Fibered manifold

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

\pi \colon E\to B\,

i.e. a surjective differentiable mapping such that at each point $y \in E$ the tangent mapping

T_{y}\pi \colon T_{y}E\to T_{\pi (y)}B

is surjective, or, equivalently, its rank equals 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 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 E was not part of the structure, but derived from it as a quotient space of 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 $(E, π, B)$ where $E$ and $B$ are differentiable manifolds and $π : E \to B$ is a surjective submersion, is called a fibered manifold.^[10] E is called the total space, 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 $(S 1 \times ℝ, π 1, S 1)$ and deleting two points in two different fibers over the base manifold $S 1$ .The result is a new fibered manifold where all the fibers except two are connected.

Properties[]

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

Fibered coordinates[]

Let $B$ (resp. $E$ ) be an $n$ -dimensional (resp. $p$ -dimensional) manifold. A fibered manifold $(E, π, B)$ admits fiber charts. We say that a chart $(V, ψ)$ on $E$ is a fiber chart, or is adapted to the surjective submersion $π : E \to B$ if there exists a chart $(U, φ)$ on $B$ such that $U = π (V)$ and

u^{1}=x^{1}\circ \pi ,\,u^{2}=x^{2}\circ \pi ,\,\dots ,\,u^{n}=x^{n}\circ \pi \,,

where

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

The above fiber chart condition may be equivalently expressed by

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

where

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

is the projection onto the first $n$ coordinates. The chart $(U, φ)$ is then obviously unique. In view of the above property, the fibered coordinates of a fiber chart $(V, ψ)$ are usually denoted by $ψ = (x i, y σ)$ where $i \in {1, ..., n}$ , $σ \in {1, ..., m}$ , $m = p - n$ the coordinates of the corresponding chart $U, φ)$ on $B$ are then denoted, with the obvious convention, by $φ = (x i)$ where $i \in {1, ..., n}$ .

Conversely, if a surjection $π : E \to B$ admits a fibered atlas, then $π : E \to B$ is a fibered manifold.

Local trivialization and fiber bundles[]

Let $E \to B$ be a fibered manifold and $V$ any manifold. Then an open covering ${U α}$ of $B$ together with maps

\psi :\pi ^{-1}(U_{\alpha })\rightarrow U_{\alpha }\times V,

called trivialization maps, such that

\mathrm {pr} _{1}\circ \psi _{\alpha }=\pi ,\forall \alpha

is a local trivialization with respect to $V$ .^[13]

A fibered manifold together with a manifold $V$ is a fiber bundle with typical fiber (or just fiber) $V$ if it admits a local trivialization with respect to $V$ . The atlas $Ψ = {(U α, ψ α)}$ is then called a bundle atlas.

Notes[]

^ Kolář 1993, p. 11
^ Seifert 1932
^ Whitney 1935
^ Whitney 1940
^ Feldbau 1939
^ Ehresman 1947a
^ Ehresman 1947b
^ Ehresman 1955
^ Serre 1951
^ Krupka & Janyška 1990, p. 47
^ Giachetta, Mangiarotti & Sardanashvily 1997, p. 11
^ Giachetta, Mangiarotti & Sardanashvily 1997, p. 15
^ 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).

[1] Kolář 1993, p. 11

[2] Seifert 1932

[3] Whitney 1935

[4] Whitney 1940

[5] Feldbau 1939

[6] Ehresman 1947a

[7] Ehresman 1947b

[8] Ehresman 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

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