Fiber Bundle
Definition. Data $(E, M, F, \pi)$: total space $E$, base $M$, fiber $F$, and a smooth surjection $\pi: E \to M$ that is locally trivial: every $x \in M$ has a neighborhood $U$ with a diffeomorphism $\Phi: \pi^{-1}(U) \to U \times F$ commuting with projection. Written $F \hookrightarrow E \xrightarrow{\pi} M$.
Intuition. A family of copies of $F$, one over each point of $M$, varying smoothly — locally an honest product $U \times F$, but possibly globally twisted (cylinder vs. Möbius band). The twist lives in the Transition Functions comparing overlapping trivializations.
Taxonomy used in this vault.
| Fiber | Structure group action | Name |
|---|---|---|
| Vector space $V$ | linear | Vector Bundle |
| The group $G$ itself | right translation | Principal Bundle |
| Any $F$ with $G$-action, built from $P$ | via a representation | Associated Bundle |
Physics anchor. Base = spacetime; fiber = internal degrees of freedom (phase, isospin, color); fields = sections. "Internal symmetry" is geometry perpendicular to spacetime.
Related: Local Trivialization · Transition Functions · Section of a Bundle · Tangent Bundle · Hopf Fibration · Module 05 - Fiber Bundles - First Definitions
Sources: Sternberg Ch. VII; Nakahara Ch. 9 (optional).