Section of a vector bundle - Imported Knowlpedia Content

Definition

Let $\pi:E\to M$ be a smooth vector bundle. A section of $E$ is a smooth section $s:M\to E$ ; equivalently,

\pi\circ s=\operatorname{id}_M, \qquad s(x)\in E_x

for every $x\in M$ . A local section over an open subset $U\subseteq M$ is defined similarly as a smooth map $s:U\to E$ with values $s(x)\in E_x$ . Because each fiber is a vector space, sections can be added and multiplied by smooth scalar functions pointwise. The resulting space of global smooth sections is denoted $\Gamma^\infty(M,E)$ , or simply $\Gamma(E)$ .

Local description

In a local trivialization $E|_U\cong U\times\mathbb F^r$ , a section is uniquely represented by a smooth map $U\to\mathbb F^r$ , hence by $r$ smooth component functions. On an overlap, these component functions transform by the bundle transition function. This compatibility is what distinguishes a section from an arbitrary collection of fiber vectors.

A section is determined by its restrictions to an open cover. Conversely, local sections that agree on overlaps glue to a unique global section. These facts make sections naturally sheaf-like, although this knowl concerns sections of a bundle rather than the abstract notion of a sheaf section.

Algebraic structure

If $s,t\in\Gamma^\infty(M,E)$ and $f\in C^\infty(M)$ , then

(s+t)(x)=s(x)+t(x), \qquad (fs)(x)=f(x)s(x).

Thus $\Gamma^\infty(M,E)$ is a module over the algebra of smooth functions. Bundle morphisms over $M$ induce module homomorphisms on sections by pointwise composition.

Examples

Sections of the tangent bundle are smooth vector fields, while sections of the cotangent bundle are smooth one-forms. Every vector bundle has the zero section, so—unlike a general fiber bundle—the existence of a global section does not imply triviality. A rank- $r$ vector bundle is trivial precisely when it admits $r$ global sections that form a basis in every fiber.

References

J. M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. DOI record. Relevant: Chapter 10, vector bundles and local frames.
L. W. Tu, Differential Geometry: Connections, Curvature, and Characteristic Classes, Springer, 2017. DOI record. Relevant: Chapter 1, vector bundles and sections.