Rigidity of a right-angle boundary-chord contact

Prove that if, along a unit-speed magnetic geodesic on a closed negatively curved surface with nonpositive magnetic curvature, the angle between the magnetic velocity and the forward boundary chord equals $\pi/2$ at one time, then it equals $\pi/2$ at every time and the surface is $\mu$-magnetically flat.

Background

The paper defines η(t)\eta(t) as the angle between a magnetic geodesic and the forward boundary chord based at its point γ(t)\gamma(t). It proves that η(t)≤π/2\eta(t)\leq\pi/2 and establishes a one-sided result: equality at a time T0T_0 implies equality for all later times T≥T0T\geq T_0. The authors conjecture the stronger two-sided rigidity statement, which would characterize magnetic flatness from a single right-angle occurrence.

References

By \cref{Lemma:angle_<_pi/2}, $\eta(t)\leq \pi/2$, and we now make the following rigidity conjecture. Suppose $(\Sigma,\mu)$ satisfies \ref{H1}. If $\eta(T_0)=\pi/2$ for some $T_0$, then $\eta(t)=\pi/2$ for all $t\in\re$, so $\Sigma$ is $\mu$-magnetically flat.

— Surfaces with nonpositive magnetic curvature  (2608.13534 - Hasselblatt et al., 13 Aug 2026) in Conjecture 2.15, Section 2, Subsection “Characterization of magnetic flatness”