can be used to determine when two Riemannian metrics cannot be

connected with a path of metrics maintaining a certain curvature

condition. We use the eta invariant of Spin

distinguish connected components of moduli spaces of Riemannian

metrics with positive Ricci curvature. We then find infinitely many

non-diffeomorphic five dimensional manifolds for which these moduli

spaces each have infinitely many components. The manifolds are total

spaces of principal S

projective space and the metrics are lifted from Ricci positive

metrics on the base.