On the existence of geodesic vector fields on closed surfaces

Vladimir S. Matveev

arxiv: 2405.09604 · v2 · submitted 2024-05-15 · 🧮 math.DG

On the existence of geodesic vector fields on closed surfaces

Vladimir S. Matveev This is my paper

Pith reviewed 2026-05-24 01:10 UTC · model grok-4.3

classification 🧮 math.DG

keywords Riemannian metric2-torusuniversal coverRiemann normal coordinatesgeodesic vector fieldsclosed surfacescounterexample

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The pith

There exists a Riemannian metric on the 2-torus whose universal cover does not admit global Riemann normal coordinates.

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

The paper constructs an example of a Riemannian metric on the 2-torus. For this metric the lifted structure on the universal cover, diffeomorphic to the plane, does not admit global Riemann normal coordinates. These coordinates would express the metric in standard form along geodesics throughout the cover. The result matters because it shows that local geodesic straightening need not extend to a global coordinate system on the cover of a compact surface.

Core claim

We construct an example of a Riemannian metric on the 2-torus such that its universal cover does not admit global Riemann normal coordinates.

What carries the argument

The explicit Riemannian metric on the 2-torus and its lift to the universal cover that prevents global normal coordinates.

Load-bearing premise

A smooth positive-definite metric can be placed on the closed 2-torus so that its lift to the plane violates the global extension property for normal coordinates at every point.

What would settle it

An explicit verification or proof that the constructed metric on the 2-torus does admit global Riemann normal coordinates throughout its universal cover would disprove the claim.

Figures

Figures reproduced from arXiv: 2405.09604 by Vladimir S. Matveev.

**Figure 1.** Figure 1: The torus made of the sphere: the dark-gray part is the 2ε-ball around the north pole. The surgery was made in the light-gray part. We consider the universal cover R 2 and denote by ˜g the lift of the metric. Let us show that (R 2 , g˜) does not admit a geodesic vector field. We assume it does, denote the geodesic vector field by v, and find a contradiction. In order to do it, consider the circle of radius… view at source ↗

read the original abstract

We construct an example of a Riemannian metric on the 2-torus such that its universal cover does not admit global Riemann normal coordinates.

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 claims a construction of a metric on the 2-torus whose lift to the plane lacks global Riemann normal coordinates, but the abstract supplies no details to check it.

read the letter

The core claim is a construction of a Riemannian metric on the closed 2-torus such that its universal cover does not admit global Riemann normal coordinates. That is the result being offered. The paper does the right thing by aiming for an explicit example rather than a general statement, since a counterexample is what is needed here. If the construction is correct and the metric is smooth and positive definite, it would show that the global normal coordinate property fails for at least one lift of a torus metric. That is the useful part. The obvious limitation is that the abstract gives no equations, no description of the metric, and no argument showing why the exponential map fails to be a global diffeomorphism on the cover. Without those steps it is impossible to check whether the example avoids contradictions with Gauss-Bonnet, completeness, or the definition of normal coordinates. The stress-test note is accurate on this point: no load-bearing assumption can be tested from what is visible. This work is aimed at differential geometers who study global properties of the exponential map and geodesic behavior on surface covers. A reader already working on similar counterexamples or on the distinction between local and global normal coordinates would get the most from seeing the actual construction. I would send it to peer review so that a referee can inspect the details and confirm whether the example holds.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to construct a Riemannian metric on the closed 2-torus such that the lifted metric on its universal cover (diffeomorphic to R^2) does not admit global Riemann normal coordinates.

Significance. If substantiated by an explicit construction, the result would supply a concrete counterexample to the global extendability of Riemann normal coordinates on the universal cover of a compact surface, potentially informing questions about geodesic completeness and coordinate charts in Riemannian geometry. No such construction, equations, or verification appear in the provided text, so significance cannot be assessed.

major comments (1)

[Abstract] Abstract: the central existence claim is asserted without any metric definition, curvature computation, or argument showing that the exponential map fails to be a global diffeomorphism from every point in the cover. This absence is load-bearing for the result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for reviewing the manuscript. The report correctly identifies that the provided text consists solely of an abstract asserting an existence result without any supporting construction or verification.

read point-by-point responses

Referee: [Abstract] Abstract: the central existence claim is asserted without any metric definition, curvature computation, or argument showing that the exponential map fails to be a global diffeomorphism from every point in the cover. This absence is load-bearing for the result.

Authors: We agree that the manuscript text contains only the existence claim in the abstract, with no metric definition, curvature computations, or argument regarding the exponential map. This omission means the result cannot be substantiated from the given text. The manuscript will be revised to include an explicit Riemannian metric on the 2-torus, the necessary computations, and a demonstration that the lifted metric on the universal cover fails to admit global Riemann normal coordinates from every point. revision: yes

Circularity Check

0 steps flagged

No significant circularity; construction claim is self-contained

full rationale

The paper states an existence result via explicit construction of a Riemannian metric on the 2-torus whose universal cover lacks global Riemann normal coordinates. No derivation chain, equations, fitted parameters, or self-citations are present in the abstract or context that would reduce the claimed example to its own inputs by definition. An existence claim by construction carries no load-bearing deductive steps that could exhibit circularity of the enumerated kinds. The result is independent of any prior fitted data or author-specific uniqueness theorems.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No information is available from the abstract to identify free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5523 in / 1085 out tokens · 29094 ms · 2026-05-24T01:10:00.033793+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

[1]

Integrable geodesic flows on surfaces

Misha Bialy. “Integrable geodesic flows on surfaces”. In: Geom. Funct. Anal. 20.2 (2010), pp. 357–367. issn: 1016-443X,1420-8970. doi: 10 . 1007/s00039-010-0069-4 . url: https://doi.org/10.1007/s00039- 010-0069-4

work page doi:10.1007/s00039- 2010
[2]

Open problems, questions and challenges in ﬁnite-dimensional integrable systems

Alexey Bolsinov et al. “Open problems, questions and challenges in finite-dimensional integrable systems”. English. In: Philos. Trans. R. Soc. Lond., A, Math. Phys. Eng. Sci. 376.2131 (2018). Id/No 20170430, p. 40. issn: 1364-503X. doi: 10.1098/rsta.2017.0430 . url: https: //royalsocietypublishing.org/doi/10.1098/rsta.2017.0430

work page doi:10.1098/rsta.2017.0430 2018
[3]

Open problems and questions about geodesics

Keith Burns and Vladimir S. Matveev. “Open problems and questions about geodesics”. In: Ergodic Theory Dynam. Systems 41.3 (2021), pp. 641–

work page 2021
[4]

issn: 0143-3857,1469-4417. doi: 10 . 1017 / etds . 2019 . 73. url: https://doi.org/10.1017/etds.2019.73

work page doi:10.1017/etds.2019.73 2019
[5]

Conformal product structures on compact K¨ ahler manifolds

Andrei Moroianu and Mihaela Pilca. “Conformal product structures on compact K¨ ahler manifolds”. In:arXiv (2024). url: https://arxiv. org/abs/2405.08430. 4

work page arXiv 2024

[1] [1]

Integrable geodesic flows on surfaces

Misha Bialy. “Integrable geodesic flows on surfaces”. In: Geom. Funct. Anal. 20.2 (2010), pp. 357–367. issn: 1016-443X,1420-8970. doi: 10 . 1007/s00039-010-0069-4 . url: https://doi.org/10.1007/s00039- 010-0069-4

work page doi:10.1007/s00039- 2010

[2] [2]

Open problems, questions and challenges in ﬁnite-dimensional integrable systems

Alexey Bolsinov et al. “Open problems, questions and challenges in finite-dimensional integrable systems”. English. In: Philos. Trans. R. Soc. Lond., A, Math. Phys. Eng. Sci. 376.2131 (2018). Id/No 20170430, p. 40. issn: 1364-503X. doi: 10.1098/rsta.2017.0430 . url: https: //royalsocietypublishing.org/doi/10.1098/rsta.2017.0430

work page doi:10.1098/rsta.2017.0430 2018

[3] [3]

Open problems and questions about geodesics

Keith Burns and Vladimir S. Matveev. “Open problems and questions about geodesics”. In: Ergodic Theory Dynam. Systems 41.3 (2021), pp. 641–

work page 2021

[4] [4]

issn: 0143-3857,1469-4417. doi: 10 . 1017 / etds . 2019 . 73. url: https://doi.org/10.1017/etds.2019.73

work page doi:10.1017/etds.2019.73 2019

[5] [5]

Conformal product structures on compact K¨ ahler manifolds

Andrei Moroianu and Mihaela Pilca. “Conformal product structures on compact K¨ ahler manifolds”. In:arXiv (2024). url: https://arxiv. org/abs/2405.08430. 4

work page arXiv 2024