In most of the textbooks that I have been reading and many of the answers here, it has been an important point that we cannot compare vectors in the tangent spaces of two different points. That is, given two points on a manifold $p, q\in M$, there is no natural notion of parallel transport that can allow us to take a vector from $T_pM$ and add/subtract it to one in $T_qM$.
At the same time, we can define a vector field and then define when this function $M\to TM$ is smooth. To me, this implies that we are doing calculus with the vectors in the vector field, which would require some way to compare vectors based at different points, contradicting the first notion. In fact, in order to define a smooth vector field as a smooth section of the bundle, we need to define a structure on $TM$ to make it a smooth $2n$ dimensional manifold. In doing so, we obtain diffeomorphisms between any neighborhood in $TM$ and an open subset in $\mathbb{R}^n\times\mathbb{R}^n$. Then we can start talking about addition, subtraction, limits, and derivatives in the tangent bundle. Therefore, we have created an isomorphism between the tangent spaces and created a way to compare vectors at any two points in the neighborhood, correct?
I want to confirm that these ideas are compatible. We can exploit the local triviality of the bundle to develop a local method of parallel transport, right? And this allows us to create a local definition for a smooth vector field, circumnavigating the problem I mentioned in the first paragraph?
Best Answer
Every local chart provides a way to identify tangent vectors at one point with tangent vectors at another point. It thereby provides notions of addition, subtraction, differentiability, etc. But different local chart, related to the first one by a smooth change of coordinates, will provide a different identification, different addition, different subtraction, but the same notion of differentiability. Thus, addition and subtraction are not well-defined just on the basis of the manifold structure; they vary according to which of the many available charts you use. But all charts (from the atlas defining your manifold) give the same notion of differentiability of vector fields, so that notion is well-defined just by the manifold, with no further arbitrary choices.