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Rigidity of broken geodesic flow and inverse problems
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 132, Number 2, April 2010
- pp. 529-562
- 10.1353/ajm.0.0103
- Article
- Additional Information
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Consider broken geodesics $\alpha([0,l])$ on a compact Riemannian manifold
$(M,g)$ with boundary of dimension $n\geq 3$. The broken geodesics are
unions of two geodesics with the property that they have a common end
point. Assume that for every broken geodesic $\alpha([0,l])$ starting at
and ending to the boundary $\partial M$ we know the starting point and
direction $(\alpha(0),\alpha'(0))$, the end point and direction
$(\alpha(l),\alpha'(l))$, and the length $l$. We show that this data
determines uniquely, up to an isometry, the manifold $(M,g)$. This result
has applications in inverse problems on very heterogeneous media for
situations where there are many scattering points in the medium, and arises
in several applications including geophysics and medical imaging. As an
example we consider the inverse problem for the radiative transfer
equation (or the linear transport equation)
with a nonconstant wave speed. Assuming that the scattering kernel is
everywhere positive, we show that the boundary measurements determine the
wave speed inside the domain up to an isometry.