Indifferential geometry, in the category ofdifferentiable manifolds, afibered manifold is asurjectivesubmersion$${\displaystyle \pi :E\to B}$$that is, a surjective differentiable mapping such that at each point${\displaystyle y\in E}$ the tangent mapping$${\displaystyle T_y\pi :T_{y}E\to T_{\pi (y)}B}$$is surjective, or, equivalently, its rank equals${\displaystyle \dim B.}$^[1]

Intopology, the wordsfiber (Faser in German) andfiber space (gefaserter Raum) appeared for the first time in a paper byHerbert Seifert in 1932, but his definitions are limited to a very special case.^[2] The main difference from the present day conception of a fiber space, however, was that for Seifert what is now called thebase space (topological space) of a fiber (topological) space${\displaystyle E}$ was not part of the structure, but derived from it as a quotient space of${\displaystyle E.}$ The first definition offiber space is given byHassler Whitney in 1935 under the namesphere space, but in 1940 Whitney changed the name tosphere bundle.^[3][4]

The theory of fibered spaces, of whichvector bundles,principal bundles, topologicalfibrations and fibered manifolds are a special case, is attributed toSeifert,Hopf,Feldbau,Whitney,Steenrod,Ehresmann,Serre, and others.^{[5][6][7][8][9]}

A triple${\displaystyle (E,\pi ,B)}$ where${\displaystyle E}$ and${\displaystyle B}$ are differentiable manifolds and${\displaystyle \pi :E\to B}$ is a surjective submersion, is called afibered manifold.^[10]${\displaystyle E}$ is called thetotal space,${\displaystyle B}$ is called thebase.

• Every differentiablefiber bundle is afibered manifold.
• Every differentiablecovering space is afibered manifold with discrete fiber.
• In general, a fibered manifold need not be a fiber bundle: different fibers may have different topologies. An example of this phenomenon may be constructed by restricting projection to total space of any vector bundle over smooth manifold with removed base space embedded by global section, finitely many points or any closed submanifold not containing a fiber in general.

• Any surjective submersion${\displaystyle \pi :E\to B}$ is open: for each open${\displaystyle V\subseteq E,}$ the set${\displaystyle \pi (V)\subseteq B}$ is open in${\displaystyle B.}$
• Each fiber${\displaystyle {\pi }^{-1}(b)\subseteq E,b\in B}$ is a closed embedded submanifold of${\displaystyle E}$ of dimension${\displaystyle \dim E-\dim B.}$^[11]
• A fibered manifold admits local sections: For each${\displaystyle y\in E}$ there is an open neighborhood${\displaystyle U}$ of${\displaystyle \pi (y)}$ in${\displaystyle B}$ and a smooth mapping${\displaystyle s:U\to E}$ with${\displaystyle \pi \circ s={\mathrm{Id}}_U}$ and${\displaystyle s(\pi (y))=y.}$
• A surjection${\displaystyle \pi :E\to B}$ is a fibered manifold if and only if there exists a local section${\displaystyle s:B\to E}$ of${\displaystyle \pi }$ (with${\displaystyle \pi \circ s={\mathrm{Id}}_B}$) passing through each${\displaystyle y\in E.}$^[12]

Let${\displaystyle B}$ (resp.${\displaystyle E}$) be an${\displaystyle n}$-dimensional (resp.${\displaystyle p}$-dimensional) manifold. A fibered manifold${\displaystyle (E,\pi ,B)}$ admitsfiber charts. We say that achart${\displaystyle (V,\psi )}$ on${\displaystyle E}$ is afiber chart, or isadapted to the surjective submersion${\displaystyle \pi :E\to B}$ if there exists a chart${\displaystyle (U,\phi )}$ on${\displaystyle B}$ such that${\displaystyle U=\pi (V)}$ and$${\displaystyle u^1=x^1\circ \pi ,u^2=x^2\circ \pi ,\dots ,u^n=x^n\circ \pi ,}$$where

The above fiber chart condition may be equivalently expressed by$${\displaystyle \phi \circ \pi ={\mathrm{p}\mathrm{r}}_1\circ \psi ,}$$where$${\displaystyle {\mathrm{p}\mathrm{r}}_1:{\mathbb{R}}^n\times {\mathbb{R}}^{p-n}\to {\mathbb{R}}^n}$$is the projection onto the first${\displaystyle n}$ coordinates. The chart${\displaystyle (U,\phi )}$ is then obviously unique. In view of the above property, thefibered coordinates of a fiber chart${\displaystyle (V,\psi )}$ are usually denoted by${\displaystyle \psi =\left(x^i,y^{\sigma }\right)}$ where${\displaystyle i\in \{1,\dots ,n\},}$${\displaystyle \sigma \in \{1,\dots ,m\},}$${\displaystyle m=p-n}$ the coordinates of the corresponding chart${\displaystyle (U,\phi )}$ on${\displaystyle B}$ are then denoted, with the obvious convention, by${\displaystyle \phi =\left(x_i\right)}$ where${\displaystyle i\in \{1,\dots ,n\}.}$

Conversely, if a surjection${\displaystyle \pi :E\to B}$ admits afiberedatlas, then${\displaystyle \pi :E\to B}$ is a fibered manifold.

Let${\displaystyle E\to B}$ be a fibered manifold and${\displaystyle V}$ any manifold. Then an open covering${\displaystyle \left\{U_{\alpha }\right\}}$ of${\displaystyle B}$ together with maps$${\displaystyle \psi :{\pi }^{-1}\left(U_{\alpha }\right)\to U_{\alpha }\times V,}$$calledtrivialization maps, such that$${\displaystyle {\mathrm{p}\mathrm{r}}_1\circ {\psi }_{\alpha }=\pi ,\text{ for all }\alpha }$$is alocal trivialization with respect to${\displaystyle V.}$^[13]

A fibered manifold together with a manifold${\displaystyle V}$ is afiber bundle withtypical fiber (or justfiber)${\displaystyle V}$ if it admits a local trivialization with respect to${\displaystyle V.}$ The atlas${\displaystyle \mathrm{\Psi }=\left\{\left(U_{\alpha },{\psi }_{\alpha }\right)\right\}}$ is then called abundle atlas.

• Algebraic fiber space
• Connection (fibred manifold) – Operation on fibered manifolds
• Covering space – Type of continuous map in topology
• Fiber bundle – Continuous surjection satisfying a local triviality condition
• Fibration – Concept in algebraic topology
• Natural bundle
• Quasi-fibration – Concept from mathematics
• Seifert fiber space – Topological space

• Kolář, Ivan; Michor, Peter; Slovák, Jan (1993),Natural operators in differential geometry(PDF), Springer-Verlag, archived fromthe original(PDF) on March 30, 2017, retrievedJune 15, 2011
• Krupka, Demeter; Janyška, Josef (1990),Lectures on differential invariants, Univerzita J. E. Purkyně V Brně,ISBN 80-210-0165-8
• Saunders, D.J. (1989),The geometry of jet bundles, Cambridge University Press,ISBN 0-521-36948-7
• Giachetta, G.; Mangiarotti, L.;Sardanashvily, G. (1997).New Lagrangian and Hamiltonian Methods in Field Theory.World Scientific.ISBN 981-02-1587-8.

• Ehresmann, C. (1947a). "Sur la théorie des espaces fibrés".Coll. Top. Alg. Paris (in French). C.N.R.S.:3–15.
• Ehresmann, C. (1947b). "Sur les espaces fibrés différentiables".C. R. Acad. Sci. Paris (in French).224:1611–1612.
• Ehresmann, C. (1955). "Les prolongements d'un espace fibré différentiable".C. R. Acad. Sci. Paris (in French).240:1755–1757.
• Feldbau, J. (1939). "Sur la classification des espaces fibrés".C. R. Acad. Sci. Paris (in French).208:1621–1623.
• Seifert, H. (1932)."Topologie dreidimensionaler geschlossener Räume".Acta Math. (in French).60:147–238.doi:10.1007/bf02398271.
• Serre, J.-P. (1951). "Homologie singulière des espaces fibrés. Applications".Ann. of Math. (in French).54:425–505.doi:10.2307/1969485.JSTOR 1969485.
• Whitney, H. (1935)."Sphere spaces".Proc. Natl. Acad. Sci. USA.21 (7):464–468.Bibcode:1935PNAS...21..464W.doi:10.1073/pnas.21.7.464.PMC 1076627.PMID 16588001.
• Whitney, H. (1940)."On the theory of sphere bundles".Proc. Natl. Acad. Sci. USA.26 (2):148–153.Bibcode:1940PNAS...26..148W.doi:10.1073/pnas.26.2.148.MR 0001338.PMC 1078023.PMID 16588328.

• McCleary, J."A History of Manifolds and Fibre Spaces: Tortoises and Hares"(PDF).

• Differential geometry
• Fiber bundles
• Manifolds

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