  The aim of this paper is the geometric study of the symplectic operators
which are a perturbation of the identity by a Hilbert-Schmidt operator. This
subgroup of the symplectic group was introduced in Pierre de la Harpe's
classical book of Banach-Lie groups. Throughout this paper we will endow the
tangent spaces with different Riemannian metrics. We will use the minimal
curves of the unitary group and the positive invertible operators to compare
the length of the geodesic curves in each case. Moreover we will study the
completeness of the symplectic group with the geodesic distance.
