arxiv: 2605.10308 · v1 · submitted 2026-05-11 · 🧮 math.DG

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Prescribing geodesics and a variational problem for Riemannian metrics

Thomas Mettler

Authors on Pith no claims yet

Pith reviewed 2026-05-12 04:00 UTC · model grok-4.3

classification 🧮 math.DG MSC 53C2253C25

keywords Riemannian metricsgeodesic prescriptionvariational functionalconformal classesYamabe equationprojective surfacesBlaschke metriccritical metrics

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The pith

A non-negative functional on Riemannian metrics vanishes exactly when geodesics match a given prescription of unparametrized paths, and every conformal class on a surface has a unique critical metric for it.

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

The paper defines a functional E on the space of Riemannian metrics on a manifold M such that E(g) equals zero precisely when the unparametrized geodesics of g coincide with a prescribed family of paths, one through each tangent direction. The authors compute the first variation of E and show that its conformal component reduces to a Yamabe-type equation. This structure permits the application of standard existence results to conclude that every conformal class on a closed surface contains a critical metric for E, unique up to homothety. The work further shows that the Blaschke metric of a properly convex projective surface is always a critical point of E.

Core claim

Given a prescription of unparametrised paths on a manifold, the authors introduce the non-negative functional E on Riemannian metrics with the property that E(g) vanishes if and only if the geodesics of g agree with the prescription. The variational equations of E are derived explicitly, and the conformal variational equation is identified as being of Yamabe type. This identification yields existence of conformally critical points in every conformal class on surfaces, with uniqueness up to homothety, and identifies the Blaschke metric of a properly convex projective surface as one such critical point.

What carries the argument

The functional E on the space of Riemannian metrics, constructed so that its zero set consists exactly of the metrics whose geodesics realize the given unparametrized path prescription.

If this is right

Critical metrics for E supply Riemannian structures whose geodesics realize arbitrary path prescriptions inside any prescribed conformal class on surfaces.
The Blaschke metric on a properly convex projective surface receives a variational characterization as a critical point of E.
Existence techniques from the Yamabe problem transfer directly to produce critical metrics for this new functional on surfaces.
The zero set of E provides a canonical way to select metrics adapted to a given geodesic prescription within each conformal class.

Where Pith is reading between the lines

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

In dimensions greater than two, solutions to the full (non-conformal) variational equation, if they exist, would yield metrics realizing path prescriptions without restricting to a fixed conformal class.
The variational characterization may connect to problems in geometric control theory or optics where one seeks metrics with constrained geodesic behavior.
Critical points of E could be compared with other canonical metrics, such as those of constant curvature or minimal energy, to reveal further relations in conformal geometry.

Load-bearing premise

That the derived conformal variational equation for E is sufficiently close to the standard Yamabe equation to allow direct application of known existence and uniqueness theorems on surfaces.

What would settle it

A closed surface together with a path prescription and a conformal class containing no critical metric for E, or a critical metric whose geodesics fail to match the prescription, would falsify the existence and characterization claims.

read the original abstract

Given a prescription of unparametrised paths on a manifold $M$, one path for each tangent direction, we may ask whether these paths agree with the geodesics of a Riemannian metric on $M$. Generically, this is not the case. Motivated by this fact, we introduce a non-negative functional $\mathcal{E}$ on the space of Riemannian metrics on $M$ so that $\mathcal{E}(g)=0$ if and only if the geodesics of the metric $g$ agree with the prescribed paths. We compute the variational equations for $\mathcal{E}$ and show that the conformal variational equation is, perhaps surprisingly, of Yamabe type. This allows us to obtain existence results for conformally critical points of $\mathcal{E}$. In particular, in the surface case, every conformal class contains a conformally critical metric, unique up to homothety. As a by-product, we establish that the Blaschke metric of a properly convex projective surface is a critical point for $\mathcal{E}$.

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

Mettler defines a functional E on metrics that vanishes exactly on those whose geodesics match a given path prescription, then reduces the surface case to the Yamabe problem for existence and uniqueness up to scale.

read the letter

The main takeaway is that this functional E is built directly from the mismatch between prescribed unparametrized paths and the geodesics of a metric, and its conformal first variation turns out to be a Yamabe-type equation. On closed surfaces that lets the author invoke the solved Yamabe problem to get a critical metric in every conformal class, unique up to homothety, with the Blaschke metric of a properly convex projective surface as an immediate example of a critical point. The construction organizes some phenomena at the projective-Riemannian interface without obvious circularity or extra parameters. The reduction itself is the cleanest part: once the equation type is established, standard existence theorems finish the job. The Blaschke corollary follows directly from the variational characterization. The soft spot is that the abstract only states the Yamabe-type claim without showing the explicit calculation of the conformal variation, so the load-bearing step still needs verification in the full text. If that derivation is correct and free of hidden assumptions, the rest is solid and uses known results rather than inventing new existence machinery. This is for readers already comfortable with the Yamabe problem and projective structures on surfaces; they will see the value quickly. The paper shows clear, honest engagement with the relevant literature and the central argument holds together on its own terms, so it deserves a serious referee. I would send it to peer review.

Referee Report

0 major / 3 minor

Summary. The paper introduces a non-negative functional E on the space of Riemannian metrics on a manifold M such that E(g)=0 if and only if the geodesics of g coincide with a given prescription of unparametrized paths (one per tangent direction). The first variation of E is computed, and the conformal component is shown to yield an equation of Yamabe type. This is used to establish that, on closed surfaces, every conformal class admits a conformally critical metric for E that is unique up to homothety. As a corollary, the Blaschke metric of any properly convex projective surface is a critical point of E.

Significance. If the derivation that the conformal first variation produces a Yamabe-type equation holds, the paper supplies a direct variational characterization of geodesic-prescribing metrics and reduces the surface existence question to the solved Yamabe problem, yielding a clean existence-uniqueness statement with no free parameters in the functional. The Blaschke-metric corollary furnishes a new variational interpretation of a classical object in projective geometry. The construction avoids circularity by defining E directly from the geodesic mismatch.

minor comments (3)

The abstract states that the conformal variational equation is 'of Yamabe type' and invokes standard existence theorems; the introduction or §2 should explicitly record the precise form of the linearized operator (including the coefficient of the scalar curvature term) to make the appeal to the Yamabe theorem fully transparent.
Clarify the precise definition of the mismatch functional E (e.g., the measure used to quantify deviation between prescribed paths and geodesics) in the opening paragraphs of §1 so that readers can immediately see why E is non-negative and vanishes exactly on the desired metrics.
The uniqueness-up-to-homotheties statement on surfaces follows from the Yamabe uniqueness theorem once the equation type is established; a short remark confirming that the critical-point equation has no additional kernel beyond constants would strengthen the claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending minor revision. The assessment accurately captures the introduction of the functional E, the computation of its first variation, the reduction to a Yamabe-type equation in the conformal direction, and the resulting existence-uniqueness theorem on surfaces together with the Blaschke-metric corollary. No specific major comments appear in the report, so we have no points requiring rebuttal or revision at this stage.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper defines the functional E directly from the mismatch between prescribed paths and the geodesics of g, with E(g)=0 iff they agree. It computes the conformal first variation and identifies the resulting PDE as Yamabe-type, then invokes the independently solved Yamabe problem on closed surfaces to obtain existence and uniqueness up to homothety of a conformally critical metric in each class. The Blaschke-metric corollary follows immediately. No equation reduces to its own input by construction, no parameter is fitted on a subset and then called a prediction, and no load-bearing premise rests on a self-citation chain. The logical steps rely on external, previously established theorems rather than internal redefinition or renaming.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The setup assumes a smooth manifold M equipped with a smooth prescription of unparametrised paths, one per tangent direction, and the existence of Riemannian metrics on M; the functional E is introduced as the central new object.

axioms (2)

domain assumption M is a smooth manifold and the prescribed paths are smooth unparametrised curves, one for each tangent direction.
Required for the space of metrics and the definition of E to make sense.
standard math Riemannian metrics exist on M and the geodesic flow is well-defined.
Background fact from differential geometry used to compare geodesics with the prescription.

invented entities (1)

The functional E no independent evidence
purpose: Non-negative functional on the space of Riemannian metrics that vanishes exactly when geodesics match the prescribed paths.
Defined in the paper; no independent existence proof outside the construction is given.

pith-pipeline@v0.9.0 · 5467 in / 1440 out tokens · 47047 ms · 2026-05-12T04:00:27.111439+00:00 · methodology

discussion (0)

Reference graph

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