Pith. sign in

REVIEW 1 major objections 2 minor 25 references

Curvature of Lie Groupoids with Source-Fibre Metric and Riemannian Lie Algebroids

T0 review · 1 major / 2 minor · reviewed 2026-06-28 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Lie groupoids with right-invariant source-fibre metrics carry sectional curvatures that reduce to the Arnold-Milnor formula on Lie groups and to O'Neill's formula on Riemannian submersions.

desk verdict The paper sketches a groupoid unification of Arnold-Milnor and O'Neill curvature formulas under right-invariance, but the abstract alone leaves the derivations and exact cancellation unverified. read the letter →

arxiv 2606.04568 v1 pith:GLFJCLSV submitted 2026-06-03 math.DG

René Lang{o}en This is my paper · ORCID
classification math.DG
keywords Liegroupoidssource-fibremetricssectionalcurvatureRiemannianalgebroidsArnold-MilnorformulaO'Neillformulassubmersions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper shows that curvature formulas long treated as separate theories for Lie groups, principal bundles, and Riemannian submersions all arise from one geometric object: a Lie groupoid equipped with a right-invariant source-fibre metric. It derives an explicit sectional curvature expression for such groupoids that extends the classical Arnold-Milnor 1-2-3-4 formula, and it produces the corresponding Lie algebroid version of O'Neill's curvature formulas for Riemannian submersion Lie algebroids. The construction therefore supplies a single setting in which the older results appear as special cases. Examples include the action groupoid of diffeomorphisms on densities, which links the Euler equation to Wasserstein geometry, and a model for the rotational configuration space of the Earth.

What carries the argument

The source-fibre metric on a Lie groupoid, taken to be right-invariant under the groupoid multiplication, which makes the curvature reduce exactly to the classical Arnold-Milnor and O'Neill expressions without extra correction terms.

What would settle it

A direct computation of sectional curvature on a concrete Lie groupoid whose source-fibre metric fails to be right-invariant would show whether the derived formulas still hold or whether additional terms appear.

Watch

Extended reading notes

Core claim

A Lie groupoid with a right-invariant source-fibre metric admits a sectional curvature formula that extends the Arnold-Milnor formula for Lie groups, while the associated Riemannian Lie algebroid satisfies a version of O'Neill's curvature formulas for Riemannian submersions, thereby placing the classical results inside a common groupoid framework.

Load-bearing premise

The source-fibre metric is right-invariant under the groupoid multiplication.

Editorial extensions

If this is right

  • The curvature formula applies to the action groupoid of diffeomorphisms on densities and therefore relates the Euler equation to the geometry of Wasserstein space.
  • The same framework produces a geometric model for the rotational configuration space of the Earth.
  • Curvature results previously obtained separately for Lie groups and for Riemannian submersions become direct special cases of the groupoid formulas.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If right-invariance of the metric is dropped, the curvature expressions would acquire extra terms that the paper does not compute.
  • The unification indicates that other classical curvature identities might admit similar groupoid lifts once an appropriate invariance condition is identified.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 2 minor

Summary. The paper develops a unified framework for classical curvature formulas using Lie groupoids equipped with right-invariant source-fibre metrics and their infinitesimal counterparts (Riemannian Lie algebroids). It derives a sectional curvature formula for such groupoids that extends the Arnold-Milnor 1-2-3-4 formula from Lie groups, and establishes a Lie algebroid version of O'Neill's curvature formulas for Riemannian submersion Lie algebroids. Examples include the action of diffeomorphisms on densities (linking Euler equations to Wasserstein space) and a geometric model for the rotational configuration space of the Earth.

Significance. If the derivations hold without residual terms, the work supplies a common geometric setting that recovers the Arnold-Milnor and O'Neill formulas exactly under the stated invariance assumption. This unification could streamline treatments in geometric mechanics and optimal transport; the concrete examples tying the Euler equation to Wasserstein geometry on densities and to Earth rotation are noteworthy strengths.

major comments (1)
  1. [Section on sectional curvature formula for Lie groupoids (and the corresponding Lie algebroid section)] The load-bearing step for the central claim (exact extension of the Arnold-Milnor formula and exact recovery of O'Neill formulas) is the cancellation of all groupoid-specific correction terms under right-invariance of the source-fibre metric. The manuscript must exhibit the explicit computation (likely in the derivation of the sectional curvature formula) showing that the source map, algebroid bracket, and any non-invariance contributions vanish identically, rather than merely stating that the assumption produces the classical expressions.
minor comments (2)
  1. [Introduction] The abstract refers to the '1-2-3-4' formula; the introduction should briefly recall the classical Arnold-Milnor expression (with equation numbers) for immediate comparison.
  2. [Preliminaries] Notation for the source-fibre metric and its right-invariance should be introduced with a clear definition before the main theorems to avoid ambiguity in later sections.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. The positive assessment of the unification framework and examples is appreciated. We address the major comment below by committing to an explicit expansion of the key derivations.

read point-by-point responses

  1. Referee: [Section on sectional curvature formula for Lie groupoids (and the corresponding Lie algebroid section)] The load-bearing step for the central claim (exact extension of the Arnold-Milnor formula and exact recovery of O'Neill formulas) is the cancellation of all groupoid-specific correction terms under right-invariance of the source-fibre metric. The manuscript must exhibit the explicit computation (likely in the derivation of the sectional curvature formula) showing that the source map, algebroid bracket, and any non-invariance contributions vanish identically, rather than merely stating that the assumption produces the classical expressions.

    Authors: We agree that the explicit verification of term cancellation is essential to substantiate the exact recovery of the classical formulas. In the revised manuscript we will insert a detailed, line-by-line computation in the sectional-curvature section (and the parallel Lie-algebroid section) that isolates each groupoid-specific correction term, applies the right-invariance hypothesis on the source-fibre metric, and demonstrates that the contributions involving the source map, the algebroid bracket, and any non-invariant remainder all cancel identically. The same level of detail will be supplied for the O'Neill-type formulas. These additions will not change the stated theorems but will make the load-bearing cancellation transparent. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; extensions rest on external classical formulas under stated invariance assumption.

full rationale

The provided abstract and context describe a derivation that extends the Arnold-Milnor 1-2-3-4 formula and O'Neill formulas to Lie groupoids and algebroids under the explicit assumption of right-invariance of the source-fibre metric. This assumption is introduced to ensure exact reduction to the classical cases without extra terms, but the derivation itself is not shown to reduce by construction to its inputs, self-citations, or fitted parameters. No load-bearing self-citation chains, self-definitional steps, or renaming of known results appear in the text. The central claims therefore retain independent content from the classical results and the groupoid structure.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The abstract introduces no free parameters, no new postulated entities, and relies only on the standard axioms of Lie groupoid and Lie algebroid theory already present in the prior literature.

assumptions (1)
  • standard math Standard axioms and definitions of Lie groupoids, Lie algebroids, and Riemannian metrics on them
    The framework is built directly on established differential-geometric structures without additional unproved assumptions stated in the abstract.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Curvature of Lie Groupoids with Source-Fibre Metric and Riemannian Lie Algebroids." pith.science (2026). https://pith.science/paper/GLFJCLSV

@misc{pith2026260604568,
  author       = {Pith},
  title        = {Pith review of: Curvature of Lie Groupoids with Source-Fibre Metric and Riemannian Lie Algebroids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GLFJCLSV}},
  note         = {Machine review of arXiv:2606.04568}
}
read the original abstract

Classical curvature formulas for Lie groups, principal bundles, and Riemannian submersions are usually treated as separate theories. This paper shows that they can be understood within a common framework using Lie groupoids with source-fibre metrics and their infinitesimal counterparts, Riemannian Lie algebroids. We derive a sectional curvature formula for Lie groupoids with right-invariant source metrics, extending the Arnold-Milnor ``1-2-3-4'' formula for Lie groups, and we establish a Lie algebroid version of O'Neill's curvature formulas for Riemannian submersion Lie algebroids. The examples include the action of diffeomorphisms on densities on the torus, linking the Euler equation to the Wasserstein space of densities and optimal transport, as well as a geometric model for the rotational configuration space of the Earth.

Figures

Figures reproduced from arXiv: 2606.04568 by the authors.

Figure 1
Figure 1. A basic visualization of a groupoid, with a source fibre Gx, a target fibre G y and g in their intersection. The inclusion of the base B is shown in red. The intersection of Gx with G y is here shown by one point, but is in general a set G y x , which is the right￾translation of the isotropy group at y: G y x = G y y g. (2) The Cartesian product B × B ⇒ B is a Lie groupoid called the pair groupoid, with src(y, x) = … view at source ↗
Figure 2
Figure 2. An illustration of a Lie groupoid G, with a right-invariant vector field. It is “constant” under right-translation along the target fibres G y . The vectors are source￾tangents, thus has no component in “the source direction”. The vector field projects under d trg to a vector field on T B. The red diagonal is the identity inclusion idB, and the tangent spaces along it constitute the Lie algebroid A. The restriction … view at source ↗
Figure 3
Figure 3. The 2-sphere S 2 with the action Lie algebroid so(3) ⋉ S2 attached. 5. Examples 5.1. Rotations of the 2-sphere. Consider the Lie group SO(3) of orthogonal matrices with determinant equal to 1, SO(3) := {A ∈ GL(3, R) | A ⊺A = I, det A = 1} . It acts transitively on the unit sphere S 2 in R 3 by rotations, and preserves the Euclidean product: ⟨Ax, Ay⟩R3 = ⟨x, A⊺Ay⟩R3 = ⟨x, y⟩R3 , for x, y ∈ S 2 . The action of SO(3) o… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: The real projective line RP1 as the collection of lines through the origin in R 2 . One can also think of RP1 as the circle S 1 with antipodal points identified. The vector fields shown are the S 1 vector field S[x, y] = (y, −x), and the vertical vector field V [x, y] …
Figure 5
Figure 5. Figure 5: Left: the earth modelled as the WGS84 ellipsoid E, isometrically embedded in R 3 , with eccentricity ε (greatly exaggerated for visualization). Right: the 2-sphere S 2 in R 3 , diffeomorphic to E by F : S 2 → E. The diffeomorphism is made into an isometry by pulling ba…
Figure 6
Figure 6. Figure 6: Left: The normalized vector field σ2 on the sphere, with an integral curve. The meridian integral curve is a geodesic, and rotations about the x3-axis act by isome￾tries, generating all meridian geodesics. Right: An artistic illustration of the 2-sphere with the rotati…
Figure 6
Figure 6. Figure 6: 6.3. Computation of curvature. We apply the machinery developed in Section 3 to compute the curvature of the earth modelled as (S 2 ,⟨·, ·⟩S 2 ) together with the associated curvature of the action Lie algebroid so(3) ⋉ S2 . By a direct computation involving equation (…
Figure 7
Figure 7. Figure 7: Sectional curvature on Earth as a function of projection onto the rotational axis. In the trg-fibre with rotations ending at the north or south pole, the sectional curvature is minimal and equal to 1 4−ε 2 = 0.243, and for rotations ending at the equator, it is maximal…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

25 extracted references · 1 canonical work pages

  1. [1]

    Vladimir I. Arnold,Sur la g´ eom´ etrie diff´ erentielle des groupes de Lie de dimension infinie et ses applications ` a l’hydrodynamique des fluides parfaits, Annales de l’Institut Fourier16(1966), no. 1, 319–361 (fr)

    1966

  2. [2]

    ,Mathematical Methods of Classical Mechanics, Second Edition, Graduate Texts in Mathematics, Springer, 1989

    1989

  3. [3]

    Arnold and Boris A

    Vladimir I. Arnold and Boris A. Khesin,Topological Methods in Hydrodynamics, Applied Mathematical Sciences, vol. 125, Springer International Publishing, Cham, 2021 (en)

    2021

  4. [4]

    M. F. Atiyah,Complex Analytic Connections in Fibre Bundles, Transactions of the American Mathematical Society 85(1957), no. 1, 181–207

    1957

  5. [5]

    2, 575– 620

    Marius Crainic and Rui Loja Fernandes,Integrability of Lie Brackets, Annals of Mathematics157(2003), no. 2, 575– 620

    2003

  6. [6]

    1, 71–137

    ,Integrability of Poisson brackets, Journal of Differential Geometry66(2004), no. 1, 71–137

    2004

  7. [7]

    735, 143–173 (en)

    Matias del Hoyo and Rui Loja Fernandes,Riemannian metrics on Lie groupoids, Journal f¨ ur die reine und angewandte Mathematik (Crelles Journal)2018(February 2018), no. 735, 143–173 (en)

    2018

  8. [8]

    Z.292(June 2019), no

    ,Riemannian metrics on differentiable stacks, Math. Z.292(June 2019), no. 1, 103–132 (en)

    2019

Show all 25 references
  1. [9]

    Ebin and Jerrold Marsden,Groups of Diffeomorphisms and the Motion of an Incompressible Fluid, The Annals of Mathematics92(July 1970), no

    David G. Ebin and Jerrold Marsden,Groups of Diffeomorphisms and the Motion of an Incompressible Fluid, The Annals of Mathematics92(July 1970), no. 1, 102. 52 REN ´E LANGØEN

    1970

  2. [10]

    1, 119–179

    Rui Loja Fernandes,Lie Algebroids, Holonomy and Characteristic Classes, Advances in Mathematics170(September 2002), no. 1, 119–179

    2002

  3. [11]

    Rui Loja Fernandes and Marius Crainic,Lectures on Integrability of Lie Brackets, Lectures on Poisson Geometry, 2011 (en)

    2011

  4. [12]

    Anton Izosimov and Boris Khesin,Vortex sheets and diffeomorphism groupoids, Advances in Mathematics338(2018), 447–501

    2018

  5. [13]

    Var.63(November 2023), no

    ,Geometry of generalized fluid flows, Calc. Var.63(November 2023), no. 1, 3 (en)

    2023

  6. [14]

    2, 15 (en)

    Boris Khesin, Gerard Misio lek, and Alexander Shnirelman,Geometric Hydrodynamics in Open Problems, Arch Rational Mech Anal247(April 2023), no. 2, 15 (en)

    2023

  7. [15]

    I, John Wiley & Sons, 1963

    Shoshichi Kobayashi and Katsumi Nomizu,Foundations of Differential Geometry, Vol. I, John Wiley & Sons, 1963

    1963

  8. [16]

    Lee,Introduction to Smooth Manifolds, Graduate Texts in Mathematics, vol

    John M. Lee,Introduction to Smooth Manifolds, Graduate Texts in Mathematics, vol. 218, Springer New York, New York, NY, 2012 (en)

    2012

  9. [17]

    John Lott,Some Geometric Calculations on Wasserstein Space, Commun. Math. Phys.277(November 2007), no. 2, 423–437 (en)

    2007

  10. [18]

    Kirill C. H. Mackenzie,Lie groupoids and Lie algebroids in differential geometry, Vol. 124, Cambridge university press, 1987

    1987

  11. [19]

    ,General Theory of Lie Groupoids and Lie Algebroids, London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 2005

    2005

  12. [20]

    3, 293–329

    John Milnor,Curvatures of left invariant metrics on lie groups, Advances in Mathematics21(September 1976), no. 3, 293–329

    1976

  13. [21]

    1, 215–235

    Gerard Misio lek,Stability of Flows of Ideal Fluids and the Geometry of the Group of Diffeomorphisms, Indiana University Mathematics Journal42(1993), no. 1, 215–235

    1993

  14. [22]

    3, 335–390 (en)

    Klas Modin,Geometry of matrix decompositions seen through optimal transport and information geometry, JGM9 (August 2017), no. 3, 335–390 (en)

    2017

  15. [23]

    National Geospatial-Intelligence Agency,Department of Defense World Geodetic System 1984: Its Definition and Relationships with Local Geodetic Systems, Technical Report NGA.STND.0036 1.0.0 WGS84, National Geospatial- Intelligence Agency, Washington, DC, 2014

    1984

  16. [24]

    4, 459–469 (en)

    Barrett O’Neill,The fundamental equations of a submersion., Michigan Mathematical Journal13(December 1966), no. 4, 459–469 (en)

    1966

  17. [25]

    1-2, 101–174

    Felix Otto,The Geometry of Dissipative Evolution Equations: The Porous Medium Equation, Communications in Partial Differential Equations26(January 2001), no. 1-2, 101–174. eprint: https://doi.org/10.1081/PDE-100002243. R.L.: Department of Mathematics, University of Bergen, Nor...

    2001 doi

Pith tools

Reviewed June 28, 2026 · model on record in the stance chip above.