Preprint · arXiv · April 19, 2012

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

Yernat M. Assylbekov, N. S. Dairbekov

Read on arXiv PDFDOI 10.48550/arxiv.1204.4383

Abstract

We consider a general family of curves $Γ$ on a compact oriented Finsler surface $(M,F)$ with boundary $\partial M$. Let $φ\in C^{\infty}(M)$ and $ω$ a smooth 1-form on $M$. We show that $$\int_{γ(t)}\{φ(γ(t))+ω_{γ(t)}(\dotγ(t))\}\,dt=0$$ holds for every $γ\inΓ$ whose endpoints belong to $\partial M$, $γ(a)\in\partial M$, $γ(b)\in\partial M$ if and only if $φ=0$ and $ω$ 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.

Advanced Differential Geometry ResearchMathematical Dynamics and FractalsGeometric Analysis and Curvature FlowsX-ray transformcurves on surfacescohomological equation

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

Abstract

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