We consider the fast diffusion equation (FDE) ut = Δum (0 < m < 1) on a nonparabolic Riemannian manifold M. Existence of weak solutions holds. Then we show that the validity of Euclidean–type Sobolev inequalities implies that certain Lp−Lq smoothing effects of the type ∥u(t)∥q ≤ Ct−α ∥u0∥γp, the case q = ∞ being included. The converse holds if m is sufficiently close to one. We then consider the case in which the manifold has the addition gap property min σ(−Δ) > 0. In that case solutions vanish in finite time, and we estimate from below and from above the extinction time.