My question is why terms in differential geometry that are apriori complicated can be completely computed by linear tensors; details below.
Let $M$ be a manifold with a bundle $E$ and a connection $\nabla$, for simplicity we can take $E$ to be the tangent bundle.
Consider the two linear tensors:
- Torsion; sending vector fields $X,Y$ to $\nabla_X Y - \nabla _Y X - [X,Y]$
- Curvature: sending $X,Y,Z$ to $[\nabla_X, \nabla_Y](Z)$
Those both formalize intuitions that are expressed as local coordinates. Let us take a chart $U$, and say we have two tangent vectors $x,y$ at $0$.
- We can either go $\epsilon x$ and then go the parallel transport of $\epsilon y$, or the other way around. The difference (the main term) is the torsion.
- Draw the parallelogram of the vectors $\epsilon x$, $\epsilon y$, then parallel transporting $z$ along the two paths gives different results whose difference (the main term) is the curvature.
I am much more fond of the latter two explanations, being more geometric. But it's unclear to me how without a tedious computation (worse, a good explanation) one can show they really depend only on the vectors $x,y$, and not on vector FIELDS $X,Y$ or the underlying chart. At the moment I use the geometric intuition but prove statements via the vector field formulation.
Getting to the essence of my question, it is clear to me that given a connection, it is natural to consider the functor sending loops around a point to linear transformations (via parallel transport), but apriori this functor is complicated, and it turns out we can completely capture it using linear information at each poitn (I do agree that once you are able to locally approximate the functor, if you have a contractible loop you can integrate the curvature inside to obtain the total transport, but even locally I don't see apriori why there should be a linear notion).
I have also heard about fancier terms like Ehresmann connection, I can look into that if it gives a natural definition