vector bundles

A vector bundle is a mathematical structure that smoothly assigns a vector space to every point of a geometric object, like a surface or shape. Imagine each point on a shape has a small, attached space of directions and magnitudes (vectors), which varies smoothly as you move across the shape. These bundles provide a framework to study how such spaces evolve, capturing complex geometric and topological properties. They are fundamental in areas like geometry, physics, and engineering, where understanding how data or directions change across a space is essential.