The copshere bundle of a differentiable
manifold $M$ (denoted by $S^*(M)$) is the quotient of its
cotangent bundle without the zero section with respect to the
action by multiplications of $\RR^+$ which covers the identity on
$M$. It is a contact manifold which has the same privileged
position in contact geometry that cotangent bundles have in
symplectic geometry. Using a Riemannian metric on $M$, we can
identify $S^*(M)$ with its unitary tangent bundle and its Reeb
vector field with the geodesic field on $M$. If $M$ is endowed
with the proper action of a Lie group $G$, the lift of this action
on $S^*(M)$ respects the contact structure and admits an
equivariant momentum map $J$. We will study the topological and
geometrical properties of the reduced space of $S^*(M)$ at zero
momentum, i.e. $\left(S^*(M)\right)_0 :=J^{-1}(0)/G$. In particular,
we will annalyse the behaviour of the geodesic flow with respect to the
Lie group symmetries of $M$.