Section of a fiber bundle - Imported Knowlpedia Content

Definition

Let $\pi:E\to M$ be a smooth fiber bundle. A smooth global section of $\pi$ is a smooth map $s:M\to E$ satisfying

\pi\circ s=\operatorname{id}_M.

Equivalently, $s(x)\in E_x=\pi^{-1}(x)$ for every $x$ , and this choice varies smoothly with $x$ . A local section over an open set $U\subseteq M$ is a smooth map $s:U\to E$ with $\pi\circ s=\operatorname{id}_U$ . A fiber bundle always has local sections near each base point, but it need not have a global section.

Local description and constructions

Under a local trivialization $\Phi:\pi^{-1}(U)\to U\times F$ , every local section has the form

\Phi(s(x))=(x,f(x))

for a smooth map $f:U\to F$ , and every such $f$ defines a local section. Local representatives on overlaps satisfy the transition-function rule for the bundle.

Sections pull back along smooth maps: if $f:N\to M$ , then $x\mapsto(x,s(f(x)))$ is a section of the pullback bundle $f^*E\to N$ .

Examples and obstructions

For the product bundle $M\times F\to M$ , every smooth map $f:M\to F$ gives the section $x\mapsto(x,f(x))$ .

For a vector bundle, the zero vector in each fiber defines a canonical global section. Sections of the tangent bundle are vector fields.

A principal bundle admits a global section exactly when it is trivial, as expressed by the global-section triviality theorem and its converse. The Hopf principal circle bundle is a standard bundle with no global section.

Conventions and scope

“Section” and “cross-section” are synonymous here. The adjective “global” distinguishes a section on all of $M$ from one defined only on an open subset.

References

J. M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. DOI record. Relevant: chapter 10, smooth bundles and sections.
D. Husemoller, Fibre Bundles, 3rd ed., Springer, 1994. DOI record. Relevant: chapters 1–2, bundle maps, local triviality, and sections.