Section (fiber bundle)

In the mathematical field of topology, a section (or cross section)[1] of a fiber bundle ${\displaystyle E}$ is a continuous right inverse of the projection function ${\displaystyle \pi }$. In other words, if ${\displaystyle E}$ is a fiber bundle over a base space, ${\displaystyle B}$:

${\displaystyle \pi \colon E\to B}$

A section ${\displaystyle s}$ of a bundle ${\displaystyle p\colon E\to B}$. A section ${\displaystyle s}$ allows the base space ${\displaystyle B}$ to be identified with a subspace ${\displaystyle s(B)}$ of ${\displaystyle E}$.

A vector field on ${\displaystyle \mathbb {R} ^{2}}$. A section of a tangent vector bundle is a vector field.

A vector bundle ${\displaystyle E}$ over a base ${\displaystyle M}$ with section ${\displaystyle s}$.

then a section of that fiber bundle is a continuous map,

${\displaystyle \sigma \colon B\to E}$

such that

${\displaystyle \pi (\sigma (x))=x}$ for all ${\displaystyle x\in B}$.

A section is an abstract characterization of what it means to be a graph. The graph of a function ${\displaystyle g\colon B\to Y}$ can be identified with a function taking its values in the Cartesian product ${\displaystyle E=B\times Y}$, of ${\displaystyle B}$ and ${\displaystyle Y}$:

${\displaystyle \sigma \colon B\to E,\quad \sigma (x)=(x,g(x))\in E.}$

Let ${\displaystyle \pi \colon E\to B}$ be the projection onto the first factor: ${\displaystyle \pi (x,y)=x}$. Then a graph is any function ${\displaystyle \sigma }$ for which ${\displaystyle \pi (\sigma (x))=x}$.

The language of fibre bundles allows this notion of a section to be generalized to the case when ${\displaystyle E}$ is not necessarily a Cartesian product. If ${\displaystyle \pi \colon E\to B}$ is a fibre bundle, then a section is a choice of point ${\displaystyle \sigma (x)}$ in each of the fibres. The condition ${\displaystyle \pi (\sigma (x))=x}$ simply means that the section at a point ${\displaystyle x}$ must lie over ${\displaystyle x}$. (See image.)

For example, when ${\displaystyle E}$ is a vector bundle a section of ${\displaystyle E}$ is an element of the vector space ${\displaystyle E_{x}}$ lying over each point ${\displaystyle x\in B}$. In particular, a vector field on a smooth manifold ${\displaystyle M}$ is a choice of tangent vector at each point of ${\displaystyle M}$: this is a section of the tangent bundle of ${\displaystyle M}$. Likewise, a 1-form on ${\displaystyle M}$ is a section of the cotangent bundle.

Sections, particularly of principal bundles and vector bundles, are also very important tools in differential geometry. In this setting, the base space ${\displaystyle B}$ is a smooth manifold ${\displaystyle M}$, and ${\displaystyle E}$ is assumed to be a smooth fiber bundle over ${\displaystyle M}$ (i.e., ${\displaystyle E}$ is a smooth manifold and ${\displaystyle \pi \colon E\to M}$ is a smooth map). In this case, one considers the space of smooth sections of ${\displaystyle E}$ over an open set ${\displaystyle U}$, denoted ${\displaystyle C^{\infty }(U,E)}$. It is also useful in geometric analysis to consider spaces of sections with intermediate regularity (e.g., ${\displaystyle C^{k}}$ sections, or sections with regularity in the sense of Hölder conditions or Sobolev spaces).