Fiber and Vector Bundles

A fiber bundle is a space that looks locally like a product $U\times F$ but may be globally twisted. Bundles are how geometry keeps track of data attached to each point of a manifold — a tangent space, a vector, an internal charge — and how that data can be glued nontrivially. This is the setting for every field in physics. This page builds the general and vector-bundle cases; the group-structured principal case is principal-bundles.md. It uses the manifold language of tensors-and-forms.md.

Definition

A fiber bundle is a smooth surjection $\pi :E\to M$ (the total space over the base $M$, with projection $\pi$) together with a typical fiber $F$, such that $E$ is locally trivial: every point of $M$ has a neighbourhood $U$ and a diffeomorphism $${\varphi }_U:{\pi }^{-1}(U)\stackrel{\sim }{\to }U\times F$$ commuting with projection to $U$. The fiber over $p$ is $E_p={\pi }^{-1}(p)\cong F$. When $E=M\times F$ globally the bundle is trivial; the interest is in bundles that are only locally trivial.

Transition functions and the cocycle condition

On an overlap $U_{\alpha }\cap U_{\beta }$, two trivializations differ by a fiber- preserving map $${\varphi }_{\alpha }\circ {\varphi }_{\beta }^{-1}(p,f)=(p,g_{\alpha \beta }(p)\cdot f),$$ the transition functions $g_{\alpha \beta }:U_{\alpha }\cap U_{\beta }\to G$, valued in a group $G$ of fiber symmetries (the structure group). They satisfy the cocycle condition $$g_{\alpha \beta }{g}_{\beta \gamma }{g}_{\gamma \alpha }=e\text{on triple overlaps},g_{\alpha \alpha }=e.$$ Remarkably, the cocycle data $\{g_{\alpha \beta }\}$ reconstructs the bundle: a bundle is nothing but a cover plus transition functions modulo relabelling. The global twisting lives entirely in how these functions fail to be simultaneously trivializable — the seed of the classification in principal-bundles.md.

Sections

A section is a smooth right inverse $s:M\to E$ with $\pi \circ s={\mathrm{id}}_M$ — a smooth choice of one point in each fiber. Sections are the fields: a vector field is a section of the tangent bundle, a wavefunction is a section of an associated bundle. A key fact distinguishing trivial from twisted bundles:

• A trivial bundle always has global sections (e.g. constant ones).
• Many bundles have no nonvanishing global section — the hairy ball theorem says $T{S}^2$ has no nowhere-zero section; the Möbius band's tautological line bundle has no nonvanishing section. Obstructions to sections are measured by characteristic classes.

Vector bundles

A vector bundle is a fiber bundle whose fiber is a vector space ${\mathbb{F}}^k$ and whose transition functions are linear, $g_{\alpha \beta }\in GL(k,\mathbb{F})$; $k$ is the rank. Then each fiber $E_p$ is a $k$-dimensional vector space and sections can be added and scaled pointwise. Canonical examples:

• the tangent bundle $TM$ and cotangent bundle $T^{\ast }M$ (rank $=\dim M$);
• the tensor bundles $T^{(r,s)}M$ and form bundles ${\Lambda }^p{T}^{\ast }M$;
• line bundles ($k=1$): the Möbius band over $S^1$, and the tautological bundle over ${\mathbb{CP}}^n$;
• in physics, the bundle whose sections are a charged matter field (an associated bundle, associated-bundles.md).

Operations on vector spaces (dual, $\oplus$, $\otimes$, $\wedge$) extend fiberwise to operations on vector bundles.

The frame bundle

To a rank-$k$ vector bundle one associates its frame bundle $FE$: the fiber over $p$ is the set of ordered bases (frames) of $E_p$, on which $GL(k,\mathbb{F})$ acts freely and transitively by change of basis. This is a principal $GL(k,\mathbb{F})$-bundle — the bridge from vector bundles to the principal bundles of principal-bundles.md. Conversely every vector bundle is associated to its frame bundle via the defining representation. Reducing the structure group of $FM$ (the frame bundle of $TM$) encodes geometric structure: to $O(n)$ a Riemannian metric, to $SO(n)$ an orientation, to $G{L}^{+}$ orientability — the theme of associated-bundles.md and riemannian-bridge.md.

References

• Steenrod, The Topology of Fibre Bundles — the classic.
• Lee, Introduction to Smooth Manifolds, Ch. 10.
• Nakahara, Geometry, Topology and Physics, Ch. 9.