Deformative Magnetic Marked Length Spectrum Rigidity

TL;DR

This paper proves that under certain conditions, a family of magnetic flows on a closed surface with constant periodic orbit lengths implies the metrics and magnetic functions are smoothly equivalent via diffeomorphisms, extending previous rigidity results.

Contribution

It generalizes the magnetic flow rigidity theorem of Guillemin and Kazhdan to cases with variable magnetic functions and metrics, under negative magnetic curvature.

Findings

Constant periodic orbit lengths imply metric and magnetic function equivalence.

The result extends classical rigidity to magnetic flows with variable magnetic fields.

Provides conditions for smooth conjugacy of magnetic flows.

Abstract

Let $M$ be a closed surface and let $\{g_s \ | \ s \in (-\epsilon, \epsilon)\}$ be a smooth one-parameter family of Riemannian metrics on $M$. Also let $\{\kappa_s : M \rightarrow \mathbb{R} \ | \ s \in (-\epsilon, \epsilon)\}$ be a smooth one-parameter family of functions on $M$. Then the family $\{(g_s, \kappa_s) \ | \ s \in (-\epsilon, \epsilon)\}$ gives rise to a family of magnetic flows on $TM$. We show that if the magnetic curvatures are negative for $s \in (-\epsilon, \epsilon)$ and the lengths of each periodic orbit remains constant as the parameter $s$ varies, then there exists a smooth family of diffeomorphisms $\{f_s : M \rightarrow M \ | \ s \in (-\epsilon, \epsilon)\}$ such that $f_s^*(g_s) = g_0$ and $f_s^*(\kappa_s) = \kappa_0$. This generalizes a result of Guillemin and Kazhdan to the setting of magnetic flows.