Quote by mathwonk
if you really want to understand tensors, i sugest you try to understand the curvature tensor. this concept was apparently the reason for the invention of tensors and the first tensor invented by riemann.

I think the curvature tensor let us see that covariant en contravariant components refers to the same object.
You can describe the curvature by a (1,3)tensor but the (0,4) tensor contains the same information about the curvation of the manifold