Section of a vector bundle

Builds on Fiber bundle Smooth manifold Smooth function Vector space

Definition

Let $M$ be a smooth manifold and let $k \ge 0$ . A smooth vector bundle of rank $k$ over $M$ is a smooth fiber bundle $\pi \colon E \to M$ with fiber $\R^k$ and the following structure. Each fiber $E_p = \pi^{-1}(p)$ is a real vector space, and the local trivializations $\pi^{-1}(U) \to U \times \R^k$ can be chosen linear on each fiber. A smooth section of $E$ is a smooth map $s \colon M \to E$ such that $\pi \circ s = \id_M,$ that is, $s(p) \in E_p$ for every $p \in M$ . The set of smooth sections is written $\Gamma(E)$ .

Remarks

The zero section $p \mapsto 0_p$ is a section of every vector bundle. For the trivial bundle $E = M \times \R^k$ , the sections are the maps $p \mapsto (p, u(p))$ with $u \colon M \to \R^k$ smooth. A smooth vector field on $M$ is a section of the tangent bundle $TM$ .

Non-example: on $E = S^1 \times \R$ , the map $s(p) = (-p, 0)$ is smooth, but $\pi(s(p)) = -p \neq p$ , so it is not a section.

Sections are added and multiplied by smooth functions pointwise, so $\Gamma(E)$ is a module over $C^\infty(M)$ . The same definition, without linearity, gives sections of any fiber bundle; continuous sections are defined with continuous maps.

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Tag	`XDYMCB41NG`
URI	`m2://XDYMCB41NG`
Kind	definition
Writing	W0 stub
MSC	55R25, 58A05
Defines	Section of a vector bundle
Uses	Fiber bundle, Smooth manifold, Smooth function, Vector space
Changed	2 November 2026

Section of a vector bundle

Definition

Remarks

Reading path

Further reading

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