  Consider a 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
non-constant 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.
