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We prove a Poincar\'e, and a general Sobolev type inequalities for functionswith compact support defined on a $k$-rectifiable varifold $V$ defined on acomplete Riemannian manifold with positive injectivity radius and sectionalcurvature bounded above. Our techniques allow us to consider Riemannianmanifolds $(M^n,g)$ with $g$ of class $C^2$ or more regular, avoiding the useof Nash's isometric embedding theorem. Our analysis permits to do some quiteimportant fragments of geometric measure theory also for those Riemannianmanifolds 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 firstvariation $\delta V$ lies in an appropriate Lebesgue space $L^p$ with respectto its weight measure $\|V\|$ with the exponent $p\in\mathbb{R}$ satisfying$p>k$.

Poincar\'e and Sobolev type inequalities for intrinsic rectifiable varifolds