Broken ray tensor tomography with one reflecting obstacle

Ilmavirta, Joonas; Paternain, Gabriel P.

by Joonas Ilmavirta, Gabriel P. Paternain

Abstract:

We show that a tensor field of any rank integrates to zero over all broken rays if and only if it is a symmetrized covariant derivative of a lower order tensor which satisfies a symmetry condition at the reflecting part of the boundary and vanishes on the rest. This is done in a geometry with non-positive sectional curvature and a strictly convex obstacle in any dimension. We give two proofs, both of which contain new features also in the absence of reflections. The result is new even for scalars in dimensions above two.

View PDF

Reference:

Broken ray tensor tomography with one reflecting obstacle (Joonas Ilmavirta, Gabriel P. Paternain), Communications in Analysis and Geometry, 2018. (To appear.) [show abstract] [hide abstract] We show that a tensor field of any rank integrates to zero over all broken rays if and only if it is a symmetrized covariant derivative of a lower order tensor which satisfies a symmetry condition at the reflecting part of the boundary and vanishes on the rest. This is done in a geometry with non-positive sectional curvature and a strictly convex obstacle in any dimension. We give two proofs, both of which contain new features also in the absence of reflections. The result is new even for scalars in dimensions above two. [arXiv]

Bibtex Entry:

@article{brt-tensor-hd,
	author = {Joonas Ilmavirta and Gabriel P. Paternain},
	title = {{Broken ray tensor tomography with one reflecting obstacle}},
        journal = {Communications in Analysis and Geometry},
        note = {To appear.},
	month = may,
	year = {2018},
	arxiv = {1805.04947},
        gsid = {5013269271163380398},
	url={http://users.jyu.fi/~jojapeil/pub/tensor-hd-1obst-v5.pdf},
	abstract = {We show that a tensor field of any rank integrates to zero over all broken rays if and only if it is a symmetrized covariant derivative of a lower order tensor which satisfies a symmetry condition at the reflecting part of the boundary and vanishes on the rest. This is done in a geometry with non-positive sectional curvature and a strictly convex obstacle in any dimension. We give two proofs, both of which contain new features also in the absence of reflections. The result is new even for scalars in dimensions above two.}
}