Faserbundel

Faserbundel is a German term that translates to "fiber bundle" in English. In mathematics, particularly in differential geometry, a fiber bundle is a topological space that locally resembles a product space, but globally may not. More formally, it is a topological space P, a topological space B, and a continuous surjective map p: P -> B such that for every point b in B, the preimage p^{-1}(b) has a specific structure. This preimage, called the fiber, is homeomorphic to a fixed topological space F, and the map p is locally trivial. This means that in a neighborhood of any point in B, the total space P looks like the product of the neighborhood and the fiber F.

The space B is called the base space, and the space F is called the fiber. The map p is known as the projection map. Fiber bundles are fundamental objects in geometry and physics. They provide a framework for understanding concepts like tangent bundles, which are crucial for differential geometry, and have applications in areas such as gauge theory in theoretical physics. The local triviality condition allows mathematicians to "straighten out" the bundle in small regions, making it easier to analyze. There are various types of fiber bundles, distinguished by the structure of the fiber and the group acting on it, such as vector bundles, principal bundles, and tangent bundles.