  We prove a Poincar\'e, and a general Sobolev type inequalities for functions
with compact support defined on a $k$-rectifiable varifold $V$ defined on a
complete Riemannian manifold with positive injectivity radius and sectional
curvature bounded above. Our techniques allow us to consider Riemannian
manifolds $(M^n,g)$ with $g$ of class $C^2$ or more regular, avoiding the use
of Nash's isometric embedding theorem. Our analysis permits to do some quite
important fragments of geometric measure theory also for those Riemannian
manifolds carrying a $C^2$ metric $g$, that is not $C^{k+\alpha}$ with
$k+\alpha>2$. The class of varifolds we consider are those which first
variation $\delta V$ lies in an appropriate Lebesgue space $L^p$ with respect
to its weight measure $\|V\|$ with the exponent $p\in\mathbb{R}$ satisfying
$p>k$.
