  Let X be a right Hilbert C*-module over A. We study the geometry and the
topology of the projective space P(X) of X, consisting of the orthocomplemented
submodules of X which are generated by a single element. We also study the
geometry of the p-sphere S_p(X) and the natural fibration S_p(X) \to P(X),
where S_p(X)={x\in X: <x,x>=p}, for p in A a projection. The projective space
and the p-sphere are shown to be homogeneous differentiable spaces of the
unitary group of the algebra L_A(X) of adjointable operators of X. The homotopy
theory of these spaces is examined.
