Connection between covariant and contravariant components o tensor What is the general proof of the relation between covariant and contravariant components of a tensor using the metric tensor?
$${g^{mr}g_{rn}=\delta^{m}_{n}}$$

I guess you are interested in the connection between covariant and covariant tensors on a Riemannian manifold. Let $(M,g)$ be a Riemannian manifold and let $\alpha$ be a $(0,1)$- covariant tensor (a section of $T^*M$). Then there exists a unique $(1,0)$-contravariant tensor $v$  (a section of $TM$) such that $\alpha=g(v,\cdot)$.  
Example: $g(grad\, f,\cdot)=df$
So  for any (r,s)- tensore $T$ (a section of $\otimes^rT^M{\otimes^s}T^*M$) you can find a unique (s,r)-tensor $T^*$ associated with $T$ via the metric $g$.
