Weak Riemannian manifolds from finite index subfactors

Let N subset of M be a finite Jones' index inclusion of II1 factors and denote by U-N subset of U-M their unitary groups. In this article, we study the homogeneous space U-M/U-N, which is a (infinite dimensional) differentiable manifold, diffeomorphic to the orbit O(p) = {u p u* : u is an element of U-M} of the Jones projection p of the inclusion. We endow O(p) with a Riemannian metric, by means of the trace on each tangent space. These are pre-Hilbert spaces (the tangent spaces are not complete); therefore, O(p) is a weak Riemannian manifold. We show that O(p) enjoys certain properties similar to classic Hilbert-Riemann manifolds. Among them are metric completeness of the geodesic distance, uniqueness of geodesics of the Levi-Civita connection as minimal curves, and partial results on the existence of minimal geodesics. For instance, around each point p(1) of O(p), there is a ball {q is an element of O(p) : parallel to q - p(1)parallel to < r} (of uniform radius r) of the usual norm of M, such that any point p(2) in the ball is joined to p(1) by a unique geodesic, which is shorter than any other piecewise smooth curve lying inside this ball. We also give an intrinsic (algebraic) characterization of the directions of degeneracy of the submanifold inclusion O(p) subset of P(M-1), where the last set denotes the Grassmann manifold of the von Neumann algebra generated by M and p.