  We investigate the validity, as well as the failure, of Sobolev-type
inequalities on Cartan-Hadamard manifolds under suitable bounds on the
sectional and the Ricci curvatures. We prove that if the sectional curvatures
are bounded from above by a negative power of the distance from a fixed pole
(times a negative constant), then all the $ L^p $ inequalities that interpolate
between Poincar\'e and Sobolev hold for radial functions provided the power
lies in the interval $ (-2,0) $. The Poincar\'e inequality was established by
H.P. McKean under a constant negative bound from above on the sectional
curvatures. If the power is equal to the critical value $ -2 $ we show that $ p
$ must necessarily be bounded away from $ 2 $. Upon assuming that the Ricci
curvature vanishes at infinity, the nonradial version of such inequalities
turns out to fail, except in the Sobolev case. Finally, we discuss applications
of the here-established Sobolev-type inequalities to optimal smoothing effects
for radial porous medium equations.
