  We study Sobolev estimates for the solutions of parabolic equations acting on
a vector bundle, in a complete, compact or non compact, riemannian manifold
$M.$ The idea is to introduce geometric weights on $M.$ We get global Sobolev
estimates with these weights. As applications, we find and improve "classical
results", i.e. results without weights, by use of a Theorem by Hebey and
Herzlich. As an example we get Sobolev estimates for the solutions of the heat
equation on $p$-forms when the manifold has "weak bounded geometry " of order
$1$.
