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- Section_(fiber_bundle) abstract "In the mathematical field of topology, a section (or cross section) of a fiber bundle π is a continuous right inverse of the function π. In other words, if E is a fiber bundle over a base space, B:π : E → Bthen a section of that fiber bundle is a continuous map,s : B → Esuch thatfor all 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 can be identified with a function taking its values in the Cartesian product E = B×Y of B and Y:Let π : E → X be the projection onto the first factor: π(x,y) = x. Then a graph is any function s for which π(s(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 s(x) in each of the fibres. The condition π(s(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 Ex lying over each point x ∈ B. In particular, a vector field on a smooth manifold M is a choice of tangent vector at each point of M: this is a section of the tangent bundle of M. Likewise, a 1-form on 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 is a smooth 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 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 Hölder conditions or Sobolev spaces).".
- Section_(fiber_bundle) thumbnail Bundle_section.svg?width=300.
- Section_(fiber_bundle) wikiPageExternalLink FiberBundle.html.
- Section_(fiber_bundle) wikiPageID "367621".
- Section_(fiber_bundle) wikiPageRevisionID "603570742".
- Section_(fiber_bundle) hasPhotoCollection Section_(fiber_bundle).
- Section_(fiber_bundle) title "Fiber Bundle".
- Section_(fiber_bundle) urlname "FiberBundle".
- Section_(fiber_bundle) subject Category:Algebraic_topology.
- Section_(fiber_bundle) subject Category:Differential_topology.
- Section_(fiber_bundle) subject Category:Fiber_bundles.
- Section_(fiber_bundle) subject Category:Homotopy_theory.
- Section_(fiber_bundle) type AnimalTissue105267548.
- Section_(fiber_bundle) type BodyPart105220461.
- Section_(fiber_bundle) type FiberBundle105475681.
- Section_(fiber_bundle) type FiberBundles.
- Section_(fiber_bundle) type NervousTissue105296775.
- Section_(fiber_bundle) type Part109385911.
- Section_(fiber_bundle) type PhysicalEntity100001930.
- Section_(fiber_bundle) type Thing100002452.
- Section_(fiber_bundle) type Tissue105267345.
- Section_(fiber_bundle) comment "In the mathematical field of topology, a section (or cross section) of a fiber bundle π is a continuous right inverse of the function π. In other words, if E is a fiber bundle over a base space, B:π : E → Bthen a section of that fiber bundle is a continuous map,s : B → Esuch thatfor all x in B.A section is an abstract characterization of what it means to be a graph.".
- Section_(fiber_bundle) label "Schnitt (Faserbündel)".
- Section_(fiber_bundle) label "Sección (matemática)".
- Section_(fiber_bundle) label "Section (fiber bundle)".
- Section_(fiber_bundle) label "Section d'un fibré".
- Section_(fiber_bundle) label "截面 (纤维丛)".
- Section_(fiber_bundle) label "断面 (位相幾何学)".
- Section_(fiber_bundle) sameAs Schnitt_(Faserbündel).
- Section_(fiber_bundle) sameAs Sección_(matemática).
- Section_(fiber_bundle) sameAs Section_d'un_fibré.
- Section_(fiber_bundle) sameAs 断面_(位相幾何学).
- Section_(fiber_bundle) sameAs 단면_(올다발).
- Section_(fiber_bundle) sameAs m.02p1n_g.
- Section_(fiber_bundle) sameAs Q11703678.
- Section_(fiber_bundle) sameAs Q11703678.
- Section_(fiber_bundle) sameAs Section_(fiber_bundle).
- Section_(fiber_bundle) wasDerivedFrom Section_(fiber_bundle)?oldid=603570742.
- Section_(fiber_bundle) depiction Bundle_section.svg.
- Section_(fiber_bundle) isPrimaryTopicOf Section_(fiber_bundle).
