Two-Point Connectivity by Fixed-Energy Magnetic Geodesics

Definitions and mathematical statement

A closed manifold means a compact smooth manifold without boundary. The supplied record asks about fixed-energy magnetic geodesics on a finite-dimensional closed manifold, but does not specify the magnetic data or the hypotheses needed for its intended connectivity assertion.

For a conventional formulation, suppose that a magnetic system consists of a Riemannian manifold $(M,g)$ and a closed real differential $2$-form $\sigma$. This is an explicit interpretation of the terminology, rather than additional hypotheses established by the supplied passage. Define the Lorentz-force endomorphism $Y:TM\to TM$ by \[ g_x(Y_xv,w)=\sigma_x(v,w),\qquad x\in M,\quad v,w\in T_xM. \] Writing $\nabla$ for the Levi--Civita connection of $g$, a magnetic geodesic in this convention is a $C^2$ curve $\gamma:[0,T]\to M$ satisfying \[ \nabla_{\dot\gamma}\dot\gamma=Y_{\gamma}\dot\gamma. \] Its kinetic energy is $E(\gamma,\dot\gamma)=\tfrac12 g_{\gamma}(\dot\gamma,\dot\gamma)$.

The extracted question asks under which hypotheses on the magnetic system and energy the following endpoint-connectivity property holds: \[ \begin{gathered} \forall p,q\in M\quad \forall k>0\quad \exists T>0\quad \exists\gamma\in C^2([0,T],M):\\ \gamma(0)=p,\qquad \gamma(T)=q,\\ \nabla_{\dot\gamma(t)}\dot\gamma(t)=Y_{\gamma(t)}\dot\gamma(t),\qquad \tfrac12 g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t))=k \quad(0\le t\le T). \end{gathered} \] Here the travel time is allowed to vary. The displayed formula expresses the unrestricted positive-energy question; it is not asserted to hold. The source also asks how the admissible energy range depends on Mañé critical values. Its supplied dated assessment singles out a supercritical threshold question, but neither identifies the critical value nor specifies a corresponding inequality or the hypotheses under which that value is defined. Consequently, a precise supercritical specialization cannot be recovered from this record.

Further qualifications remain unresolved. Connectedness is not stated: endpoints in distinct connected components cannot be joined by a curve, so any intended universal assertion needs a connectedness hypothesis or a restriction on endpoints. The passage also does not say whether the two endpoints must be distinct; including $p=q$ requires a positive-energy return to the same point. These choices must be settled before the displayed quantifiers can serve as the definitive problem statement.

Short English statement

For a magnetic system on a closed manifold, ask when every prescribed pair of endpoints can be joined by a magnetic geodesic with prescribed positive kinetic energy and freely chosen travel time. Determine how this property depends on the relevant Mañé critical values. The magnetic hypotheses, endpoint restrictions, and intended critical threshold are missing, so the formulation remains provisional. The supplied open status is dated curation and requires independent review.

Sources

• Local registry: data/unsolvedmath/problems.json, supplied record for permanent UnsolvedMath ID $30006368$, exact source problem code catalogue entry, titled "Two-Point Connectivity by Fixed-Energy Magnetic Geodesics." No individual proposer is supplied; the proposed year is recorded as 2025.
• Supplied report citation: Dynamische Systeme (2026), Oberwolfach Reports, DOI 10.4171/owr/2025/31, pp. 1647--1706. https://doi.org/10.4171/owr/2025/31. The record attributes the extracted question to this workshop-report context.
• Supplied literature source, labeled "Albers--Maier primary paper": https://doi.org/10.1007/s00220-026-05651-z. No title or precise supporting result is supplied.
• Supplied literature source, labeled "Oberwolfach report": https://doi.org/10.4171/owr/2025/31.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the finite-dimensional supercritical Mañé-threshold connectivity question on closed manifolds open and says the extracted passage lacks hypotheses sufficient for a stronger classification. This is supplied dated curation, not an independently verified assessment.
• The original passage refers to reference [14] for Mañé critical values; its bibliographic identity is not supplied. The registry labels the statement corrected_verified and reports that the official report was checked and notation repaired. These are supplied verification claims, not verification performed for this draft.