Tangent spaces at different points on manifolds

Question

Why are tangent spaces on a general manifold associated to single points on the manifold? I've heard that it has to do with not being able to subtract/ add one point from/to another on a manifold (ignoring the concept of a connection at the moment), but I'm not sure I fully understand this - is it simply because, even if two points lie in the same coordinate patch the coordinate map will not be Cartesian (i.e. the identity map) in general, and so subtracting/adding their coordinate values will not correspond to subtracting/adding one point from/to another on the manifold?

Answer 1

The tangent space can be thought of as a linear approximation to the manifold at a given point, this is the same as locally approximating a curve by its tangent line at a point.

We'll continue with the analogy of the tangent line to a curve. Imagine adding two vectors that lie along the same tangent line together, we get another vector which is also approximating the function in some sense at that point on the curve. However if I consider two tangent lines at different points in a curve then it won't be useful to try and add a vector from each together, I could add them componentwise if I wanted to but what would that give me? Potentially a new vector pointing off in a direction that bears no relation to the function at either of the two points from which we took tangent vectors. Hence if we did start adding vectors from the tangent spaces of a curve/ manifold then we would be ending up with objects that no longer represent anything about the original object we are studying, hence there is nothing very useful to gain by doing this.

I hope this helps clarify things.