Geometry of the sphere of a Hilbert module

The sphere S-X = {x is an element of X: [x, x] = 1} of a right Hilbert C*-m odule X over a unital C*-algebra B is studied using differential geometric techniques. An action of the unitary group of the algebra L-B(X) of adjoint able B-module operators of X makes S-X a homogeneous space of this group, A reductive structure is introduced, as well as a Finsler metric. Metric pro perties of the geodesic curves are established. In the case B a von Neumann algebra and X self-dual, the fundamental group of S-X is computed.