Geodesic
The ray transform of symmetric tensor fields with incomplete projection data, I: The kernel of the ray transform
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 18 (2021) no. 2, pp. 1219-1237

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We consider the ray transform $I_\Gamma$ that integrates symmetric rank $m$ tensor fields on ${\mathbb{R}}^n$ supported in a bounded convex domain $D\subset{\mathbb{R}}^n$ over lines. The integrals are known for the family $\Gamma$ of lines $l$ such that endpoints of the segment $l\cap D$ belong to a given part $\gamma=\partial D\cap{\mathbb{R}}^n_+$ of the boundary, for some half-space ${\mathbb{R}}^n_+\subset{\mathbb{R}}^n$. We prove that the kernel of the operator $I_\Gamma$ coincides with the space of $\gamma$-potential tensor fields.
Keywords: tomography with incomplete data, ray transform, tensor analysis. 
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     author = {V. A. Sharafutdinov},
     title = {The ray transform of symmetric tensor fields with incomplete projection data, {I:} {The} kernel of the ray transform},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {1219--1237},
     publisher = {mathdoc},
     volume = {18},
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     year = {2021},
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     url = {http://geodesic.mathdoc.fr/item/SEMR_2021_18_2_a68/}
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V. A. Sharafutdinov. The ray transform of symmetric tensor fields with incomplete projection data, I: The kernel of the ray transform. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 18 (2021) no. 2, pp. 1219-1237. http://geodesic.mathdoc.fr/item/SEMR_2021_18_2_a68/