Tensor tomography in periodic slabs (bibtex)
by Joonas Ilmavirta, Gunther Uhlmann
Abstract:
The X-ray transform on the periodic slab $[0,1]\times\mathbb T^n$, $n\geq0$, has a non-trivial kernel due to the symmetry of the manifold and presence of trapped geodesics. For tensor fields gauge freedom increases the kernel further, and the X-ray transform is not solenoidally injective unless $n=0$. We characterize the kernel of the geodesic X-ray transform for $L^2$-regular $m$-tensors for any $m\geq0$. The characterization extends to more general manifolds, twisted slabs, including the M\"obius strip as the simplest example.
Reference:
Tensor tomography in periodic slabs (Joonas Ilmavirta, Gunther Uhlmann), Journal of Functional Analysis, 2017. (To appear.) [show abstract] [hide abstract] The X-ray transform on the periodic slab $[0,1]\times\mathbb T^n$, $n\geq0$, has a non-trivial kernel due to the symmetry of the manifold and presence of trapped geodesics. For tensor fields gauge freedom increases the kernel further, and the X-ray transform is not solenoidally injective unless $n=0$. We characterize the kernel of the geodesic X-ray transform for $L^2$-regular $m$-tensors for any $m\geq0$. The characterization extends to more general manifolds, twisted slabs, including the Möbius strip as the simplest example. [arXiv]
Bibtex Entry:
@article{x-ray-slab,
	author = {Joonas Ilmavirta and Gunther Uhlmann},
	title = {{Tensor tomography in periodic slabs}},
	month = jul,
	year = {2017},
    	journal = {Journal of Functional Analysis},
	note = {To appear.},
	arxiv = {1707.01343},
	doi = {10.1016/j.jfa.2018.04.004},
	url={http://users.jyu.fi/~jojapeil/pub/x-ray-slab.pdf},
	abstract = {The X-ray transform on the periodic slab $[0,1]\times\mathbb T^n$, $n\geq0$, has a non-trivial kernel due to the symmetry of the manifold and presence of trapped geodesics. For tensor fields gauge freedom increases the kernel further, and the X-ray transform is not solenoidally injective unless $n=0$. We characterize the kernel of the geodesic X-ray transform for $L^2$-regular $m$-tensors for any $m\geq0$. The characterization extends to more general manifolds, twisted slabs, including the M\"obius strip as the simplest example.}
}
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