  In this article, we study the classical chaotic scattering of a He atom off a
harmonically vibrating Cu surface. The three degrees of freedom (3- dof) model
is studied by first considering the non-vibrating 2-dof model for different
values of the energy. We calculate the set of singularities of the scattering
functions and study its connection with the tangle between the stable and
unstable manifolds of the fixed point at an infinite distance to the Cu surface
in the Poincar\'e map for different values of the initial energy. With these
manifolds, it is possible to construct the stable and unstable manifolds for
the 3-dof coupled model considering the extra closed degree of freedom and the
deformation of a stack of maps of the 2-dof system calculated at different
values of the energy. Also, for the 3-dof system, the resulting invariant
manifolds have the correct dimension to divide the constant total energy
manifold. By this construction, it is possible to understand the chaotic
scattering phenomena for the 3-dof system from a geometric point of view. We
explain the connection between the set of singularities of the scattering
function, the Jacobian determinant of the scattering function, the relevant
invariant manifolds in the scattering problem, and the cross-section, as well
as their behavior when the coupling due to the surface vibration is switched
on. In particular, we present in detail the connection between the changes in
the structure of the caustics in the cross-section and the changes in the zero
level set of the Jacobian determinant of the scattering function.
