The first obstruction in differential geometry is that a [vector space](/page/Vector%20Space) attached to a point should vary with the point, but the spaces over different points usually have no canonical way to be identified. A tangent vector at one point of a manifold is not naturally a tangent vector at another point. A field of tangent vectors therefore needs a structure that says what it means for these pointwise vector spaces to vary smoothly, without pretending that they are all the same vector space.

The naive model is a product $M \times \mathbb{R}^r$, where every point $p \in M$ carries the copy $\{p\} \times \mathbb{R}^r$. This model is too rigid: the tangent bundle of a sphere, the normal bundle of an embedded submanifold, and the line bundle over a circle obtained by gluing with a sign change all behave locally like products but need not be products globally. Smooth vector bundles isolate exactly this local-product behavior.

[example: The Tangent Bundle of the Circle] Let $S^1 \subset \mathbb{R}^2$ be the unit circle, and write $p=(\cos t,\sin t)$. The curve $c(t)=(\cos t,\sin t)$ has velocity \begin{align*} c'(t)=(-\sin t,\cos t). \end{align*} This vector is tangent to $S^1$ at $p$, and it is perpendicular to the radial vector $p$ because \begin{align*} (\cos t,\sin t)\cdot(-\sin t,\cos t)=-\cos t\sin t+\sin t\cos t=0. \end{align*} Thus the tangent line at $p$ is \begin{align*} T_pS^1=\{a(-\sin t,\cos t):a\in\mathbb{R}\}. \end{align*} The total space is \begin{align*} TS^1=\bigcup_{p\in S^1}\{p\}\times T_pS^1\subset S^1\times\mathbb{R}^2. \end{align*} Define \begin{align*} \Phi:S^1\times\mathbb{R}\to TS^1 \end{align*} by \begin{align*} \Phi((\cos t,\sin t),a)=((\cos t,\sin t),a(-\sin t,\cos t)). \end{align*} If $t$ is replaced by $t+2\pi k$, then \begin{align*} \cos(t+2\pi k)=\cos t \end{align*} and \begin{align*} \sin(t+2\pi k)=\sin t, \end{align*} so the formula gives the same point of $TS^1$. Hence $\Phi$ is well-defined. For each $p=(\cos t,\sin t)$, every vector in $T_pS^1$ has a unique form $a(-\sin t,\cos t)$, since $(-\sin t,\cos t)\neq(0,0)$. Therefore the inverse map is \begin{align*} \Phi^{-1}((p,a(-\sin t,\cos t)))=(p,a). \end{align*} So $TS^1$ is globally described as the product $S^1\times\mathbb{R}$. This example is special because the tangent bundle of the circle is globally a product; in general, a tangent bundle is only required to have such product descriptions locally. [/example]

The example hides the essential rule: a [vector bundle](/page/Vector%20Bundle) is not just a family of vector spaces. It must include a total space, a projection to the base, and smooth local coordinates in which the projection looks like the projection from a product. The coordinate changes must preserve the vector-space structure in each fiber.

## Definition

The definition must remember both the smooth geometry of the total space and the linear algebra inside each fiber. The rank records the dimension of the model fiber, while the projection records which vector space belongs to which base point.

[definition: Smooth Vector Bundle] Let $M$ be a [smooth manifold](/page/Smooth%20Manifold). A smooth vector bundle of rank $r$ over $M$ is a smooth manifold $E$ of dimension $\dim M + r$, a smooth surjective submersion \begin{align*} \pi: E &\to M, \end{align*} a real vector space structure on each fiber $E_p = \pi^{-1}(\{p\})$, and for every $p \in M$ an open neighbourhood $U \subset M$ together with a diffeomorphism \begin{align*} \Phi: \pi^{-1}(U) &\to U \times \mathbb{R}^r \end{align*} such that $\operatorname{pr}_1 \circ \Phi = \pi$ on $\pi^{-1}(U)$ and, for every $q \in U$, the restricted map \begin{align*} \Phi_q: E_q &\to \{q\} \times \mathbb{R}^r \end{align*} is a linear isomorphism after identifying $\{q\} \times \mathbb{R}^r$ with $\mathbb{R}^r$. [/definition]

The map $\pi: E \to M$ is the bundle projection, $M$ is the base, $E$ is the total space, and $E_p$ is the fiber over $p$. The local diffeomorphisms $\Phi$ are the data that convert geometric variation into ordinary smooth functions on open sets of $M$.

## Product Bundles and Local Models

To separate local product behavior from global twisting, we first need the reference case where no twisting is present. The product bundle is the benchmark against which all twisting is measured: it fixes the model for local coordinates and gives the test case that later transition functions will modify.

[definition: Product Smooth Vector Bundle] Let $M$ be a smooth manifold and let $V$ be a real vector space of dimension $r$. The product smooth vector bundle with fiber $V$ is the projection \begin{align*} \operatorname{pr}_1: M \times V &\to M \end{align*} with rule \begin{align*} (p,v) &\mapsto p, \end{align*} where each fiber $\{p\} \times V$ carries the vector space structure transported from $V$. [/definition]

The local definition says that every smooth vector bundle is a product after restricting to small enough open sets. The global question is how those local products are glued.

## Local Trivialisations and Transition Functions

### Comparing Local Products

A single local product chart does not describe a bundle over all of $M$. The information lives in how two such descriptions compare on an overlap.

[definition: Local Trivialisation] Let $\pi: E \to M$ be a smooth vector bundle of rank $r$. A local trivialisation over an [open set](/page/Open%20Set) $U \subset M$ is a diffeomorphism \begin{align*} \Phi: \pi^{-1}(U) &\to U \times \mathbb{R}^r \end{align*} such that $\operatorname{pr}_1 \circ \Phi = \pi$ and each fiber map $\Phi_q: E_q \to \mathbb{R}^r$ is linear. [/definition]

Suppose $\Phi_i$ is a trivialisation over $U_i$ and $\Phi_j$ is a trivialisation over $U_j$. On $U_i \cap U_j$, both charts assign coordinates in $\mathbb{R}^r$ to the same vector in the same fiber. Their comparison is therefore linear in the fiber coordinate and smooth in the base point, and the next definition records this comparison as a matrix-valued function.

[definition: Transition Function of a Smooth Vector Bundle] Let $\Phi_i: \pi^{-1}(U_i) \to U_i \times \mathbb{R}^r$ and $\Phi_j: \pi^{-1}(U_j) \to U_j \times \mathbb{R}^r$ be local trivialisations of a smooth vector bundle $\pi: E \to M$. The transition function from $\Phi_i$ to $\Phi_j$ is the smooth map \begin{align*} g_{ji}: U_i \cap U_j &\to \operatorname{GL}_r(\mathbb{R}) \end{align*} defined by \begin{align*} \Phi_j \circ \Phi_i^{-1}(p,v) &= (p, g_{ji}(p)v). \end{align*} [/definition]

### Cocycle Compatibility

The transition functions carry all of the non-product behavior. They are smooth matrix-valued functions, so the bundle is smooth precisely because the gluing matrices change smoothly with the base point. A collection of such matrices can glue local products only when all coordinate changes agree on triple overlaps.