Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Scattering rigidity with trapped geodesics   We prove that the flat product metric on $D^n\times S^1$ is scattering rigid
where $D^n$ is the unit ball in $\R^n$ and $n\geq 2$. The scattering data
(loosely speaking) of a Riemannian manifold with boundary is map $S:U^+\partial
M\to U^-\partial M$ from unit vectors $V$ at the boundary that point inward to
unit vectors at the boundary that point outwards. The map (where defined) takes
$V$ to $\gamma'_V(T_0)$ where $\gamma_V$ is the unit speed geodesic determined
by $V$ and $T_0$ is the first positive value of $t$ (when it exists) such that
$\gamma_V(t)$ again lies in the boundary. We show that any other Riemannian
manifold $(M,\partial M,g)$ with boundary $\partial M$ isometric to
$\partial(D^n\times S^1)$ and with the same scattering data must be isometric
to $D^n\times S^1$. This is the first scattering rigidity result for a manifold
that has a trapped geodesic. The main issue is to show that the unit vectors
tangent to trapped geodesics in $(M,\partial M,g)$ have measure 0 in the unit
tangent bundle.

36
The categories of this paper are: cmp-lg
22
The categories of this paper are: chao-dyn
32
The categories of this paper are: atom-ph
25
The categories of this paper are: patt-sol
3
The categories of this paper are: physics
10
The categories of this paper are: q-bio
30
The categories of this paper are: q-alg
18
The categories of this paper are: econ
23
The categories of this paper are: solv-int
20
The categories of this paper are: acc-phys
7
The categories of this paper are: hep-th
37
The categories of this paper are: supr-con
9
The categories of this paper are: nlin
27
The categories of this paper are: comp-gas
16
The categories of this paper are: stat
13
The categories of this paper are: quant-ph