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I am now understanding a bit -- just a bit: still struggling - about the tangent bundle.

But I have no idea WHY this is important.

As I understand, at every point on a manifold (or, more appropriately: at the coordinates placed on a manifold by a mapping), we study the union of the POINT and the TANGENT SPACE for all points. I sort of get that.

(I do not know why we make a thing of it, though.)

But then they go into this idea of TWO sub-manifolds: 1) fiber and 2) base.

And then things get hazy: it looks like the base is the space of all tangent vectors that interesect with tangent vectors from other points. And the fiber is the non-intersecting components of the tangent vectors.

I think. Now I am not so much asking for clarification on the tangent bundle (although I would not mind any clarifications).

*** I am asking: WHY ARE WE DOING THIS? *** What is to be gained? I SORT of think it might have to do with position vectors and momentum vectors (in physics, at least).

But NO book I see is even taking the time to say WHY we are splitting this up... if we are... and WHY we need to do it for all the points in a union (does that justify the "CONNECTION" in the covariant derivaitve?)

Please try not to be too mathematical if you are so kind as to respond. I am flinging words around here and not entirely so sure I know what I am talking about.