The X-ray transform on a general family of curves on Finsler surfaces

Abstract

We consider a general family of curves $\Gamma$ on a compact oriented Finsler surface $(M,F)$ with boundary $\partial M$. Let $\phi \in C^{\infty }(M)$ and $\omega$ a smooth 1-form on $M$. We show that $${\int }_{\gamma (t)}\{\phi (\gamma (t))+{\omega }_{\gamma (t)}(\dot{\gamma }(t))\}dt=0$$ holds for every $\gamma \in \Gamma$ whose endpoints belong to $\partial M$, $\gamma (a)\in \partial M$, $\gamma (b)\in \partial M$ if and only if $\phi =0$ and $\omega$ is exact. Similar results were proved when $M$ is closed and some additional conditions on Gaussian curvature are imposed. We also study the cohomological equations of Anosov generelized thermostats on a closed Finsler surface. Finally, we gave conditions when thermostat is of Anosov type.

25 pages

References in corpus (1)

• Hopf type rigidity for thermostats

Cited by in corpus (2)

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