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...s the base for a set of transversal lines that are [[homeomorphic]] to the fiber and that each touch the edge of the band twice.]] ...d, open-bounded, compact, even [[Line (mathematics)#Ray|ray]]s, can be the fiber. An I-bundle is said to be twisted if it is not trivial. ...

...g = g \circ \pi</math> with some extra requirement depending on a typical fiber. [[Category:Fiber bundles]] ...

...ogical perspective, there is no difference between sphere bundles and disk bundles: this is a consequence of the [[Alexander trick]], which implies <math>\ope ...f extension|one-point compactification]] of <math>X</math>, the [[homotopy fiber]] of <math>\operatorname{BTop}(X) \to \operatorname{BTop}(X^+)</math> is <m ...

...er a [[Topological space|topological]] [[Hausdorff space]], such that each fiber has the structure of a [[Banach space]]. ...open function|open]] [[surjective function|surjection]], such that each '''fiber''' <math>B_x := \pi^{-1}(x)</math> is a Banach space. Which satisfies the f ...

In algebraic geometry, the '''moduli stack of rank-''n'' vector bundles''' Vect_''n'' is the [[stack (mathematics)|stack]] parametrizing ...ctor bundles over ''U'' where every morphism is an isomorphism (i.e., each fiber of ''p'' is a groupoid). ...

{{short description|Type of fiber bundle}} In mathematics, an '''affine bundle''' is a [[fiber bundle]] whose typical fiber, fibers, trivialization morphisms and transition functions are affine.<ref> ...

...ortant relations between [[connection (fibred manifold)|connections]] on [[fiber bundle]]s <math>Y\to X</math>, <math>Y\to \Sigma</math> and <math>\Sigma\to of fiber bundles ...

...mology, corresponding in de Rham cohomology to "pointwise product with the fiber". The composition of the "wrong way" map with the usual map gives a map fro ...or the [[Euler characteristic#Fibration property|Euler characteristic of a fiber bundle]].<ref>{{citation ...

...al linear group]]; i.e., the moduli stack was the [[moduli stack of vector bundles]], but, today, the term refers to any of generalizations. The scheme-theore ...heme over <math>\mathbf{F}_q</math> whose fibers are connected and generic fiber is a semisimple group. Given a ''G''-bundle ''P'' on ''X'', the homomorphis ...

...logy]], the '''Milnor–Wood inequality''' is an obstruction to endow circle bundles over surfaces with a flat structure. It is named after [[John Milnor]] and == Flat bundles == ...

...mations are defined as particular [[automorphisms]] of so-called natural [[fiber bundle]]s. However, there is a category of above mentioned natural bundles <math>T\to X</math> which admit the functorial lift <math>\widetilde\tau</m ...

...incipal ''G''-bundle]], just as a [[fibration]] is a generalization of a [[fiber bundle]]. By definition,<ref>{{cite book|last=James|first=I.M.|title=Handbo [[Category:Fiber bundles]] ...

...over a paracompact base, first note the construction is clear for trivial bundles. For the general case, if the base is compact, choose {{mvar|E{{'}}}} such ...(''L'')}} is trivial. Thus, the set of the isomorphism classes of all line bundles on some topological space {{mvar|X}} forms an abelian group called the [[Pi ...

...e to [[André Weil]] and will be needed to define essentially finite vector bundles: ==Finite vector bundles== ...

In [[algebraic geometry]], '''Procesi bundles''' are [[vector bundle]]s of rank <math>n!</math> on certain [[symplectic r ...math>i > 0</math>. The isomorphism ensures that each [[fiber (mathematics)|fiber]] of <math>\mathcal{P}</math> is naturally the [[regular representation]] o ...

...rivatives up to order at most <math>s</math>.<ref>{{citation|title=Natural bundles have finite order|last=Palais|first=Richard|author-link=Richard Palais |las ...of an arbitrary bundle of geometric objects.<ref>{{citation|title=Natural bundles and their general properties|author=A. Nijenhuis |publisher=Diff. Geom. in ...

...ics)|basis]] of <math>T_pM\,</math>, where <math>T_pM\,</math> denotes the fiber over <math>p\,</math> of the [[tangent vector bundle]] <math>TM\,</math>. [[Category:Fiber bundles]] ...

==Spinor bundles== which is given by left multiplication on each fiber. The spinor bundle ''S''(''M'') is therefore a bundle of [[Clifford module] ...

...f name="simpson">{{cite journal |last1=Simpson |first1=Carlos |title=Higgs bundles and local systems |journal=Publications Mathématiques de l'IHÉS |date=1992 ...t [[gauge theory]] with flat connections by working with the simpler Higgs bundles. ...

...licitly, it is given by the following [[short exact sequence]] of [[vector bundles]] over ''<math> M</math>'': ...t=Atiyah|first=M. F.|date=1957|title=Complex analytic connections in fibre bundles|url=https://www.ams.org/tran/1957-085-01/S0002-9947-1957-0086359-5/|journal ...