Hello everyone, I'm new here !
I haven't had much chance looking all over the internet and going through my bookshelf to answer my questions. I will try to keep it short :

  1. What is the purpose in diff. geometry to have so many derivatives ( Directional derivative of the tangent vector, exterior derivative, covariant exterior derivative, covariant derivative, Lie derivative, Ehresman connection). I understand how they work, on what objects they act, but I can't get my head around why we need so many. In the case of exterior derivative of differential forms , I have seen the treatment of it as a list of properties we'd like the derivative to have and trying to find it. But is that really rigorous, I mean for the directional derivative for example we formulate what quantity we want to study ( namely the variation of whatever in a particular direction ) and derive properties from the operator not the other way around.
  2. The intuition behind a tangent vector being a line segment and the differential forms being the length of the line segment. By the mechanics of the operations, I understand the dual object acting on the vector to extract his component is basically "projecting" the vector to the basis. But in the context of integration where we have no metric on a manifold, the intuition of Riemann sums is applied to Tangent spaces where since we have no way to compare point on the manifold we end up working in the Linear Tangent space. We consider there Tangent vector as being line segment and since we have a canonical isomorphisme or the euclidean space with the tangent vector we can validate this identification (Tangent vector = line segment). Then out of nowhere we see that differential forms transform properly on a change of charts and that's another clue that it is the objects to integrate on a manifold. But for me there is little intuition to it. The tangent vector is a directional derivative operator that act on function, how can it be a line segment ? it's at best a direction "intuitively" but again not formally. To sum up I understand the motivation of integration of n-forms on manifolds but I don't understant why when we reduce the problem to a local chart with euclidean structure, we take the leap to identify tangent vector to segments and n-forms to evaluating the length of them to make sense of the Riemann sum.

I hope I have been clear enough on my issues and hope to have nice food for thought from you