The Closed Curve Problem

Definitions and mathematical statement

The phrase "an integral curve defined by two periodic functions" does not specify how those functions determine a curve. The supplied triage identifies the intended standard closed-curve problem as the periodic curvature-and-torsion problem, whereas the background describes coordinatewise integration. These are different mathematical questions and are retained separately.

For the classical regular space-curve interpretation, a precise setting to be confirmed is a period $T>0$ and smooth functions $\kappa,\tau:{\mathbb{R}}\to{\mathbb{R}}$ satisfying \[ \kappa(s+T)=\kappa(s)>0,\qquad \tau(s+T)=\tau(s). \] Here $s$ is arc length, $\kappa$ is curvature, and $\tau$ is torsion. The Frenet curve and orthonormal frame $(\mathbf t,\mathbf n,\mathbf b)$ are determined up to an initial Euclidean motion by \[ \gamma'=\mathbf t,\qquad \mathbf t'=\kappa\mathbf n,\qquad \mathbf n'=-\kappa\mathbf t+\tau\mathbf b,\qquad \mathbf b'=-\tau\mathbf n. \] Normalize $\gamma(0)=0$ and $F(0)=I_3$, where $F=(\mathbf t\ \mathbf n\ \mathbf b)$ is the frame matrix. Its one-period Euclidean-motion monodromy is $(F(T),\gamma(T))$. Position closure over the selected period requires $\gamma(T)=0$; position and tangent matching additionally require $\mathbf t(T)=\mathbf t(0)$; full frame matching requires $F(T)=I_3$. Full position-and-frame closure is therefore the formal condition \[ \gamma(T)=0,\qquad F(T)=I_3. \] The intended research request, according to the supplied assessment, is for explicit, usable necessary and sufficient conditions on the periodic data $\kappa,\tau$ beyond merely solving this differential system and testing its monodromy. "Explicit" and "readily checked" are not given a formal computational meaning.

This is only a specified classical setting: the stored statement does not impose regularity, positive curvature, an ambient dimension, a closure convention, or whether closure after a multiple of $T$ suffices. Curves with vanishing curvature or singularities require additional definitions, and the record does not choose such definitions. These omissions prevent a unique unconditional formulation.

For comparison, under the background's coordinatewise interpretation with continuous $T$-periodic functions $f,g$ and \[ \eta(s)=\left(\int_0^s f(u)\,{\,\mathrm{d}} u,\int_0^s g(u)\,{\,\mathrm{d}} u\right), \] the supplied triage notes the elementary zero-mean closure criterion $\int_0^T f(u)\,{\,\mathrm{d}} u=\int_0^T g(u)\,{\,\mathrm{d}} u=0$. This is not the standard Efimov--Fenchel curvature-and-torsion problem.

The supplied literature review, checked 2026-08-17, labels the intended problem partially solved. It reports the formal Frenet-monodromy criterion, Bamba and Ogata's rotationally symmetric special-family solution with extensions to several singular-curve settings, and Arroyo, Garay and Mencia's explicit sufficient and nonclosure results for the planar one-curvature-function case. It reports no verified general explicit usable characterization for periodic curvature-torsion pairs. The source also mentions motivation from dynamical systems, Hamiltonian mechanics and periodic orbits. These are dated source descriptions, not a new literature assessment.

Short English statement

Find usable tests for when periodic curvature and torsion generate a closed space curve. The stored wording does not identify the functions as curvature and torsion or specify what kind of closure is required; coordinatewise integration gives a different, elementary zero-mean problem. The supplied assessment records monodromy and special-family progress, with the general intended characterization still open at its check date.

Sources

• UnsolvedMath local registry, permanent ID 1208, source code catalogue entry, "The Closed Curve Problem". Dataset attribution: Ulam AI and the UnsolvedMath Contributors; curation and metadata under CC BY 4.0.
• K. Bamba and Y. Ogata, Closed space curves with singularities, generated by periodic curvature and torsion, Journal of Geometry 117 (2026), article 12, https://doi.org/10.1007/s00022-025-00792-3. The supplied triage cites a rotationally symmetric special case and singular extensions.
• Josu Arroyo, Oscar J. Garay and Jose J. Mencia, When is a periodic function the curvature of a closed plane curve?, American Mathematical Monthly 115 (2008), 405--414, https://doi.org/10.1080/00029890.2008.11920543. The supplied triage cites planar closure and nonclosure criteria.
• Eric W. Weisstein, Closed Curve Problem, MathWorld, https://mathworld.wolfram.com/ClosedCurveProblem.html. The supplied assessment cites its curvature-and-torsion formulation.
• The local literature triage records its checks as 2026-08-17; the above references were not independently visited for this draft.