  For the cotangent bundle of a smooth Riemannian manifold acted upon by the
lift of a smooth and proper action by isometries of a Lie group, we
characterize the symplectic normal space at any point. We show that this space
splits as the direct sum of the cotangent bundle of a linear space and a
symplectic linear space coming from reduction of a coadjoint orbit. This
characterization of the symplectic normal space can be expressed solely in
terms of the group action on the base manifold and the coadjoint
representation. Some relevant particular cases are explored.
