Metric geometry in infinite dimensional Stiefel manifolds

Let II be a separable Banach ideal in the space of bounded operators acting in a Hilbert space HH and II the set of partial isometries in HH. Fix v∈Iv∈I. In this paper we study metric properties of the II-Stiefel manifold associated to v, namelyStI(v)={v0∈I:v−v0∈I,j(v0∗v0,v∗v)=0}, where j(,)j(,) is the Fredholm index of a pair of projections. Let UI(H)UI(H) be the Banach–Lie group of unitary operators which are perturbations of the identity by elements in II. Then StI(v)StI(v) coincides with the orbit of v   under the action of UI(H)×UI(H)UI(H)×UI(H) on II given by (u,w)⋅v0=uv0w∗(u,w)⋅v0=uv0w∗, u,w∈UI(H)u,w∈UI(H) and v0∈StI(v)v0∈StI(v). We endow StI(v)StI(v) with a quotient Finsler metric by means of the Banach quotient norm of the Lie algebra of UI(H)×UI(H)UI(H)×UI(H) by the Lie algebra of the isotropy group. We give a characterization of the rectifiable distance induced by this metric. In fact, we show that the rectifiable distance coincides with the quotient distance of UI(H)×UI(H)UI(H)×UI(H) by the isotropy group. Hence this metric defines the quotient topology in StI(v)StI(v).The other results concern with minimal curves in II-Stiefel manifolds when the ideal II is fixed as the compact operators in HH. The initial value problem is solved when the partial isometry v has finite rank. In addition, we use a length-reducing map into the Grassmannian to find some special partial isometries that can be joined with a curve of minimal length.