  We consider the porous medium equation with a power-like reaction term, posed
on Riemannian manifolds. Under certain assumptions on $p$ and $m$ in (1.1), and
for small enough nonnegative initial data, we prove existence of global in time
solutions, provided that the Sobolev inequality holds on the manifold.
Furthermore, when both the Sobolev and the Poincar\'e inequality hold, similar
results hold under weaker assumptions on the forcing term. By the same
functional analytic methods, we investigate global existence for solutions to
the porous medium equation with source term and variable density in ${\mathbb
R}^n$.
