A general model for time-minimizing navigation on a mountain slope under gravity

Nicoleta Aldea; Piotr Kopacz

REVIEW 2 major objections 5 minor 23 references

A general model for time-minimizing navigation on a mountain slope under gravity

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Time-minimal paths on any slippery mountain slope under gravity are the geodesics of a single two-parameter Finsler metric, unifying Matsumoto and Zermelo navigation with explicit strong-convexity bounds and geodesic equations.

desk verdict A coherent two-parameter unification of the slippery-slope navigation metrics, genuinely useful for the Finsler-navigation crowd, though the novelty is partly a reparametrization and the physical model rests on an unvalidated linear traction assumption. read the letter →

arxiv 2506.06720 v1 pith:BETCDN7C submitted 2025-06-07 math.DG

Nicoleta Aldea , Piotr Kopacz This is my paper

classification math.DG MSC 53B4053C60

keywords FinslergeometryZermelonavigationMatsumotoslopeproblemgeneral(αβ)-metricstimegeodesicsgravitationalwindslipperytractioncoefficients

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 claims that the fastest paths (in time) for a walker or craft moving at unit self-speed on a slippery mountain slope under gravity are precisely the geodesics of a new Finsler metric, the $(\eta,\tilde{\eta})$-slope metric $\tilde F_{\eta\tilde\eta}$, for every pair of cross- and along-traction coefficients in $[0,1]^2$. The claim matters because it turns a collection of optimal-control problems into a single geometric family: the classical Matsumoto slope problem, Zermelo navigation under a gravitational wind, the cross-slope problem, and the newly admitted mixed-slide problems all appear as boundary or interior points of the same problem square. The paper also states how strong the gravitational wind may be before the metric ceases to be strongly convex, and it writes out the ODE system whose solutions are the time-minimizing trajectories.

What carries the argument

The load-bearing object is a two-step geometric construction of the metric. First, the background Riemannian metric $h$ is anisotropically deformed by the direction-dependent vector field $(\eta-\tilde\eta)G_{\mathrm{MAT}}$, the difference-weighted projection of gravity along the direction of motion; this produces the Matsumoto-type metric $F(x,y)=\alpha^2/(\alpha-(\eta-\tilde\eta)\bar g\beta)$, which is a Finsler metric exactly when $\|G_T\|_h<1/(2|\eta-\tilde\eta|)$. Second, following Zermelo navigation, the indicatrix of this $F$ (or of $h$ itself when $\eta=\tilde\eta$) is rigidly translated by the rescaled wind $(1-\eta)G_T$ under the condition $F(x,-(1-\eta)G_T)<1$; this translation yields the implicit equations (3) and (4) for $\tilde F_{\eta\tilde\eta}$. The partition of the problem square into regions $D_1,\dots,D_4$ determines which of the two bounds, $1/(1-\tilde\eta)$ or $1/(2|\eta-\tilde\eta|)$, controls strong convexity of the translated indicatrix. Finally, the general $(\alpha,\beta)$-metric convexity and spray formulas convert the implicit metric into the explicit geodesic ODE system of Theorem 1.2.

What would settle it

On the inclined plane $z=x/2$ with $\|G_T\|_h=0.9$ and $(\eta,\tilde\eta)=(0.5,0.2)$, numerically solve the optimal control $v=u+G_{\eta\tilde\eta}$ with $\|u\|_h=1$ and compare the fastest path between two points with the geodesic of $\tilde F_{\eta\tilde\eta}$ predicted by Theorem 1.1 (in this setting the theory says both are straight lines whose direction is fixed by the indicatrix). A systematic angular mismatch larger than the numerical discretization error would falsify the claimed equivalence.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 1.1: on an $n$-dimensional Riemannian manifold $(M,h)$ modeling the slope, with gravitational wind $G_T$ and active wind $G_{\eta\tilde\eta}=(\eta-\tilde\eta)G_{\mathrm{MAT}}+(1-\eta)G_T$, the time-minimal paths are the geodesics of the metric $\tilde F_{\eta\tilde\eta}$ defined implicitly by $$\tilde F_{\eta\tilde\eta}\sqrt{\$alpha^{2}$+2(1-\eta)\bar g\$\beta$\tilde F_{\eta\tilde\eta}+(1-\eta)^2\|G_T\|$_h^{2}$\tilde F_{\eta\tilde\eta}^2}=\$alpha^{2}$+(2-\eta-\tilde\eta)\bar g\$\beta$\tilde F_{\eta\tilde\eta}+(1-\eta)(1-\tilde\eta)\|G_T\|$_h^{2}$\tilde F_{\eta\tilde\eta}^2,$$ with $\alpha,\beta$ built from $h$ and $G_T$. This metric is a general $(\alpha,\beta)$-metric and reduces to a Randers metric at $(0,0)$, a Matsumoto metric at $(1,0)$, a cross-slope metric at $(0,1)$, and the Riemannian metric $h$ at $(1,1)$. The strong convexity conditions are either $\|G_T\|_h<1/(1-\tilde\eta)$ on regions $D_1\cup D_2$, or $\|G_T\|_h<1/(2|\eta-\tilde\eta|)$ on regions $D_3\cup D_4$, with no restriction at the Riemannian corner. Theorem 1.2 then records the spray coefficients and the second-order ODE system (5) whose solutions, restricted to the indicatrix $\tilde F_{\eta\tilde\eta}=1$, are exactly the time-minimizing paths.

Load-bearing premise

The construction assumes the active wind is exactly the linear, direction-dependent combination $G_{\eta\tilde\eta}=(\eta-\tilde\eta)G_{\mathrm{MAT}}+(1-\eta)G_T$ with constant traction coefficients and unit self-speed; if traction coefficients depend on position, velocity, or time, or the wind is nonlinear, the Finsler metric formula does not describe the true fastest paths.

Editorial extensions

If this is right

Every fixed pair $(\eta,\tilde\eta)\in[0,1]^2$ defines a legitimate navigation problem whose time-minimal paths can be found by integrating the geodesic equations of $\tilde F_{\eta\tilde\eta}$, so the whole square of mixed-slide problems is reduced to Finsler geometry.
The boundary cases recover the known metrics: Randers for full Zermelo navigation, Matsumoto for the classical slope problem, the cross-slope metric, and the Riemannian metric when all gravity compensation is complete.
The admissible gravitational wind can exceed the self-speed: near the Riemannian corner the bound $\tilde b_0$ tends to infinity, and in regions $D_1\cup D_2$ the bound is $1/(1-\tilde\eta)$, which is larger than 1 whenever $\tilde\eta>0$; the classical weak-wind limitation $\|G_T\|_h<1$ applies only on the diagonal.
An interior problem with $\eta>\tilde\eta$ is equivalent to a standard slippery-slope problem with rescaled wind and cross-traction $c_1=(\eta-\tilde\eta)/(1-\tilde\eta)$, while $\eta<\tilde\eta$ corresponds to a slippery-cross-slope problem with along-traction $c_2=(\tilde\eta-\eta)/(1-\eta)$.
On any two-dimensional slope the time fronts and time geodesics can be computed from the paper's formulas; on an inclined plane the fastest paths are straight lines, and on the Gaussian triple hill they are the solutions of the explicit system (76).

Reading between the lines

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

A practical route to numeric path planning on such slopes is to solve the implicit quartic (38) once per direction to tabulate $\tilde F_{\eta\tilde\eta}$, then integrate the geodesic ODE; the paper leaves this implementation implicit.
The strong-convexity bound $\tilde b_0$ suggests a calibration experiment: by measuring the largest gravitational wind for which fastest paths remain well defined, one could infer the effective traction pair $(\eta,\tilde\eta)$ of a given surface material.
If traction coefficients or self-speed vary with position or time, the static general $(\alpha,\beta)$ framework no longer applies; the paper's closing remarks point toward time-dependent Finsler or Lorentz-Finsler models as the natural next step.
Because all interior problems are equivalent to boundary problems with rescaled gravity, experiments for mixed-slide scenarios could be performed with simpler setups by rescaling the gravitational acceleration rather than engineering arbitrary side friction.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Rough take: this is a sound, useful unification paper. The new thing is the two-parameter (η,η̃) family and the explicit partition D1–D4 of the traction square where strong convexity holds, with the different thresholds 1/(1−η̃) versus 1/(2|η−η̃|). The two-step proof—Matsumoto-type deformation of h by (η−η̃)G_MAT, then Zermelo translation by (1−η)G_T—is clean and matches the prior SLIPPERY/S-CROSS machinery. Section 5's spray coefficients are heavy but they come from the standard general (α,β)-metric formula, and I did not see any obvious gap in the denominator arguments. The examples on the inclined plane and triple Gaussian hill are consistent with the indicatrix equations and convexity bounds. This paper deserves a serious referee.

Where it is softer: Corollary 4.4 is the honesty checkpoint. It shows that any (η,η̃) problem is just SLIPPERY or S-CROSS with a rescaled gravitational wind and a redefined traction coefficient (c1=(η−η̃)/(1−η̃), c2=(η̃−η)/(1−η)). That means the two-parameter metric is not a genuinely new family—it is a repackaging of the one-parameter models with a rescaled wind. This is not fatal, but it should have been stated up front rather than buried in a corollary. The D1–D4 partition and the unified treatment of the full square are still new and useful; the examples show how the indicatrices interpolate. But 'general model' should be read as 'general parameterization' rather than 'new geometry.'

The bigger physical caveat is equation (18): the active wind is assumed to be exactly the linear combination (η−η̃)G_MAT + (1−η)G_T with constant coefficients and ||u||_h = 1. That is a modelling assumption, not derived from a traction law. If real traction is state-dependent or nonlinear (Coulomb friction, velocity-dependent slip), the Finsler geodesics of F̃_{ηη̃} will not be the true time-minimal paths. The paper does not test this against a more realistic control model, and the title's 'general model' oversells that point. A short remark acknowledging this limitation would fix most of the damage. The conditional-accept verdict in the reader report is about right.

Who is it for? Finsler geometers working on navigation problems, especially the Matsumoto/Zermelo lineage. It will be a useful reference for the D1–D4 conditions and the spray formulas. I would cite it if I worked in that corner, and I would send it to an expert referee in Finsler geometry; the mathematics is checkable and the unification is worth having in the literature.

Referee Report

2 major / 5 minor

Summary. The paper introduces a two-parameter family of navigation problems on a Riemannian manifold (M,h) modeling a slippery mountain slope. The active wind is defined in Eq. (18) as G_{\eta\tilde\eta} = (\eta-\tilde\eta)\mathrm{Proj}_u G_T + (1-\eta)G_T, where \eta,\tilde\eta\in[0,1] are cross- and along-traction coefficients. Theorem 1.1 states that time-minimal paths under this wind are geodesics of an (\eta,\tilde\eta)-slope metric \tilde F_{\eta\tilde\eta}, defined implicitly by Eq. (3), which is a general (\alpha,\beta)-metric under explicit strong-convexity bounds on \|G_T\|_h. Theorem 1.2 gives the corresponding spray coefficients and the ODE system for time geodesics. The proof proceeds in two steps: a Matsumoto-type deformation by (\eta-\tilde\eta)\mathrm{Proj}_u G_T, followed by a Zermelo navigation with wind (1-\eta)G_T. The paper also recovers the limiting cases MAT, ZNP, CROSS, SLIPPERY, S-CROSS, R-MAT, R-ZNP, and presents two-dimensional examples on an inclined plane and a triple Gaussian hill.

Significance. If the modeling assumption is accepted, the main theorem is a coherent and useful generalization of the authors' earlier slippery-slope models, subsuming Randers, Matsumoto, and cross-slope metrics. The two-step deformation-plus-translation construction is elegant and the strong-convexity analysis is mostly explicit. The examples with indicatrices and time fronts are informative and the computational detail is substantial. However, the paper claims to solve the 'general' mountain-slope navigation problem, whereas the active wind in Eq. (18) is an assumption, not a consequence of a physical traction law. The mathematical reduction is sound as a conditional statement, but the physical generality is not established.

major comments (2)

[Section 3.2, Eq. (18)] The active wind G_{\eta\tilde\eta} is introduced as a linear combination of the tangential gravity components with constant coefficients (\eta,\tilde\eta), but no derivation from a traction or friction law is given. The abstract and conclusions describe this as a 'general model' and claim 'substantial physical applicability.' This is an overclaim: real traction on a slope may depend on position, velocity, or direction (e.g., Coulomb or velocity-dependent friction, granular slip). Please state explicitly that (18) is a modeling hypothesis, and either temper the generality claims or add a discussion of the physical regimes in which the constant-traction linear model is a reasonable approximation.
[Lemma 4.3, part (iii)] The claimed equivalence (ii) \Leftrightarrow (iii) is false. For example, (\eta,\tilde\eta)=(0.9,0.95) lies in D_4, where (ii) requires \|G_T\|_h<1/(2|\eta-\tilde\eta|)=10. However, \max_\theta\|G_{\eta\tilde\eta}\|_h=(1-\eta)\|G_T\|_h=0.1\|G_T\|_h, so \|G_T\|_h=50 satisfies (iii) (since \|G\|=5<10) but violates (ii). The proof's statement that the maximum of \|G_{\eta\tilde\eta}\| coincides with \|G_T\| is true only on the boundary \eta=0 or \tilde\eta=0. Although Theorem 1.1 itself uses condition (ii), the lemma as stated is part of the proof of the main theorem and must be corrected or its part (iii) removed.

minor comments (5)

[Throughout] There are several typos and grammatical slips, e.g., 'particluar' (Section 1.1), 'Precesily' (Section 4), 'distingusihing' (Section 1.2), 'untill' (Section 3.2), 'Furthemore' (Section 3.2), and 'signficantly' (Section 3.1.3). A careful proofreading pass is recommended.
[Theorem 1.2, Eq. (6)] The last term in the spray formula is typeset ambiguously as '- \tilde R w^i_{|j} \alpha^2 w^j \bar g^2' with unclear placement of the denominator. It should be written unambiguously, e.g., as '- \tilde R \, (w^i_{|j} w^j) \alpha^2 / \bar g^2', or the auxiliary quantity r^i from Eq. (62) should be used consistently.
[Section 3.2] The discussion of the 'dead wind' and the identity (19) is terse; in particular, the geometric notation \overrightarrow{PC} and the conversion P_{\eta_0,\tilde\eta_0} \leftrightarrow P'_{1-\eta_0,1-\tilde\eta_0} would benefit from a short explanatory paragraph or a reference to Figure 4.
[Section 6, Figures 5-15] The color coding in several figures (e.g., Figure 5 right and Figure 9) is not fully described in the captions; please add a brief legend or description in the caption text, since the figures are central to the claimed visualization of the strong-convexity regions and time fronts.
[Lemma 4.1] The proof of Lemma 4.1 is correct as written, but it would be helpful to indicate explicitly why the n=2 case is covered by the same argument; the current text cites Proposition 2.2, but the second inequality alone is used in the proof.

Circularity Check

0 steps flagged · score 0.0 of 10

Derivation is a conditional mathematical construction using standard Zermelo navigation; no load-bearing step reduces by construction to its inputs.

full rationale

The paper's central claim is a conditional theorem: if the active wind is exactly G_{ηη̃} = (η−η̃)G_MAT + (1−η)G_T as in (18), then time-minimal paths are geodesics of the (η,η̃)-slope metric F̃_{ηη̃} defined implicitly by (36)–(38). The derivation is a two-step geometric construction. Step I uses Okubo's method to obtain the Matsumoto-type metric F = α²/(α−(η−η̃)ḡβ) from the first-stage equation v = u + (η−η̃)G_MAT. Step II applies the Zermelo navigation theorem (Proposition 2.1, cited to external sources [19, 8, 11]) with navigation data (F, (1−η)G_T) to obtain F̃_{ηη̃}. No parameter is fitted to data, and no quantity called a prediction is defined from the quantity it is supposed to predict. The new metric is not defined as 'the geodesic metric' but as the unique positive solution of an implicit equation, and the geodesic property is then derived from standard Finsler spray formulas (Propositions 2.2–2.3, [21]). Self-citations to [4, 2, 1] are used mainly to identify boundary cases (η=0 or η̃=0) and in Corollary 4.4 to verify algebraically that the general equation reduces to earlier equations under substitution; these are consistency checks, not load-bearing circular reductions. Lemma 4.3 invokes [4, Lemma 4.3] for technical identities about gradient wind, but that is auxiliary and does not constitute the central claim. The physical generality of the linear constant-traction active wind (18) is an assumed model, not a derived consequence; that is a limitation or correctness risk regarding real-world traction, but it is not circularity. Hence no circular step is present.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard Finsler-geometry tools plus physical modeling assumptions about constant self-speed and linear traction. No numerical parameters are fitted to data; the only inputs are the Riemannian slope (M,h), the gradient wind G_T, and the chosen traction coefficients.

assumptions (6)

domain assumption The gravitational wind G_T = -ḡ ω♯ is a gradient vector field, so the 1-form β = h(y,ω♯) is closed (s_ij = 0).
Used throughout Sections 4 and 5 to simplify covariant derivatives and to apply Proposition 2.3; see equation (25) and the comment after it.
domain assumption The walker's self-velocity satisfies ||u||_h = 1 at every point.
Standard Zermelo navigation normalization used in Section 3 and in the proof of Theorem 1.1; it fixes the speed scale.
domain assumption Traction coefficients η and η̃ are constant on M and lie in [0,1].
The model in Sections 1.2 and 3.2 assumes fixed coefficients; position-dependent coefficients are deferred to future work in Remark 5.6.
standard math Zermelo navigation theorem (Proposition 2.1) is valid for the background Finsler metric F under the weak-wind condition F(x,-(1-η)G_T)<1.
This is the external theorem that converts the equation of motion into the implicit metric equation (36), the basis of Theorem 1.1.
standard math The Yu-Zhu characterization of general (α,β)-metrics applies to the implicitly defined function.
Proposition 2.2 is used in Lemma 4.1 and Section 5 to certify strong convexity and to compute spray coefficients.
standard math Okubo's method yields a Finsler function from the norm equations in Section 4.
Used to pass from equations such as g1(x,v/F)=0 to the explicit Matsumoto-type function F=α²/(α-(η-η̃)ḡβ).

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Pith. "Pith review of A general model for time-minimizing navigation on a mountain slope under gravity." pith.science (2026). https://pith.science/paper/BETCDN7C

@misc{pith2026250606720,
  author       = {Pith},
  title        = {Pith review of: A general model for time-minimizing navigation on a mountain slope under gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BETCDN7C}},
  note         = {Machine review of arXiv:2506.06720}
}

abstract

In this work, we solve the generalized Matsumoto's slope-of-a-mountain problem by means of Riemann-Finsler geometry, making close links with the Zermelo navigation problem. The time-minimizing navigation under gravity is analyzed in the general model of a slippery mountain slope being a Riemannian manifold. Both the transverse and longitudinal gravity-additives with respect to the direction of motion are admitted to vary simultaneously in the full ranges, showing the impact of cross- and along-traction on the slippery slope. By the anisotropic deformation of the background Riemannian metric and rigid translation with the use of the rescaled gravitational wind, we obtain the purely geometric solution for optimal navigation, which is given by a new Finsler metric belonging to the class of general $(\alpha, \beta)$-metrics. The related strong convexity conditions are established and time geodesics are described. Moreover, the evolution of time fronts and the behavior of time-minimizing trajectories in relation to various gravity effects on the slippery slope, gravitational wind force and direction of motion are thoroughly discussed and visualized by several two-dimensional examples.

Figures

Figures reproduced from arXiv: 2506.06720 by the authors.

**Figure 4.** Now we are in position to apply the Zermelo’s navigation technique as in [11, 6], with the navigation data (h, W), namely, deforming the Riemannian metric h by a weak vector field W = (1 − η)GT , i.e. ||W||h = (1 − η)||GT ||h < 1, which stands for the strong convexity condition in this case. Namely, by using the assumption on the self-speed ||u||h = 1 and (11), we get ||vR−ZNP ||2 h − 2(1 − η)h(vR−ZNP , GT ) − 1 + (… view at source ↗

**Figure 2.** The extended problem diagram (square) including the new cases: R-ZNP, R-MAT and [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

Figures from the paper (14 more)

**Figure 3.** Figure 3: The complete problem square diagram S˜ = [0, 1]×[0, 1] including all navigation problems on the slippery slope under gravity with fixed (i.e. a specific point P) and varying (i.e. a transition T P ′ P ) cross- and along-traction coefficients, where P = (η, η˜),P ′ = (η…

**Figure 4.** Figure 4: Left: A model of a slippery (η, η˜)-slope as an inclined plane M of the slope angle ε in R 3 , under the gravity field G = GT + G⊥, acting perpendicularly to the horizontal plane (the base of the slope). The gravitational wind GT “blows” tangentially to M in the steepe…

**Figure 5.** Figure 5: Left: The partition of the problem square diagram [PITH_FULL_IMAGE:figures/full_fig_p024_5.png]

**Figure 6.** Figure 6: A comparison of the orthographically projected images of the surface [PITH_FULL_IMAGE:figures/full_fig_p024_6.png]

**Figure 7.** Figure 7: The Finslerian indicatrices (the limaçons) centered at [PITH_FULL_IMAGE:figures/full_fig_p034_7.png]

**Figure 9.** Figure 9: Notice that in general we can arrive at the Riemannian case twofold. First, no matter the gravitational wind force, while both parameters are equal to 1. This means that there is the greatest compensation of the gravitational wind due to traction(s) (i.e. the only non-…

**Figure 8.** Figure 8: The navigation problems related to the (η, η˜)-slopes, i.e. P1 = (0.5, 0.5), P2 = (0.2, 1/3), P3 = (0.4, 0.2), P4 = (0.3, 0.8), P5 = (0.8, 0.3), selected from different subareas of the problem square diagram S˜ (see also [PITH_FULL_IMAGE:figures/full_fig_p035_8.png]

**Figure 9.** Figure 9: The evolution of the Finslerian indicatrices centered at the origin of the coordinate [PITH_FULL_IMAGE:figures/full_fig_p036_9.png]

**Figure 10.** Figure 10: Left: A comparison of all specific types of the Finslerian indicatrices (the color-coded [PITH_FULL_IMAGE:figures/full_fig_p037_10.png]

**Figure 11.** Figure 11: Left: The time geodesics (dashed black, with a step of [PITH_FULL_IMAGE:figures/full_fig_p038_11.png]

**Figure 12.** Figure 12: Left: The color-coded graph of the gravitational wind force [PITH_FULL_IMAGE:figures/full_fig_p039_12.png]

**Figure 13.** Figure 13: The triple Gaussian bell-shaped hill G (left) and its color-coded maps of height (top right) and the gravitational wind force (also presenting the steepness of the slope; bottom right), including time fronts centered at (0, 0) for the cases: MAT (green), ZNP (blue), R…

**Figure 14.** Figure 14: Left: The evolution of the unit time front on the slippery slope of the surface [PITH_FULL_IMAGE:figures/full_fig_p041_14.png]

**Figure 15.** Figure 15: Left: A comparison of the minimum (inner) and maximum (outer) ranges (boundaries) [PITH_FULL_IMAGE:figures/full_fig_p041_15.png]

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Nonlinear Anal

Aldea, N., Kopacz, P.: Time geodesics on a slippery cross slope under gravitational wind. Nonlinear Anal. Real World Appl. 81 (2025) 104177

work page 2025

[2] [2]

Non- linear Anal.-Theor

Aldea, N., Kopacz, P.: The slope-of-a-mountain problem in a cross gravitational wind. Non- linear Anal.-Theor. 235 (2023) 113329

work page 2023

[3] [3]

Aldea, N., Kopacz, P., Wolak, R.: Randers metrics based on deformations by gradient wind. Period. Math. Hung. 86(3) (2023) 266-280

work page 2023

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