In the mathematical field of topology, a section (or cross section) of a fiber bundle E {\displaystyle E} is a continuous right inverse of the projection function π {\displaystyle \pi } . In other words, if E {\displaystyle E} is a fiber bundle over a base space, B {\displaystyle B} :
π : E → B {\displaystyle \pi \colon E\to B}
then a section of that fiber bundle is a continuous map,
σ : B → E {\displaystyle \sigma \colon B\to E}
such that
π ( σ ( x ) ) = x {\displaystyle \pi (\sigma (x))=x} for all x ∈ B {\displaystyle x\in B} . A section is an abstract characterization of what it means to be a graph. The graph of a function g : B → Y {\displaystyle g\colon B\to Y} can be identified with a function taking its values in the Cartesian product E = B × Y {\displaystyle E=B\times Y} , of B {\displaystyle B} and Y {\displaystyle Y} :
σ : B → E , σ ( x ) = ( x , g ( x ) ) ∈ E . {\displaystyle \sigma \colon B\to E,\quad \sigma (x)=(x,g(x))\in E.}
Let π : E → B {\displaystyle \pi \colon E\to B} be the projection onto the first factor: π ( x , y ) = x {\displaystyle \pi (x,y)=x} . Then a graph is any function σ {\displaystyle \sigma } for which π ( σ ( x ) ) = x {\displaystyle \pi (\sigma (x))=x} . The language of fibre bundles allows this notion of a section to be generalized to the case when E {\displaystyle E} is not necessarily a Cartesian product. If π : E → B {\displaystyle \pi \colon E\to B} is a fibre bundle, then a section is a choice of point σ ( x ) {\displaystyle \sigma (x)} in each of the fibres. The condition π ( σ ( x ) ) = x {\displaystyle \pi (\sigma (x))=x} simply means that the section at a point x {\displaystyle x} must lie over x {\displaystyle x} . (See image.) For example, when E {\displaystyle E} is a vector bundle a section of E {\displaystyle E} is an element of the vector space E x {\displaystyle E_{x}} lying over each point x ∈ B {\displaystyle x\in B} . In particular, a vector field on a smooth manifold M {\displaystyle M} is a choice of tangent vector at each point of M {\displaystyle M} : this is a section of the tangent bundle of M {\displaystyle M} . Likewise, a 1-form on M {\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 B {\displaystyle B} is a smooth manifold M {\displaystyle M} , and E {\displaystyle E} is assumed to be a smooth fiber bundle over M {\displaystyle M} (i.e., E {\displaystyle E} is a smooth manifold and π : E → M {\displaystyle \pi \colon E\to M} is a smooth map). In this case, one considers the space of smooth sections of E {\displaystyle E} over an open set U {\displaystyle U} , denoted C ∞ ( U , E ) {\displaystyle C^{\infty }(U,E)} . It is also useful in geometric analysis to consider spaces of sections with intermediate regularity (e.g., C k {\displaystyle C^{k}} sections, or sections with regularity in the sense of Hölder conditions or Sobolev spaces).
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