Vector bundles answer a basic question that appears whenever geometry carries data from point to point: how can a space have a [vector space](/page/Vector%20Space) attached at every point, while still allowing those vector spaces to twist as the point moves? A tangent plane to a surface, a line of possible phases over each point, or a family of solution spaces over parameters all look locally like a product, but the local products may glue together in a way that has global content.

The first failure to keep in mind is that local coordinates do not give global coordinates for free. A circle has a tangent line at every point, and each small arc makes the tangent lines look like the arc times $\mathbb{R}$. On a more complicated base, the local identifications can rotate, reflect, or mix coordinates as they pass from one chart to another. The theory of vector bundles records exactly this local-product structure and the rules by which the local products are compared.

[example: The Mobius Line Bundle] Start with $[0,1]\times \mathbb{R}$ and impose the [equivalence relation](/page/Equivalence%20Relation) generated by \begin{align*} (0,v)\sim (1,-v). \end{align*} The base is obtained by identifying $0$ with $1$, so it is a circle, and the fiber over the identified endpoint is not two copies of $\mathbb{R}$: the point represented by $(0,v)$ is the same fiber vector as the point represented by $(1,-v)$. On an open arc that does not contain the identified endpoint, the bundle is just the product arc $\times \mathbb{R}$. On an arc crossing the endpoint, use the coordinate $t$ near $0$ on one side and $t-1$ near $1$ on the other side; the fiber coordinate changes by \begin{align*} v\mapsto -v. \end{align*} Thus the transition function on the overlap is the constant element $-1\in \operatorname{GL}_1(\mathbb{R})$. If a continuous nonzero section existed, cutting the circle open at the endpoint would represent it by a [continuous function](/page/Continuous%20Function) \begin{align*} f:[0,1]\to \mathbb{R}\setminus\{0\} \end{align*} satisfying the endpoint compatibility condition \begin{align*} f(1)=-f(0). \end{align*} Since $[0,1]$ is connected and $f$ never equals $0$, the image of $f$ lies entirely in $(0,\infty)$ or entirely in $(-\infty,0)$. Hence $f(0)$ and $f(1)$ have the same sign, while the equation $f(1)=-f(0)$ forces them to have opposite signs. This contradiction shows that the bundle is locally a product but globally twisted. [/example]

The Mobius example is not a defect in the definition. It is the point of the definition: a vector bundle should forget unnecessary coordinates inside each fiber while remembering how those coordinates transform from one local patch to another. The base space controls where the fibers live, the total space contains all vectors in all fibers, and the projection records the point under each vector.

## Definition

The central definition separates the geometric object into three pieces. The total space is where the moving vectors live, the base is the space over which they move, and the projection assigns every vector to its base point.

[definition: Real Vector Bundle] A real vector bundle of rank $r\in \mathbb{N}$ over a [topological space](/page/Topological%20Space) $B$ consists of a topological space $E$, a continuous map $\pi:E\to B$, and for each point $b\in B$ a real vector space structure on the fiber \begin{align*} E_b&=\pi^{-1}(\{b\}) \end{align*} such that every $b\in B$ has an open neighbourhood $U\subset B$ and a homeomorphism \begin{align*} \Phi_U:\pi^{-1}(U)&\to U\times \mathbb{R}^r \end{align*} with the property that the first projection satisfies $\operatorname{pr}_1\circ \Phi_U=\pi$ on $\pi^{-1}(U)$ and each restricted map $E_b\to \{b\}\times \mathbb{R}^r$ is a linear isomorphism. [/definition]

The definition is deliberately local. It does not say that all fibers have been identified with one fixed copy of $\mathbb{R}^r$ once and for all; it says only that this can be done near each base point. The rest of the theory studies how these temporary identifications compare.

## Local Coordinates and Gluing

### Local Trivializations

The bundle definition guarantees product coordinates only one [open set](/page/Open%20Set) at a time. To compare fibers, write sections locally, or build new bundles, one must refer to a chosen product coordinate system over a particular patch, not merely to the existence of such a system. The following definition names this chosen coordinate system.

[definition: Local Trivialization] Let $\pi:E\to B$ be a real vector bundle of rank $r$. A local trivialization over an open set $U\subset B$ is a homeomorphism \begin{align*} \Phi_U:\pi^{-1}(U)&\to U\times \mathbb{R}^r \end{align*} such that $\operatorname{pr}_1\circ \Phi_U=\pi$ and, for every $b\in U$, the map $E_b\to \mathbb{R}^r$ obtained by projecting to the second factor is linear. [/definition]

### Transition Functions

Local trivializations are not unique, so the same vector in a fiber may have two different coordinate columns on an overlap. The obstruction to treating all local products as one global product is hidden in how these coordinate columns change from one patch to another. The following definition packages that change of coordinates as a matrix-valued function.

[definition: Transition Function] Let $\Phi_U$ and $\Phi_V$ be local trivializations of a rank $r$ real vector bundle $\pi:E\to B$. On an overlap $U\cap V$, the transition function $g_{VU}:U\cap V\to \operatorname{GL}_r(\mathbb{R})$ is defined by the identity \begin{align*} \Phi_V\circ \Phi_U^{-1}(b,x)&=(b,g_{VU}(b)x) \end{align*} for $b\in U\cap V$ and $x\in \mathbb{R}^r$. [/definition]

The transition functions are not extra decoration. If local products are to glue into one total space, then changing coordinates from $U$ to $V$ and then from $V$ to $W$ must agree with changing directly from $U$ to $W$. Thus the essential question is which algebraic rules on overlaps are exactly strong enough to reconstruct a vector bundle from local product pieces.

[quotetheorem:6080]

The theorem says that vector bundles can be encoded by linear gluing data. This viewpoint is often the most efficient one: construct a bundle by writing compatible matrices on overlaps, then recover the total space by gluing the local products.

### Tangent Lines as Local Products

The easiest geometric test case is a bundle whose fibers we already understand: tangent lines to a curve. The point of the example is not that anything mysterious happens on the circle, but that the transition functions arise from comparing ordinary coordinate descriptions of the same moving line.