TITLE: Tangent spaces at different points and the concept of connection
QUESTION [8 upvotes]: If $M$ is a smooth manifold and $TM$ is the tangent bundle clearly $T_pM\cong T_qM$ (as vector spaces) for every $p,q\in M$. Nobody ensures that the previous vector spaces isomorphism is natural (or canonical). In $\mathbb R^n$ we have that $T_p\mathbb R^n$ and $T_q\mathbb R^n$ are naturally isomorphic to $\mathbb R^n$ so we can differentiate a vector field along a direction, in the usual way so taking the directional derivatives of each component.
If the isomorphism between tangent spaces in different point isn't natural, why can't we differentiate a vector field in the usual way? The problem is comparing vectors belonging in different (isomorphic) vector spaces; but we can send the two vectors, with an isomorphism, in a common vector space and then subtract them. Where is the importance of a natural isomorphism?

REPLY [2 votes]: Why don't you let $M = S^2 \subset \mathbb{R}^3$ ?   The equation for the Tangent plane is
$$T_{(a,b,c)}S^2 = \{ (x-a)a+(y-b)b+(c-z)c = 0 \}$$
This is a collection of planes parameterized by points $(a,b,c): a^2 + b^2 + c^2 = 1$.  
We can define an initial vector $\vec{x}=(x,y,z) \in T_{(a,b,c)}S^2$.  Then we can ask $$\vec{x}(t + \delta t) = A(t) \vec{x}(t) \in T_{(a,b,c) + \vec{v}\delta t}S^2$$
For any time-paramterized matrix $A$, we should get a connection on the sphere that necessarily Levi-Civita connection, since this flow is not the geodesic flow.