Fibered manifold

In differential geometry, in the category of differentiable manifolds, a fibered manifold is a surjective submersion

$${\displaystyle \pi :E\to B\,}$$

that is, a surjective differentiable mapping such that at each point ${\displaystyle y\in U}$ 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]

History

In topology, the words fiber (Faser in German) and fiber space (gefaserter Raum) appeared for the first time in a paper by Herbert 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 the base 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 of fiber space is given by Hassler Whitney in 1935 under the name sphere space, but in 1940 Whitney changed the name to sphere bundle. ^{[3] [4]}

The theory of fibered spaces, of which vector bundles, principal bundles, topological fibrations and fibered manifolds are a special case, is attributed to Seifert, Hopf, Feldbau, Whitney, Steenrod, Ehresmann, Serre, and others. ^{[5] [6] [7] [8] [9]}

Formal definition

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 a fibered manifold. ^[10] ${\displaystyle E}$ is called the total space, ${\displaystyle B}$ is called the base.

Examples

• Every differentiable fiber bundle is a fibered manifold.
• Every differentiable covering space is a fibered manifold with discrete fiber.
• In general, a fibered manifold needs not to be a fiber bundle: different fibers may have different topologies. An example of this phenomenon may be constructed by taking the trivial bundle ${\displaystyle \left(S^{1}\times \mathbb {R} ,\pi _{1},S^{1}\right)}$ and deleting two points in two different fibers over the base manifold ${\displaystyle S^{1}.}$ The result is a new fibered manifold where all the fibers except two are connected.

Properties

• 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 ${\displaystyle \pi \circ s=\operatorname {Id} _{U}}$ 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 ${\displaystyle \pi \circ s=\operatorname {Id} _{B}}$) passing through each ${\displaystyle y\in E.}$ ^[12]

Fibered coordinates

Let ${\displaystyle B}$ (resp. ${\displaystyle E}$) be an ${\displaystyle n}$-dimensional (resp. ${\displaystyle p}$-dimensional) manifold. A fibered manifold ${\displaystyle (E,\pi ,B)}$ admits fiber charts. We say that a chart ${\displaystyle (V,\psi )}$ on ${\displaystyle E}$ is a fiber chart, or is adapted to the surjective submersion ${\displaystyle \pi :E\to B}$ if there exists a chart ${\displaystyle (U,\varphi )}$ 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

$${\displaystyle {\begin{aligned}\psi &=\left(u^{1},\dots ,u^{n},y^{1},\dots ,y^{p-n}\right).\quad y_{0}\in V,\\\varphi &=\left(x^{1},\dots ,x^{n}\right),\quad \pi \left(y_{0}\right)\in U.\end{aligned}}}$$

The above fiber chart condition may be equivalently expressed by

$${\displaystyle \varphi \circ \pi =\mathrm {pr} _{1}\circ \psi ,}$$

where

$${\displaystyle {\mathrm {pr} _{1}}:{\mathbb {R} ^{n}}\times {\mathbb {R} ^{p-n}}\to {\mathbb {R} ^{n}}\,}$$

is the projection onto the first ${\displaystyle n}$ coordinates. The chart ${\displaystyle (U,\varphi )}$ is then obviously unique. In view of the above property, the fibered 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,\ldots ,n\},}$${\displaystyle \sigma \in \{1,\ldots ,m\},}$${\displaystyle m=p-n}$ the coordinates of the corresponding chart ${\displaystyle (U,\varphi )}$ on ${\displaystyle B}$ are then denoted, with the obvious convention, by ${\displaystyle \varphi =\left(x_{i}\right)}$ where ${\displaystyle i\in \{1,\ldots ,n\}.}$

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

Local trivialization and fiber bundles

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,}

called trivialization maps, such that

$${\displaystyle \mathrm {pr} _{1}\circ \psi _{\alpha }=\pi ,{\text{ for all }}\alpha }$$

is a local trivialization with respect to ${\displaystyle V.}$ ^[13]

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

See also

• Algebraic fiber space
• Connection (fibred manifold)  – Operation on fibered manifolds
• Covering space  – type of continuous map in topologyPages displaying wikidata descriptions as a fallback
• 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

Notes

1. Kolář, Michor & Slovák 1993 , p. 11
2. Seifert 1932
3. Whitney 1935
4. Whitney 1940
5. Feldbau 1939
6. Ehresmann 1947a
7. Ehresmann 1947b
8. Ehresmann 1955
9. Serre 1951
10. Krupka & Janyška 1990 , p. 47
11. Giachetta, Mangiarotti & Sardanashvily 1997 , p. 11
12. Giachetta, Mangiarotti & Sardanashvily 1997 , p. 15
13. Giachetta, Mangiarotti & Sardanashvily 1997 , p. 13

References

• Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural operators in differential geometry (PDF), Springer-Verlag, archived from the original (PDF) on March 30, 2017, retrieved June 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.

Historical

• 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.

External links

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