- Section (fiber bundle)
In the

mathematical field oftopology , a**section**(or**cross section**) of afiber bundle , "π": "E" → "B", over atopological space , "B", is acontinuous map , "f" : "B" → "E", such that "π"("f"("x"))="x" for all "x" in "B".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 theCartesian product "E" = "X"×"Y" of "X" and "Y"::$f(x)\; =\; (x,g(x))\; in\; E,quad\; f:X\; o\; E.$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.)

For example, when "E" is a

vector bundle a section of "E" is an element of the vector space "E"_{x}lying over each point "x" ∈ "B". In particular, avector field on asmooth manifold "M" is a choice oftangent vector at each point of "M": this is a "section" of thetangent bundle of "M". Likewise, a1-form on "M" is a section of thecotangent bundle .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 anopen set in "B" and "π"("f"("x"))="x" for all "x" in "U". If ("U", "φ") is alocal 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 are studied in

homotopy theory andalgebraic topology , where one of the main goals is to account for the existence or non-existence of global sections. This leads tosheaf cohomology and the theory ofcharacteristic class es. For example, aprincipal bundle has a global section if and only if it is trivial. On the other hand, avector bundle always has a global section, namely thezero section . However, it only admits a nowhere vanishing section if itsEuler class is zero.Sections, particularly of principal bundles and vector bundles, are also very important tools in

differential geometry . In this setting, the base space "B" is asmooth manifold "M", and "E" is assumed to be a smooth fiber bundle over "M" (i.e., "E" is a smooth manifold and "π": "E" → "M" is asmooth 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 ingeometric analysis to consider spaces of sections with intermediate regularity (e.g. "C"^{"k"}sections, or sections with regularity in the sense ofHolder condition s orSobolev spaces ).**See also***

Fibration

*Gauge theory

*Principal bundle

*Pullback bundle

*Vector bundle **References***

Norman Steenrod , "The Topology of Fibre Bundles", Princeton University Press (1951). ISBN 0-691-00548-6.

* David Bleecker, "Gauge Theory and Variational Principles", Addison-Wesley publishing, Reading, Mass (1981). ISBN 0-201-10096-7.**External links*** [

*http://planetmath.org/encyclopedia/FiberBundle.html Fiber Bundle*] , PlanetMath

*

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

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