I am having trouble visualizing what a line subbundle of the Tangent Bundle of the sphere looks like. As I understand it, all line bundles are trivial on the sphere.
I was under the mistaken impression that such a trivial line bundle would imply the existence of a nowhere vanishing vector field. So clearly trivial bundles should not be thought of as a scalar multiplied by a tangent vector at each point.
I can visualize that for a great circle on the sphere that we can draw a line tangent to each point on that great circle. However, it seems that the mapping from the line to specific tangent vectors on that line reverses at one point on the great circle.
I am hoping someone can provide an intuitive way to thing about line bundles more clearly.