Cotangent bundle

Solution 1:

One fruitful way to think about it, if you have physics background, is as phase space. Your manifold is the configuration space for some system of particles, and the cotangent bundle is then the phases, so the cotangent directions are velocities. This is helpful also with the symplectic structure on $T^*M$.

Solution 2:

You might be interested in this MathOverflow post: https://mathoverflow.net/questions/17325/why-is-cotangent-more-canonical-than-tangent

(Sorry, I'd leave this as a comment but I just joined this site and don't have enough reputation.)

Solution 3:

I'm not completely sure what you mean by this: "It seems odd to think of it as the dual of the tangent bundle (I am finding it odd to reconcile the notions of "maps to the ground field" with this object)," but maybe the following will help you see why it is natural to consider the dual space of the tangent bundle.

Given a function f on our manifold, we want to associate something like the gradient of f. Well, in calculus, what characterized the gradient of a function? Its the vector field such that when we take its dot product with a vector v at some point p, we get the directional derivative, at p, of f along v. In a general manifold we don't have a dot product (which is a metric) but we can form a covector field (something which gives an element of the cotangent bundle at any point) such that, when applied to a vector v, we get the directional derivative of f along v. This covector field is denoted df and is called the exterior derivative of f.