Vector Bundles and Transition Functions

A rank-$r$ vector bundle looks locally like a product with $\mathbb R^r$, yet its fibers can be glued globally with a twist. Transition functions encode that gluing. Their cocycle condition is the compatibility needed for local vectors and local sections to define global objects, while changes of trivialization identify different cocycles describing the same bundle. The cylinder and Möbius line bundle make the local-versus-global distinction visible without specialized machinery.

Required background. Tensor Fields and Natural Operations supplies tensor bundles and natural bundle maps as motivating examples; Tangent and Cotangent Spaces supplies the tangent-space model, and Smooth Maps, Pushforwards, and Pullbacks supplies smooth maps. Exterior Algebra and Differential Forms supplies cotangent and exterior-power bundles.

Helpful background. Calculus and Linear Algebra supplies vector spaces, bases, and invertible matrices; The Riemann Curvature Tensor motivates the later need for connections on bundles.

Local products and transition cocycles

A smooth real rank-$r$ vector bundle consists of a manifold $E$, a projection $\pi:E\to M$, and a vector-space structure on each fiber $E_x=\pi^{-1}(x)$ such that every $x$ has a neighborhood $U_i$ and a diffeomorphism

$$\psi_i:\pi^{-1}(U_i)\longrightarrow U_i\times\mathbb R^r$$

that is linear on each fiber. On an overlap $U_i\cap U_j$, two trivializations differ by

$$\psi_i\psi_j^{-1}(x,v)=(x,g_{ij}(x)v), \qquad g_{ij}:U_i\cap U_j\to GL(r,\mathbb R).$$

Consistency on double and triple overlaps requires

$$g_{ii}=I, \qquad g_{ij}=g_{ji}^{-1}, \qquad g_{ij}g_{jk}g_{ki}=I.$$

Conversely, smooth functions obeying these cocycle relations glue the disjoint local products into a vector bundle. This construction and its equivalence under changes of trivialization are given in Hatcher 2017, §1.1, pp. 6–17.

Replacing $\psi_i$ by $h_i\psi_i$, with $h_i:U_i\to GL(r,\mathbb R)$, changes the cocycle to

$$g'_{ij}=h_i g_{ij}h_j^{-1}.$$

This is the same bundle in a different local frame. Individual transition matrices are not invariants; their gluing class is.

Cylinder and Möbius line bundles

Build a real line bundle over $S^1$ by taking $[0,2\pi]\times\mathbb R$ and identifying the ends. The identification

$$(0,v)\sim(2\pi,v)$$

gives the cylinder $S^1\times\mathbb R$. The identification

$$(0,v)\sim(2\pi,-v)$$

gives the Möbius line bundle. Both are locally $U\times\mathbb R$, and locally a change of sign can remove the difference. The endpoint identification reverses the sign of the fiber coordinate in the Möbius case. This is a gluing statement: a rule for parallel transport is additional connection data, not part of the definition of a bundle.

The cylinder has a nowhere-zero global section $s(\theta)=(\theta,1)$. If the Möbius bundle had one, represent it by a continuous function $f$ on the interval with $f(2\pi)=-f(0)$. A nowhere-zero $f$ would have constant sign, contradicting this endpoint condition by the intermediate value theorem. Thus the two bundles are not isomorphic. This is a cocycle check rather than a picture-based argument.

Sections, frames, and bundle maps

A section is a smooth map $s:M\to E$ satisfying $\pi\circ s=\operatorname{id}_M$. In a trivialization it is a vector-valued function $s_i:U_i\to\mathbb R^r$ obeying

$$s_i=g_{ij}s_j$$

on overlaps. Every vector bundle has the zero section, so the mere existence of a global section says nothing about triviality. A rank-$r$ bundle is trivial precisely when it admits a global frame of $r$ pointwise independent sections. For a line bundle, one nowhere-zero section is already a frame and hence proves triviality.

A bundle map $F:E\to E'$ over $f:M\to M'$ maps each $E_x$ linearly into $E'_{f(x)}$. Direct sums, duals, tensor products, and exterior powers are obtained by applying the corresponding linear-algebra construction to fibers and transition matrices. If $E$ has transitions $g_{ij}$, its dual has transitions $(g_{ij}^{-1})^{\mathsf T}$.

Tangent and tensor bundles

The tangent bundle $TM$ is glued by Jacobian matrices of coordinate changes. A vector field is a section of $TM$; a one-form is a section of $T^*M$; a $(k,l)$ tensor is a section of

$$(TM)^{\otimes k}\otimes(T^*M)^{\otimes l}.$$

This explains why tensor component transformation laws are bundle transition laws. A manifold can possess a tangent bundle without possessing a global coordinate frame. The two-sphere, for example, has no nowhere-zero tangent vector field, so $TS^2$ cannot be trivial, even though it is trivial over each coordinate patch.

Local triviality also does not supply a connection, metric, orientation, or spin structure. Those are additional choices or reductions, and some have global obstructions. Hatcher describes classification and clutching constructions for bundles over spheres in Hatcher 2017, §1.2, pp. 18–37.

Exercise: testing a proposed section

For the Möbius line bundle represented by $(0,v)\sim(2\pi,-v)$, decide whether $f(\theta)=\cos(\theta/2)$ and $g(\theta)=(\cos(\theta/2),\sin(\theta/2))$ define, respectively, a section of the line bundle and a two-component section of a rank-two bundle with transition $-I$. In the second case, construct a full frame and decide whether the bundle is trivial.

Solution

The line-bundle condition is $f(2\pi)=-f(0)$. Since $f(0)=1$ and $f(2\pi)=-1$, the endpoint values match the gluing. In fact $f(\theta+2\pi)=-f(\theta)$, so all derivatives match as well and the section is smooth. It vanishes at $\theta=\pi$ and therefore does not trivialize the bundle.

For the rank-two bundle, $g(0)=(1,0)$ while $g(2\pi)=(-1,0)=-g(0)$, and the same antiperiodicity makes this a smooth nowhere-zero section. One section alone does not establish rank-two triviality, but here a second one is available:

$$\widetilde g(\theta)=\bigl(-\sin(\theta/2),\cos(\theta/2)\bigr).$$

It obeys the same gluing, and the matrix with columns $g,\widetilde g$ has determinant one everywhere. They therefore form a global frame. Two copies of the Möbius line bundle have a trivial direct sum over $S^1$, although each copy is nontrivial.

References

• Hatcher, Allen. Vector Bundles and K-Theory. Version 2.2, 2017. §§1.1–1.2, pp. 6–37.