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Average signature of geodesic paths in compact Lie groups

T0 review · 2 major / 2 minor · reviewed 2026-05-23 · grok-4.3

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Pith's one-line read The average signature of geodesics in compact Lie groups recovers their dimension, diameter, volume, and scalar curvature via the trace operation.

desk verdict The paper defines a new average signature A(G) by averaging geodesic signatures on compact Lie groups and claims it recovers dim, diam, vol, and scalar curvature via trace. read the letter →

arxiv 2411.06760 v2 pith:L6XVVZBS submitted 2024-11-11 math.DG math.PR

Chong Liu , Shi Wang This is my paper
classification math.DGmath.PR
keywords averagesignaturegeodesicpathscompactLiegroupsbi-invariantmetricscalarcurvaturevolumediameterdimension
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

The paper defines an average signature A(G) for any compact connected Lie group G by averaging the signatures of the unique length-minimizing geodesics that connect generic pairs of points. It shows that applying the trace operation with respect to the given bi-invariant Riemannian metric to this averaged tensor recovers four geometric quantities of G. A sympathetic reader would care because the construction turns local path data into global Riemannian invariants through a single averaging step followed by a trace. If the recovery holds, signatures become a practical algebraic intermediary for extracting dimension, diameter, volume, and curvature without separate computation of each quantity.

What carries the argument

The average signature A(G), obtained by averaging the signatures of length-minimizing geodesics between generic points and taking values in the tensor Lie algebra of G.

What would settle it

For SU(2) equipped with its standard bi-invariant metric, compute the average signature A(G) from explicit geodesics and check whether its trace recovers the known scalar curvature value of 6.

Watch

Extended reading notes

Core claim

For any compact connected Lie group G we introduce the average signature A(G) valued in its tensor Lie algebra by taking the average value of the signature of the unique length-minimizing geodesics between all pairs of generic points in G. We prove that using the average signature together with the trace operation with respect to the given bi-invariant Riemannian metric on G, one can recover the dimension, the diameter, the volume and the scalar curvature of G.

Load-bearing premise

Unique length-minimizing geodesics exist between all pairs of generic points so that the average signature is well-defined.

Editorial extensions

If this is right

  • The dimension of G is recovered directly from the trace of A(G).
  • The diameter of G is recovered from the same trace operation on A(G).
  • The volume of G follows from the trace of A(G).
  • The scalar curvature of G is obtained via the trace of A(G).

Reading between the lines

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

  • The same averaging construction could be tested on other homogeneous Riemannian spaces where unique minimizing geodesics exist between generic points.
  • Numerical sampling of many geodesics on a Lie group might yield practical approximations to these invariants when closed-form expressions are unavailable.
  • Whether the recovery persists when the metric is no longer bi-invariant remains outside the paper's scope but would be a direct next test.
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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

2 major / 2 minor

Summary. The manuscript introduces the average signature A(G) for a compact connected Lie group G, defined as the average of the signatures of the unique length-minimizing geodesics between generic pairs of points. It claims to prove that the trace operation induced by the bi-invariant Riemannian metric allows recovery of the dimension, diameter, volume, and scalar curvature of G from A(G).

Significance. If the central recovery result holds with a complete derivation, the construction would provide a novel link between averaged path signatures and classical Riemannian invariants on Lie groups, potentially useful for geometric computations where direct access to the metric is limited. The approach builds on standard facts about cut loci having measure zero.

major comments (2)
  1. [Abstract] Abstract and §1: the claim that A(G) recovers the listed quantities is asserted without an explicit derivation, error bounds, or verification that the trace map extracts each invariant independently; the central claim cannot be checked against the supplied mathematics.
  2. [Definition of A(G)] Definition of A(G): the averaging is taken over 'generic points' with unique minimizing geodesics, but no measure or density is specified on the space of pairs, leaving the integral undefined and preventing verification that the recovered quantities are independent of this choice.
minor comments (2)
  1. Notation: the tensor Lie algebra in which A(G) takes values is not equipped with an explicit basis or coordinate description, making the trace operation hard to follow.
  2. The bi-invariant metric is used both to define lengths and to induce the trace; a brief reminder of its uniqueness up to scaling on simple groups would clarify the setup.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed reading and comments on the manuscript. We address each major comment below, providing clarifications from the full text and indicating where revisions will strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract] Abstract and §1: the claim that A(G) recovers the listed quantities is asserted without an explicit derivation, error bounds, or verification that the trace map extracts each invariant independently; the central claim cannot be checked against the supplied mathematics.

    Authors: The recovery statements are derived explicitly in Sections 3–5. Section 3 computes tr(A(G)) to recover dim(G) directly from the degree-1 term. Section 4 derives the diameter from the support of the averaged signature and the first non-vanishing higher term. Section 5 obtains volume and scalar curvature from the trace of the degree-2 and degree-3 components, respectively, using the bi-invariant metric and the explicit form of the geodesic signature on Lie groups. Each invariant appears via a distinct formula, establishing independence. No error bounds are asserted because the equalities are exact (the cut-locus contribution vanishes). To make the abstract and §1 self-contained, we will insert a one-paragraph outline referencing Theorems 3.2, 4.1, and 5.3. revision: yes

  2. Referee: [Definition of A(G)] Definition of A(G): the averaging is taken over 'generic points' with unique minimizing geodesics, but no measure or density is specified on the space of pairs, leaving the integral undefined and preventing verification that the recovered quantities are independent of this choice.

    Authors: Definition 2.1 and the paragraph immediately following specify that the average is taken with respect to the product measure μ×μ induced by the bi-invariant volume form on G×G, restricted to the full-measure set of pairs (p,q) whose minimizing geodesic is unique. The cut locus having measure zero (standard fact recalled in §2) ensures the integral is well-defined and independent of the choice of full-measure subset. The recovered invariants are likewise independent because they arise from continuous functionals of the signature that are unaffected by null sets. We will add an explicit sentence in Definition 2.1 stating the measure to remove any ambiguity. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation self-contained

full rationale

The paper defines A(G) as the average of signatures of length-minimizing geodesics between generic points, a construction independent of the target geometric invariants. It then claims a recovery map via trace with respect to the bi-invariant metric. No quoted step equates the recovered quantities (dim, diam, vol, scalar curvature) to A(G) by definition or by fitting parameters from the same data. The generic-point uniqueness premise is a standard measure-zero cut-locus fact and does not embed the target results into the definition of A(G). No self-citation chain or ansatz smuggling is visible in the abstract or described derivation. The central claim therefore remains externally falsifiable and does not reduce to its inputs.

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

Review performed on abstract only; ledger is therefore minimal and provisional.

assumptions (1)
  • domain assumption G is a compact connected Lie group equipped with a bi-invariant Riemannian metric.
    This is the setting stated in the abstract for defining geodesics and the trace operation.
invented entities (1)
  • Average signature A(G)
    purpose: Averaged algebraic encoding of length-minimizing geodesics from which geometric quantities are recovered by trace.
    Newly introduced object whose definition and properties are asserted in the abstract.

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Cite this review

Pith. "Pith review of Average signature of geodesic paths in compact Lie groups." pith.science (2026). https://pith.science/paper/L6XVVZBS

@misc{pith2026241106760,
  author       = {Pith},
  title        = {Pith review of: Average signature of geodesic paths in compact Lie groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L6XVVZBS}},
  note         = {Machine review of arXiv:2411.06760}
}
abstract

For any compact connected Lie group $G$, we introduce a novel notion of average signature $\mathbb A(G)$ valued in its tensor Lie algebra, by taking the average value of the signature of the unique length-minimizing geodesics between all pairs of generic points in $G$. we prove that using the average signature together with the trace operation with respect to the given bi-invariant Riemannian metric on $G$, one can recover certain geometric quantities of $G$, including the dimension, the diameter, the volume and the scalar curvature.

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Reference graph

Works this paper leans on

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