Tangent bundle

The tangent bundle of a smooth manifold is the disjoint union of all tangent spaces at every point, equipped with a natural topology and smooth structure that makes it a vector bundle. It is the fundamental object on which vector fields live: a vector field is precisely a smooth section of the tangent bundle. The tangent bundle transforms a manifold from a static geometric object into a dynamic one, encoding every possible direction in which one can move at every point. Without the tangent bundle, differential calculus on manifolds is impossible; with it, the manifold becomes a stage for dynamics.