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Swimming in curved space or The Baron and the cat

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J.E. Avron, O. Kenneth

Mathematical Physics · Submitted 22 Feb 2006 · 14 references

Pith's verdict: A genuinely useful geometric-mechanics paper whose main formula is coherent, but its only worked small-swimmer example disagrees with its own central equations by a factor of -2.

T0 · AI review

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arXiv IDmath-ph/0602053 (version 1)
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CitationJ.E. Avron, O. Kenneth. "Swimming in curved space or The Baron and the cat." arXiv:math-ph/0602053 (2006). https://pith.science/paper/FHF76Y22
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The paper

The paper claims

In a curved space, a small deformable body can move itself through a repeated shape change, and the distance moved per stroke equals the Riemann curvature times a moment of the swimmer's mass distribution.

The reading

In flat space, an object cannot move its center of mass by pushing and pulling on itself. A falling cat can twist its body to turn around, but it cannot make its center of mass move. This paper asks what happens when space itself is curved, like the surface of a sphere or a saddle-shaped hyperbolic space. In curved space, the usual idea of a center of mass loses its meaning, and internal forces can produce a real translation. The authors model a swimmer as a small collection of point masses with controllable internal forces. They use the fact that in spaces with lots of symmetry, there are conserved quantities associated to the isometries, generated by Killing vector fields. For a very small swimmer, they compute how much the swimmer moves after a tiny change of shape followed by a return to the original shape. Their main formula says the distance moved equals the Riemann curvature of space times a certain weighted moment of the mass distribution, contracted with the chosen shape changes. If you reverse the shape-change sequence, you swim the other way. A stroke that swims right on a sphere swims left on a hyperbolic plane, because the curvature changes sign. The authors work out the example of a triangular swimmer and find that it needs at least three masses, that too much symmetry prevents swimming, and that the optimal oars are as heavy as the payload. The whole effect is geometric, independent of how fast the swimmer moves, and requires no dissipation.
Extended reading notes

Core claim

For a small swimmer, the swimming distance in one stroke is determined by the Riemann curvature times certain moments of the swimmer. Concretely, Eq (6.1): M δx^k = R^j_lik (Σ_n m_n x^i_n η^j_(b) η^l_(c)|_{x_n}) dσ^b ∧ dσ^c. If the paper is correct, the displacement per infinitesimal stroke is a product of ambient curvature and the swimmer's mass-weighted cubic moments, contracted with the deformation fields.

Load-bearing premise

The ambient space is assumed to admit exact rigid body motions, i.e., it is homogeneous and isotropic (maximally symmetric), so that the shape-space and position coordinates (σ, τ) of Section 3.3 are valid and conserved charges P_ξ of Eq (2.5) exist for all k Killing fields. The paper explicitly restricts to this case in the introduction: 'In a generic curved space, the shape of a general swimmer completely determines its location... To be able to swim anywhere and with arbitrary orientation the ambient space must be homogeneous and isotropic.' Additionally, the small-swimmer formula Eq (6.1) uses the local Euclidean expansion (5.7), which requires RL² ≪ 1. If the space is not maximally symmetric, or the swimmer is not small, the derivation of Eq (4.7) and Eq (6.1) does not apply.

Original abstract

We study the swimming of non-relativistic deformable bodies in (empty) static curved spaces. We focus on the case where the ambient geometry allows for rigid body motions. In this case the swimming equations turn out to be geometric. For a small swimmer, the swimming distance in one stroke is determined by the Riemann curvature times certain moments of the swimmer.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,462 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.57 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.13 · compares with prior results 0.04 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We study the swimming of non-relativistic deformable bodies in (empty) static curved spaces. We focus on the case where the ambient geometry allows for rigid body motions. In this case the swimming equations turn out to be geometric. For a small swimmer, the swimming distance in one stroke is determined by the Riemann curvature times certain moments of the swimmer.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We study the swimming of non-relativistic deformable bodies in (empty) static curved spaces. contribution 0.98 0.81 0.05 asserted 0.87
s1 We focus on the case where the ambient geometry allows for rigid body motions. contribution 0.40 0.30 0.04 asserted 0.93
s2 In this case the swimming equations turn out to be geometric. result 0.89 0.85 0.19 asserted 1.00
s3 For a small swimmer, the swimming distance in one stroke is determined by the Riemann curvature times certain moments of the swimmer. result 1.00 0.86 0.74 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. The red dot is equidistant from the three identical yellow point masses and so is one natural notion of a center of mass. The blue dot is another center of mass. It is obtained by first moving the pair of points masses on the right to their center of mass and then finding the center of mass of the remaining pair of (unequal) masses. Swimming is the motion of deformable bodies. More precisely, we shall mean by this the rigid body motion affected by a periodic deformation. After completing a swimming stroke the swimmer regains its original shape and finds itself in a different location
Figure 2. Newton third law in curved space: The mutual forces are directed along a geodesic and are balanced. The constant of motion Pξ, associated to the isometry ξ, is independent of V and is given by Pξ = X n mn x˙ n · ξ(xn) (2.4) We shall assume throughout that the particles that make up the body are initially at rest, x˙ n = 0, and so Pξ = 0. Suppose now that the body, initially at rest, can control its shape by, for example, controlling V . The rigid body part of the resulting motion is determined by the conservations laws X n mn ξ(β)(xn) · dxn = 0, β = 1 . . . , k (2.5) It is evi
Figure 3. The full red circle on the left, splits into the two blue half balls which recombine at the empty red circle on the right. This allows for moving without violating Newton laws. 14
Figure 4. The red ball can displace itself to the position of the blue ball by splitting into two parts that it sends to crossing points of geodesics and then recombine satisfying Newton’s law in the process. A variation on this theme occurs whenever the space has intersecting geodesics, as shown in
Figure 5. A cone is represented by the unshaded sector of the plane. The straight lines on the boundary of the gray area are identified. Geodesics are ordinary straight lines in the unshaded area. The stationary red ball splits and then recombines to a moving (empty) ball. is then: (dℓ) 2 = |dz| 2 (1 + R|z| 2 ) 2 (B.1) R is the Gaussian curvature. The (normalized) Killing field are then ξ1 = ∂x + R
Figure 6. A swimming triangle on the sphere and the pseudosphere. The oars are long and their total weight equals the weight of the payload. B.2.2 Swimming triangles Consider an isosceles triangle made of two identical point masses m at the base so that the total mass of the swimmer is M. For a the triangle of height h and base b (whose center of mass is at the origin) one has 8 M Xmn xn y 2 n  = 4m(M − 2m) M2 h b2 ≤ 1 2 hb2 (B.10) and the optimizer is when 4m = M. The optimal weight distribution has oars whose weight balances the weight of the payload. The dependence on b may be

Figures reproduced from arXiv:math-ph/0602053 by the authors.

References

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