, independent of the target results. The derivation chain therefore remains non-circular and rests on standard G-structure axioms plus explicit 4D identities.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The construction rests on standard differential-geometry background plus one domain-specific encoding assumption; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption The triple of 2-forms associated with a spinorial SU(2)-structure can be encoded as a 2-form valued in the associated SO(3) bundle while preserving the necessary differential constraints
This encoding is the step that converts the G-structure into an SO(4)-structure and produces the functional.

pith-pipeline@v0.9.0 · 5766 in / 1305 out tokens · 26317 ms · 2026-05-24T01:33:44.418061+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 (D=3) unclear

?

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

when ˜G/G is a Lie group H, a ˜G-structure ... encoded as a 2-form with values in the associated H=SO(4)/SU(2)=SO(3) vector bundle ... unique SO(3)-invariant functional ... critical points are Einstein
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel (J-cost uniqueness) unclear

?

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

S[Σ] = −½ ∫ ϵijk Σi ∧ Aj ∧ Ak ... critical points ... Einstein

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.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

General Relativity via differential forms -- explorations in Plebanski's Formalism for GR
gr-qc 2026-04 unverdicted novelty 3.0

Plebanski's chiral 2-form formulation of GR reveals additional structure in Einstein's equations and supplies new analytical and numerical tools.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages · cited by 1 Pith paper · 3 internal anchors

[1]

Bryant, ”Recent Advances in the Theory of Holonomy,” Seminaire Bourbaki, Volume 1998/99, Asterisque 266 (2000), 351-374

R. Bryant, ”Recent Advances in the Theory of Holonomy,” Seminaire Bourbaki, Volume 1998/99, Asterisque 266 (2000), 351-374

work page 1998
[2]

Chern, ”On a generalisation of K¨ ahler geometry,” Lefschetz jubilee volume, Princeton Univ

S-s. Chern, ”On a generalisation of K¨ ahler geometry,” Lefschetz jubilee volume, Princeton Univ. Press (1957) 103-121

work page 1957
[3]

Fowdar and H

U. Fowdar and H. S´ a Earp, ”Flows of SU(2)-structures,” Mathematische Zeitschrift, vol. 311, no. 1, https://doi.org/10.1007/s00209-025-03806-7

work page doi:10.1007/s00209-025-03806-7
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Lucio Bedulli, Luigi Vezzoni, ”The Ricci tensor of SU(3)-manifolds,” J. Geom. Phys. 57 (2007), n. 4, 1125-1146,https://doi.org/10.1016/j.geomphys.2006.09.007. 28 BHOJA AND KRASNOV

work page doi:10.1016/j.geomphys.2006.09.007 2007
[5]

org/10.1142/9789812790613_0023

Spiro Karigiannis, ”Flows of Spin(7)-structures,” Proceedings of the 10th Conference on Differential Geometry and its Applications: DGA 2007; World Scientific Publishing, (2008), 263-277, https://doi. org/10.1142/9789812790613_0023

work page doi:10.1142/9789812790613_0023 2007
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work page doi:10.1063/1.523215 1977
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work page doi:10.1016/j.geomphys.2005.01.004 2005
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F. M. Cabrera and A. Swann, ”Curvature of special almost Hermitian manifolds,” Pacific Journal of Mathematics, 228 (2006), n. 1, 165-184,https://doi.org/10.48550/arXiv.math/0501062

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.math/0501062 2006
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J. C. Gonz´ alez–D´ avila, and F. M. Cabrera, ”Harmonic G-structures,” In Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 146, No. 2, pp. 435-459, https://doi.org/10.48550/arXiv. 0810.1380

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv
[11]

Fadel, E

D. Fadel, E. Loubeau, A. Moreno, and H. S´ a Earp, ”Flows of geometric structures,” Journal f¨ ur die reine und angewandte Mathematik (Crelles Journal), 2024(817), 67-152, https://doi.org/10.48550/ arXiv.2211.05197

work page arXiv 2024
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S. K. Donaldson, ”Two-forms on four-manifolds and elliptic equations,” arXiv:math/0607083 [math.DG]

work page internal anchor Pith review Pith/arXiv arXiv
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Shubham Dwivedi, Panagiotis Gianniotis, Spiro Karigiannis, ”Flows of G2-structures, II: Curvature, torsion, symbols, and functionals,” Pure and Applied Mathematics Quarterly 21 (2025), 2035-2186, https://doi.org/10.4310/PAMQ.250507180929

work page doi:10.4310/pamq.250507180929 2025
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Bryant, ”Some remarks on G2-structures,” arXiv:math/0305124 [math.DG]

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K. Krasnov, “Dynamics of Cayley Forms,” Pure and Applied Mathematics Quarterly, Volume 21 (2025), 1959-2003,https://dx.doi.org/10.4310/PAMQ.250421122254. Email address:kirill.krasnov@nottingham.ac.uk, ORCID: 0000-0003-2800-3767 School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK

work page doi:10.4310/pamq.250421122254 2025

[1] [1]

Bryant, ”Recent Advances in the Theory of Holonomy,” Seminaire Bourbaki, Volume 1998/99, Asterisque 266 (2000), 351-374

R. Bryant, ”Recent Advances in the Theory of Holonomy,” Seminaire Bourbaki, Volume 1998/99, Asterisque 266 (2000), 351-374

work page 1998

[2] [2]

Chern, ”On a generalisation of K¨ ahler geometry,” Lefschetz jubilee volume, Princeton Univ

S-s. Chern, ”On a generalisation of K¨ ahler geometry,” Lefschetz jubilee volume, Princeton Univ. Press (1957) 103-121

work page 1957

[3] [3]

Fowdar and H

U. Fowdar and H. S´ a Earp, ”Flows of SU(2)-structures,” Mathematische Zeitschrift, vol. 311, no. 1, https://doi.org/10.1007/s00209-025-03806-7

work page doi:10.1007/s00209-025-03806-7

[4] [4]

Lucio Bedulli, Luigi Vezzoni, ”The Ricci tensor of SU(3)-manifolds,” J. Geom. Phys. 57 (2007), n. 4, 1125-1146,https://doi.org/10.1016/j.geomphys.2006.09.007. 28 BHOJA AND KRASNOV

work page doi:10.1016/j.geomphys.2006.09.007 2007

[5] [5]

org/10.1142/9789812790613_0023

Spiro Karigiannis, ”Flows of Spin(7)-structures,” Proceedings of the 10th Conference on Differential Geometry and its Applications: DGA 2007; World Scientific Publishing, (2008), 263-277, https://doi. org/10.1142/9789812790613_0023

work page doi:10.1142/9789812790613_0023 2007

[6] [6]

On the separation of Einsteinian subst ructures,

J. F. Plebanski, “On the separation of Einsteinian substructures,” J. Math. Phys.18(1977), 2511-2520, https://doi:10.1063/1.523215

work page doi:10.1063/1.523215 1977

[7] [7]

Charles Boyer, Krzysztof Galicki, Sasakian Geometry, Oxford University Press, 2008

work page 2008

[8] [8]

F. M. Cabrera, ”Special almost Hermitian geometry,” J. Geom. Phys. 55 (2005), n. 4, 450-470, https: //doi.org/10.1016/j.geomphys.2005.01.004

work page doi:10.1016/j.geomphys.2005.01.004 2005

[9] [9]

F. M. Cabrera and A. Swann, ”Curvature of special almost Hermitian manifolds,” Pacific Journal of Mathematics, 228 (2006), n. 1, 165-184,https://doi.org/10.48550/arXiv.math/0501062

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.math/0501062 2006

[10] [10]

J. C. Gonz´ alez–D´ avila, and F. M. Cabrera, ”Harmonic G-structures,” In Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 146, No. 2, pp. 435-459, https://doi.org/10.48550/arXiv. 0810.1380

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv

[11] [11]

Fadel, E

D. Fadel, E. Loubeau, A. Moreno, and H. S´ a Earp, ”Flows of geometric structures,” Journal f¨ ur die reine und angewandte Mathematik (Crelles Journal), 2024(817), 67-152, https://doi.org/10.48550/ arXiv.2211.05197

work page arXiv 2024

[12] [12]

S. K. Donaldson, ”Two-forms on four-manifolds and elliptic equations,” arXiv:math/0607083 [math.DG]

work page internal anchor Pith review Pith/arXiv arXiv

[13] [13]

Shubham Dwivedi, Panagiotis Gianniotis, Spiro Karigiannis, ”Flows of G2-structures, II: Curvature, torsion, symbols, and functionals,” Pure and Applied Mathematics Quarterly 21 (2025), 2035-2186, https://doi.org/10.4310/PAMQ.250507180929

work page doi:10.4310/pamq.250507180929 2025

[14] [14]

Bryant, ”Some remarks on G2-structures,” arXiv:math/0305124 [math.DG]

R. Bryant, ”Some remarks onG 2-structures,” arXiv:math/0305124 [math.DG]

work page arXiv

[15] [15]

Dynamics of Cayley Forms,

K. Krasnov, “Dynamics of Cayley Forms,” Pure and Applied Mathematics Quarterly, Volume 21 (2025), 1959-2003,https://dx.doi.org/10.4310/PAMQ.250421122254. Email address:kirill.krasnov@nottingham.ac.uk, ORCID: 0000-0003-2800-3767 School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK

work page doi:10.4310/pamq.250421122254 2025