Vector bundle

A vector bundle is a mathematical structure that assigns a vector space (a collection of vectors) to each point on a shape or space, called a manifold. Imagine a surface where at every point, you attach a small space of arrows (vectors) that can vary smoothly as you move from point to point. These fibers (the attached vector spaces) change smoothly across the manifold, allowing mathematicians to study complex geometric and topological properties. Vector bundles are fundamental in fields like geometry and physics, providing a framework to understand phenomena such as fields and forces over space.