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Let I(∞) be the set of partial isometries with finite rank of an infinite dimensional Hilbert space H. We show that I(∞) is a smooth submanifold of the Hilbert space B2(H) of Hilbert-Schmidt operators of H, each connected component is the set IN, which consists of all partial isometries of rank N < ∞. Furthermore, I(∞) is a homogeneous space of U(∞) × U(∞), where U(∞) is the classical Banach-Lie group of unitary operators of H, which are Hilbert-Schmidt perturbations of the identity. We introduce two Riemannian metrics in I(∞). One via the ambient inner product of B2(H), the other by means of the group action. We show that both metrics are equivalent and complete