Tangent Bundles
Tangent bundles are mathematical constructs that are attached to smooth manifolds, providing a way to study the behavior of tangent vectors on the manifold at each point. The tangent bundle at a point consists of all possible tangent vectors at that point, forming a vector space. By considering the tangent bundles at every point collectively, one can gain insight into the geometric properties and differential structures of the underlying manifold. Tangent bundles play a crucial role in differential geometry, calculus of variations, and other fields of mathematics and physics.