A section is a certain generalization of the notion of the graph of a function. The graph of a function g : X → Y can be identified with a function taking its values in the Cartesian productE = X×Y of X and Y:
A section is an abstract characterization of what it means to be a graph. Let π : E → X be the projection onto the first factor: π(x,y) = x. Then a graph is any function f for which π(f(x))=x.
The language of fibre bundles allows this notion of a section to be generalized to the case when E is not necessarily a Cartesian product. If π : E → B is a fibre bundle, then a section is a choice of point f(x) in each of the fibres. The condition π(f(x)) = x simply means that the section at a point x must lie over x. (See image.)
Fiber bundles do not in general have such global sections, so it is also useful to define sections only locally. A local section of a fiber bundle is a continuous map f : U → E where U is an open set in B and π(f(x))=x for all x in U. If (U, φ) is a local trivialization of E, where φ is a homeomorphism from π-1(U) to U × F (where F is the fiber), then local sections always exist over U in bijective correspondence with continuous maps from U to F. The (local) sections form a sheaf over B called the sheaf of sections of E.
The space of continuous sections of a fiber bundle E over U is sometimes denoted C(U,E), while the space of global sections of E is often denoted Γ(E) or Γ(B,E).
Sections, particularly of principal bundles and vector bundles, are also very important tools in differential geometry. In this setting, the base space B is a smooth manifoldM, and E is assumed to be a smooth fiber bundle over M (i.e., E is a smooth manifold and π: E → M is a smooth map). In this case, one considers the space of smooth sections of E over an open set U, denoted C∞(U,E). It is also useful in geometric analysis to consider spaces of sections with intermediate regularity (e.g. Ck sections, or sections with regularity in the sense of Holder conditions or Sobolev spaces).
